research article the mann-type extragradient iterative...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 378750 19 pageshttpdxdoiorg1011552013378750

Research ArticleThe Mann-Type Extragradient Iterative Algorithms withRegularization for Solving Variational Inequality ProblemsSplit Feasibility and Fixed Point Problems

Lu-Chuan Ceng1 Himanshu Gupta2 and Ching-Feng Wen3

1 Department of Mathematics Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai UniversitiesShanghai 200234 China

2Department of Mathematics Aligarh Muslim University Aligarh 202 002 India3 Center for Fundamental Science Kaohsiung Medical University Kaohsiung 807 Taiwan

Correspondence should be addressed to Ching-Feng Wen cfwenkmuedutw

Received 3 December 2012 Accepted 31 December 2012

Academic Editor Jen-Chih Yao

Copyright copy 2013 Lu-Chuan Ceng et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization forfinding a common element of the solution set Ξ of a general system of variational inequalities the solution set Γ of a split feasibilityproblem and the fixed point set Fix(119878) of a strictly pseudocontractive mapping 119878 in the setting of the Hilbert spaces Theseiterative algorithms are based on the regularization method the Mann-type iteration method and the extragradient method dueto Nadezhkina and Takahashi (2006) Furthermore we prove that the sequences generated by the proposed algorithms convergeweakly to an element of Fix(119878) cap Ξ cap Γ under mild conditions

1 Introduction

Let H be a real Hilbert space with inner product ⟨sdot sdot⟩ andnorm sdot Let 119862 be a nonempty closed convex subset of HThe projection (nearest point ormetric projection) ofH onto119862 is denoted by 119875

119862 Let 119878 119862 rarr 119862 be a mapping and Fix(119878)

be the set of fixed points of 119878 For a given nonlinear operator119860 119862 rarr H we consider the following variational inequalityproblem (VIP) of finding 119909

lowastisin 119862 such that

⟨119860119909lowast 119909 minus 119909

lowast⟩ ge 0 forall119909 isin 119862 (1)

The solution set of VIP (1) is denoted by VI(119862 119860) Thetheory of variational inequalities has been studied quiteextensively and has emerged as an important tool in the studyof a wide class of problems from mechanics optimizationengineering science and social sciences It is well known thattheVIP is equivalent to a fixed point problemThis alternativeformulation has been used to suggest and analyze projectioniterative method for solving variational inequalities underthe conditions that the involved operator must be stronglymonotone and Lipschitz continuous In the recent past

several people have studied and proposed several iterativemethods to find a solution of variational inequalities which isalso a fixed point of a nonexpansivemapping or strict pseudo-contractive mapping see for example [1ndash9] and the refer-ences therein

For finding an element of Fix(119878) cap VI(119862 119860) when 119862 isclosed and convex 119878 is nonexpansive and 119860 is 120572-inversestrongly monotone Takahashi and Toyoda [10] introducedthe following Mann-type iterative algorithm

119909119899+1

= 120572119899119909119899+ (1 minus 120572

119899) 119878119875119862(1 minus 120582

119899119860119909119899) forall119899 ge 0 (2)

where 119875119862 is the metric projection of H onto 119862 1199090 = 119909 isin

119862 120572119899 is a sequence in (0 1) and 120582119899 is a sequence in(0 2120572)They showed that if Fix(119878)capVI(119862 119860) = 0 then the seq-uence 119909

119899 converges weakly to some 119911 isin Fix(119878) cap VI(119862 119860)

Nadezhkina and Takahashi [9] and Zeng and Yao [8] pro-posed extragradient methods motivated by Korpelevic [11]for finding a common element of the fixed point set of anonexpansive mapping and the solution set of a variationalinequality problem Further these iterative methods are

2 Abstract and Applied Analysis

extended in [12] to develop a new iterativemethod for findingelements in Fix(119878) cap VI(119862 119860)

Let 1198611 1198612

119862 rarr H be two mappings Recently Cenget al [4] introduced and considered the following problem offinding (119909

lowast 119910lowast) isin 119862 times 119862 such that

⟨12058311198611119910lowast+ 119909lowastminus 119910lowast 119909 minus 119909

lowast⟩ ge 0 forall119909 isin 119862



(3)

which is called a general system of variational inequalities(GSVI) where 120583

1gt 0 and 120583

2gt 0 are two constants The

set of solutions of problem (3) is denoted by GSVI(119862 1198611 1198612)

In particular if 1198611

= 1198612 then problem (3) reduces to the

new system of variational inequalities (NSVI) introducedand studied byVerma [13] Further if 119909lowast = 119910

lowast then theNSVIreduces to VIP (1)

Recently Ceng et al [4] transformed problem (3) into afixed point problem in the following way

Lemma 1 (see [4]) For given 119909 119910 isin 119862 (119909 119910) is a solutionof problem (3) if and only if 119909 is a fixed point of the mapping119866 119862 rarr 119862 defined by

119866 (119909) = 119875119862[119875119862(119909 minus 120583

21198612119909) minus 120583

11198611119875119862(119909 minus 120583

21198612119909)]

forall119909 isin 119862

(4)

where 119910 = 119875119862(119909 minus 120583

21198612119909)

In particular if the mapping 119861119894 119862 rarr H is 120573

119894-inverse

strongly monotone for 119894 = 1 2 then the mapping 119866 isnonexpansive provided 120583

119894isin (0 2120573

119894) for 119894 = 1 2

Utilizing Lemma 1 they introduced and studied a relaxedextragradient method for solving GSVI (3)

Throughout this paper unless otherwise specified the setof fixed points of themapping119866 is denoted byΞ Based on therelaxed extragradient method and viscosity approximationmethod Yao et al [7] proposed and analyzed an iterativealgorithm for finding a common solution of GSVI (3) andfixed point problem of a strictly pseudocontractive mapping119878 119862 rarr 119862 where 119862 is a nonempty bounded closed convexsubset of a real Hilbert spaceH

Subsequently Ceng at al [14] further presented andanalyzed an iterative scheme for finding a common elementof the solution set of VIP (1) the solution set of GSVI (3)and fixed point set of a strictly pseudocontractive mapping119878 119862 rarr 119862

Theorem 2 (see [14 Theorem 31]) Let 119862 be a nonemptyclosed convex subset of a real Hilbert spaceH Let119860 119862 rarr Hbe 120572-inverse strongly monotone and let 119861

119894 119862 rarr H be

120573119894-inverse strongly monotone for 119894 = 1 2 Let 119878 119862 rarr 119862

be a 119896-strictly pseudocontractive mapping such that Fix(119878) cap

Ξ cap VI(119862 119860) = 0 Let 119876 119862 rarr 119862 be a 120588-contraction with

120588 isin [0 12) For given 1199090

isin 119862 arbitrarily let the sequences119909119899 119910119899 and 119911

119899 be generated iteratively by

119911119899 = 119875119862 (119909119899 minus 120582119899119860119909119899)

119910119899= 120572119899119876119909119899

+ (1 minus 120572119899) 119875119862 [119875119862 (119911119899 minus 12058321198612119911119899)

minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(5)

where 120583119894 isin (0 2120573119894) for 119894 = 1 2 120582119899 sub (0 2120572] and 120572119899 120573119899120574119899 120575119899 sub [0 1] such that

(i) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0

(ii) lim119899rarrinfin

120572119899= 0 and sum

infin

119899=0120572119899= infin

(iii) 0 lt lim inf119899rarrinfin

120573119899

le lim sup119899rarrinfin

120573119899

lt 1 andlim inf119899rarrinfin120575119899 gt 0

(iv) lim119899rarrinfin(120574119899+1(1 minus 120573119899+1) minus 120574119899(1 minus 120573119899)) = 0(v) 0 lt lim inf

119899rarrinfin120582119899


120582119899

lt 2120572 andlim119899rarrinfin

|120582119899+1

minus 120582119899| = 0

Then the sequence 119909119899 generated by (5) converges strongly to

119909 = 119875Fix(119878)capΞcapVI(119862119860)119876119909 and (119909 119910) is a solution of GSVI (3)where 119910 = 119875

119862(119909 minus 120583

21198612119909)

On the other hand let 119862 and 119876 be nonempty closed con-vex subsets of real Hilbert spaces H

1and H

2 respectively

The split feasibility problem (SFP) is to find a point 119909lowast withthe following property

119909lowast

isin 119862 119860119909lowast

isin 119876 (6)

where 119860 isin 119861(H1H2) and 119861(H

1H2) denotes the family of

all bounded linear operators fromH1toH2

In 1994 the SFP was first introduced by Censor andElfving [15] in finite-dimensional Hilbert spaces for mod-eling inverse problems which arise from phase retrievalsand in medical image reconstruction A number of imagereconstruction problems can be formulated as the SFP seefor example [16] and the references therein Recently it isfound that the SFP can also be applied to study intensity-modulated radiation therapy see for example [17ndash19] andthe references therein In the recent past a wide varietyof iterative methods have been used in signal processingand image reconstruction and for solving the SFP see forexample [16ndash26] and the references therein A special caseof the SFP is the following convex constrained linear inverseproblem [27] of finding an element 119909 such that

119909 isin 119862 119860119909 = 119887 (7)

It has been extensively investigated in the literature using theprojected Landweber iterative method [28] Comparativelythe SFP has received much less attention so far due tothe complexity resulting from the set 119876 Therefore whethervarious versions of the projected Landweber iterativemethod[28] can be extended to solve the SFP remains an interesting

Abstract and Applied Analysis 3

open topic For example it is yet not clear whether the dualapproach to (7) of [29] can be extended to the SFP Theoriginal algorithm given in [15] involves the computation ofthe inverse 119860

minus1 (assuming the existence of the inverse of 119860)and thus has not become popular A seemingly more popularalgorithm that solves the SFP is the 119862119876 algorithm of Byrne[16 21] which is found to be a gradient-projection method(GPM) in convex minimization It is also a special case of theproximal forward-backward splitting method [30] The 119862119876

algorithm only involves the computation of the projections119875119862and119875119876onto the sets119862 and119876 respectively and is therefore

implementable in the case where 119875119862and 119875119876have closed-form

expressions for example 119862 and 119876 are closed balls or half-spaces However it remains a challenge how to implement the119862119876 algorithm in the case where the projections 119875

119862andor 119875

119876

fail to have closed-form expressions though theoretically wecan prove the (weak) convergence of the algorithm

Very recently Xu [20] gave a continuation of the studyon the 119862119876 algorithm and its convergence He applied Mannrsquosalgorithm to the SFP andpurposed an averaged119862119876 algorithmwhichwas proved to be weakly convergent to a solution of theSFP He also established the strong convergence result whichshows that the minimum-norm solution can be obtained

Furthermore Korpelevic [11] introduced the so-calledextragradient method for finding a solution of a saddle pointproblem He proved that the sequences generated by theproposed iterative algorithm converge to a solution of thesaddle point problem

Throughout this paper assume that the SFP is consistentthat is the solution set Γ of the SFP is nonempty Let 119891

H1 rarr R be a continuous differentiable function The mini-mization problem

min119909isin119862

119891 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2 (8)

is ill posed Therefore Xu [20] considered the followingTikhonov regularization problem

min119909isin119862

119891120572 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2+

1

21205721199092 (9)

where 120572 gt 0 is the regularization parameter The regularizedminimization (9) has a unique solution which is denoted by119909120572 The following results are easy to prove

Proposition 3 (see [31 Proposition 31]) Given 119909lowast

isin H1 thefollowing statements are equivalent

(i) 119909lowast solves the SFP

(ii) 119909lowast solves the fixed point equation

119875119862(119868 minus 120582nabla119891) 119909

lowast= 119909lowast (10)

where 120582 gt 0 nabla119891 = 119860lowast(119868 minus 119875

119876)119860 and 119860

lowast is the adjointof 119860

(iii) 119909lowast solves the variational inequality problem (VIP) of

finding 119909lowast

isin 119862 such that

⟨nabla119891 (119909lowast) 119909 minus 119909


It is clear from Proposition 3 that

Γ = Fix (119875119862 (119868 minus 120582nabla119891)) = VI (119862 nabla119891) (12)

for all 120582 gt 0 where Fix(119875119862(119868 minus 120582nabla119891)) and VI(119862 nabla119891) denotethe set of fixed points of 119875119862(119868 minus 120582nabla119891) and the solution set ofVIP (11) respectively

Proposition 4 (see [31]) The following statements hold

(i) the gradient

nabla119891120572= nabla119891 + 120572119868 = 119860

lowast(119868 minus 119875

119876) 119860 + 120572119868 (13)

is (120572+1198602)-Lipschitz continuous and 120572-strongly mon-

otone(ii) the mapping 119875

119862(119868 minus 120582nabla119891

120572) is a contraction with coef-

ficient

radic1 minus 120582 (2120572 minus 120582(1198602+ 120572)2

) (le radic1 minus 120572120582 le 1 minus1

2120572120582)

(14)

where 0 lt 120582 le 120572(1198602+ 120572)2

(iii) if the SFP is consistent then the strong lim120572rarr0

119909120572exists

and is the minimum-norm solution of the SFP

Very recently by combining the regularization methodand extragradient method due to Nadezhkina and Takahashi[32] Ceng et al [31] proposed an extragradient algorithmwith regularization and proved that the sequences generatedby the proposed algorithm converge weakly to an element ofFix(119878) cap Γ where 119878 119862 rarr 119862 is a nonexpansive mapping

Theorem 5 (see [31 Theorem 31]) Let 119878 119862 rarr 119862 be anonexpansive mapping such that Fix(119878) cap Γ = 0 Let 119909

119899 and

119910119899 be the sequences in 119862 generated by the following extra-

gradient algorithm

1199090= 119909 isin 119862 chosen arbitrarily

119910119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1 = 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119910119899)) forall119899 ge 0

(15)

where suminfin

119899=0120572119899lt infin 120582

119899 sub [119886 119887] for some 119886 119887 isin (0 1119860

2)

and 120573119899 sub [119888 119889] for some 119888 119889 isin (0 1) Then both sequences

119909119899 and 119910

119899 converge weakly to an element 119909 isin Fix(119878) cap Γ

Motivated and inspired by the research going on this areawe propose and analyze the following Mann-type extragra-dient iterative algorithms with regularization for finding acommon element of the solution set of the GSVI (3) thesolution set of the SFP (6) and the fixed point set of a strictlypseudocontractive mapping 119878 119862 rarr 119862

Algorithm 6 Let 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899 120574119899 120575119899 sub [0 1] such

that 120590119899+ 120591119899

le 1 and 120573119899+ 120574119899+ 120575119899

= 1 for all 119899 ge 0 For


given 1199090

isin 119862 arbitrarily let 119909119899 119910119899 119911119899 be the sequences

generated by the Mann-type extragradient iterative schemewith regularization

119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120590119899119909119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120590119899minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus 12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(16)

Under appropriate assumptions it is proven that all thesequences 119909

119899 119910119899 119911119899 converge weakly to an element 119909 isin

Fix(119878) cap Ξ cap Γ Furthermore (119909 119910) is a solution of the GSVI(3) where 119910 = 119875

119862(119909 minus 120583

21198612119909)


isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590119899 120573119899 120574119899 120575119899 sub [0 1] such that

120573119899+ 120574119899+ 120575119899= 1 for all 119899 ge 0 For given 119909

0isin 119862 arbitrarily let

119909119899 119906119899 119899 be the sequences generated by the Mann-type

extragradient iterative scheme with regularization

119906119899 = 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899) minus 12058311198611119875119862 (119909119899 minus 12058321198612119909119899)]

119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572

119899

(119906119899))

119910119899= 120590119899119909119899+ (1 minus 120590

119899) 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(17)

Also under mild conditions it is shown that all thesequences 119909



119862(119909 minus 120583

21198612119909)

Observe that both [20 Theorem 57] and [31 Theorem31] are weak convergence results for solving the SFP and soare our results as well But our problem of finding an ele-ment of Fix(119878)capΞcapΓ is more general than the correspondingones in [20 Theorem 57] and [31 Theorem 31] respectivelyHence there is no doubt that our weak convergence resultsare very interesting and quite valuable Because the Mann-type extragradient iterative schemes (16) and (17) with regu-larization involve two inverse strongly monotone mappings1198611 and 1198612 a 119896-strictly pseudocontractive self-mapping 119878 andseveral parameter sequences they aremore flexible andmoresubtle than the corresponding ones in [20 Theorem 57]and [31 Theorem 31] respectively Furthermore the hybridextragradient iterative scheme (5) is extended to develop theMann-type extragradient iterative schemes (16) and (17) withregularization In our results the Mann-type extragradientiterative schemes (16) and (17) with regularization lack therequirement of boundedness for the domain in which variousmappings are defined see for example Yao et al [7Theorem32] Therefore our results represent the modification sup-plementation extension and improvement of [20 Theorem57] [31 Theorem 31] [14 Theorem 31] and [7 Theorem32]

2 Preliminaries

LetH be a real Hilbert space whose inner product and normare denoted by ⟨sdot sdot⟩ and sdot respectively Let119870 be a nonemptyclosed and convex subset ofH Nowwe present some knowndefinitions and results which will be used in the sequel

The metric (or nearest point) projection fromH onto 119870

is the mapping 119875119870 H rarr 119870 which assigns to each point119909 isin H the unique point 119875

119870119909 isin 119870 satisfying the property

1003817100381710038171003817119909 minus 1198751198701199091003817100381710038171003817 = inf119910isin119870

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 119889 (119909 119870) (18)

Some important properties of projections are gathered inthe following proposition

Proposition 8 For given 119909 isin H and 119911 isin 119870

(i) 119911 = 119875119870119909 hArr ⟨119909 minus 119911 119910 minus 119911⟩ le 0 for all 119910 isin 119870

(ii) 119911 = 119875119870119909 hArr 119909 minus 119911

2le 119909 minus 119910

2minus 119910 minus 119911

2 for all119910 isin 119870

(iii) ⟨119875119870119909 minus 119875

119870119910 119909 minus 119910⟩ ge 119875

119870119909 minus 119875

1198701199102 for all 119910 isin

H which hence implies that 119875119870is nonexpansive and

monotone

Definition 9 A mapping 119879 H rarr H is said to be

(a) nonexpansive if1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin H (19)

(b) firmly nonexpansive if 2119879 minus 119868 is nonexpansive orequivalently

⟨119909 minus 119910 119879119909 minus 119879119910⟩ ge1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

2 forall119909 119910 isin H (20)

alternatively 119879 is firmly nonexpansive if and only if 119879can be expressed as

119879 =1

2(119868 + 119878) (21)

where 119878 H rarr H is nonexpansive projections arefirmly nonexpansive

Definition 10 Let 119879 be a nonlinear operator with domain119863(119879) sube H and range 119877(119879) sube H

(a) 119879 is said to be monotone if

⟨119909 minus 119910 119879119909 minus 119879119910⟩ ge 0 forall119909 119910 isin 119863 (119879) (22)

(b) Given a number 120573 gt 0 119879 is said to be 120573-stronglymonotone if

⟨119909 minus 119910 119879119909 minus 119879119910⟩ ge 1205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (23)

(c) Given a number 120584 gt 0 119879 is said to be 120584-inversestrongly monotone (120584-ism) if


1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (24)


It can be easily seen that if 119878 is nonexpansive then 119868 minus 119878 ismonotone It is also easy to see that a projection 119875

119870is 1-ism

Inverse strongly monotone (also referred to as cocoer-cive) operators have been applied widely in solving practicalproblems in various fields for instance in traffic assignmentproblems see for example [33 34]

Definition 11 A mapping 119879 H rarr H is said to be an aver-agedmapping if it can be written as the average of the identity119868 and a nonexpansive mapping that is

119879 equiv (1 minus 120572) 119868 + 120572119878 (25)

where 120572 isin (0 1) and 119878 H rarr H is nonexpansive Moreprecisely when the last equality holds we say that 119879 is 120572-averagedThus firmly nonexpansivemappings (in particularprojections) are 12-averaged maps

Proposition 12 (see [21]) Let 119879 H rarr H be a givenmapping

(i) 119879 is nonexpansive if and only if the complement 119868 minus 119879

is 12-ism

(ii) If 119879 is 120584-ism then for 120574 gt 0 120574119879 is 120584120574-ism

(iii) 119879 is averaged if and only if the complement 119868 minus 119879 is120584-ism for some 120584 gt 12 Indeed for 120572 isin (0 1) 119879 is120572-averaged if and only if 119868 minus 119879 is 12120572-ism

Proposition 13 (see [21 35]) Let 119878 119879 119881 H rarr H be givenoperators

(i) If 119879 = (1 minus 120572)119878 + 120572119881 for some 120572 isin (0 1) and if 119878 isaveraged and 119881 is nonexpansive then 119879 is averaged

(ii) 119879 is firmly nonexpansive if and only if the complement119868 minus 119879 is firmly nonexpansive

(iii) If 119879 = (1 minus 120572)119878 + 120572119881 for some 120572 isin (0 1) and if 119878 isfirmly nonexpansive and 119881 is nonexpansive then 119879 isaveraged

(iv) The composite of finitely many averaged mappings isaveraged That is if each of the mappings 119879

119894119873

119894=1is

averaged then so is the composite 1198791 ∘ 1198792 ∘ sdot sdot sdot ∘ 119879119873 Inparticular if 1198791 is 1205721-averaged and 1198792 is 1205722-averagedwhere 1205721 1205722 isin (0 1) then the composite 1198791 ∘ 1198792 is 120572-averaged where 120572 = 1205721 + 1205722 minus 12057211205722

(v) If the mappings 119879119894119873

119894=1are averaged and have a

common fixed point then

119873

⋂

119894=1

Fix (119879119894) = Fix (119879

1sdot sdot sdot 119879119873) (26)

The notation Fix(119879) denotes the set of all fixed points ofthe mapping 119879 that is Fix(119879) = 119909 isin H 119879119909 = 119909

It is clear that in a real Hilbert space H 119878 119862 rarr 119862

is 119896-strictly pseudocontractive if and only if there holds thefollowing inequality

⟨119878119909 minus 119878119910 119909 minus 119910⟩

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2minus

1 minus 119896

2

1003817100381710038171003817(119868 minus 119878) 119909 minus (119868 minus 119878) 1199101003817100381710038171003817

2

forall119909 119910 isin 119862

(27)

This immediately implies that if 119878 is a 119896-strictly pseudo-contractive mapping then 119868 minus 119878 is (1 minus 119896)2-inverse stronglymonotone for further detail we refer to [9] and the referencestherein It is well known that the class of strict pseudocontrac-tions strictly includes the class of nonexpansive mappings

The following elementary result in the real Hilbert spacesis quite well known

Lemma 14 (see [36]) LetH be a real Hilbert space Then forall 119909 119910 isin H and 120582 isin [0 1]

1003817100381710038171003817120582119909 + (1 minus 120582)1199101003817100381710038171003817

2= 120582119909

2+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2

minus 120582 (1 minus 120582)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(28)

Lemma 15 (see [37 Proposition 21]) Let 119862 be a nonemptyclosed convex subset of a real Hilbert spaceH and 119878 119862 rarr 119862

be a mapping

(i) If 119878 is a 119896-strict pseudocontractive mapping then 119878

satisfies the Lipschitz condition

1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 le

1 + 119896

1 minus 119896

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119862 (29)

(ii) If 119878 is a 119896-strict pseudocontractive mapping then themapping 119868 minus 119878 is semiclosed at 0 that is if 119909

119899 is a

sequence in 119862 such that 119909119899

rarr 119909 weakly and (119868 minus

119878)119909119899

rarr 0 strongly then (119868 minus 119878)119909 = 0(iii) If 119878 is 119896-(quasi-)strict pseudocontraction then the fixed

point set Fix(119878) of 119878 is closed and convex so that theprojection 119875Fix(119878) is well defined

The following lemma plays a key role in proving weakconvergence of the sequences generated by our algorithms

Lemma 16 (see [38 p 80]) Let 119886119899infin

119899=0 119887119899infin

119899=0 and 120575

119899infin

119899=0be

sequences of nonnegative real numbers satisfying the inequality

119886119899+1

le (1 + 120575119899) 119886119899 + 119887

119899 forall119899 ge 0 (30)

If suminfin119899=0

120575119899

lt infin and suminfin

119899=0119887119899

lt infin then lim119899rarrinfin

119886119899exists If

in addition 119886119899infin

119899=0has a subsequence which converges to zero

then lim119899rarrinfin

119886119899= 0

Corollary 17 (see [39 p 303]) Let 119886119899infin

119899=0and 119887

119899infin

119899=0be two


119886119899+1

le 119886119899+ 119887119899 forall119899 ge 0 (31)

If suminfin119899=0

119887119899converges then lim

119899rarrinfin119886119899exists


Lemma 18 (see [7]) Let119862 be a nonempty closed convex subsetof a real Hilbert space H Let 119878 119862 rarr 119862 be a 119896-strictlypseudocontractive mapping Let 120574 and 120575 be two nonnegativereal numbers such that (120574 + 120575)119896 le 120574 Then

1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119862

(32)

The following lemma is an immediate consequence of aninner product

Lemma 19 In a real Hilbert spaceH there holds the inequal-ity

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)

Let119870be a nonempty closed convex subset of a realHilbertspace H and let 119865 119870 rarr H be a monotone mapping Thevariational inequality problem (VIP) is to find 119909 isin 119870 suchthat

⟨119865119909 119910 minus 119909⟩ ge 0 forall119910 isin 119870 (34)

The solution set of the VIP is denoted by VI(119870 119865) It iswell known that

119909 isin VI (119870 119865) lArrrArr 119909 = 119875119870 (119909 minus 120582119865119909) forall120582 gt 0 (35)

A set-valued mapping 119879 H rarr 2H is called monotone

if for all 119909 119910 isin H 119891 isin 119879119909 and 119892 isin 119879119910 imply that ⟨119909 minus 119910 119891 minus

119892⟩ ge 0 A monotone set-valued mapping 119879 H rarr 2H is

called maximal if its graph Gph(119879) is not properly containedin the graph of any other monotone set-valued mapping It isknown that a monotone set-valued mapping 119879 H rarr 2

H ismaximal if and only if for (119909 119891) isin H timesH ⟨119909 minus 119910 119891 minus 119892⟩ ge 0

for every (119910 119892) isin Gph(119879) implies that 119891 isin 119879119909 Let 119865 119870 rarr

H be a monotone and Lipschitz continuous mapping and let119873119870119907 be the normal cone to 119870 at 119907 isin 119870 that is

119873119870119907 = 119908 isin H ⟨119907 minus 119906 119908⟩ ge 0 forall119906 isin 119870 (36)

Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)

It is known that in this case the mapping 119879 is maximalmonotone and 0 isin 119879119907 if and only if 119907 isin VI(119870 119865) for furtherdetails we refer to [40] and the references therein

3 Main Results

In this section we first prove the weak convergence of thesequences generated by theMann-type extragradient iterativealgorithm (16) with regularization

Theorem 20 Let 119862 be a nonempty closed convex subset of areal Hilbert spaceH

1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1

be 120573119894-inverse strongly monotone for 119894 = 1 2 Let 119878 119862 rarr 119862


ΞcapΓ = 0 For given 1199090isin 119862 arbitrarily let 119909

119899 119910119899 119911119899 be the

sequences generated by the Mann-type extragradient iterativealgorithm (16) with regularization where 120583

119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899 + 120574119899 + 120575119899 = 1 and (120574119899 + 120575119899)119896 le 120574119899 for all 119899 ge 0(iii) 120590

119899+ 120591119899le 1 for all 119899 ge 0

(iv) 0 lt lim inf119899rarrinfin

120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1

(v) 0 lt lim inf119899rarrinfin

120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0

(vi) 0 lt lim inf119899rarrinfin

120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2

Then all the sequences 119909119899 119910119899 119911119899 converge weakly to an

element 119909 isin Fix(119878) capΞcapΓ Furthermore (119909 119910) is a solution ofGSVI (3) where 119910 = 119875

119862(119909 minus 120583

21198612119909)

Proof First taking into account 0 lt lim inf119899rarrinfin

120582119899

le

lim sup119899rarrinfin

120582119899

lt 11198602 without loss of generality we may

assume that 120582119899 sub [119886 119887] for some 119886 119887 isin (0 1119860

2)

Now let us show that 119875119862(119868 minus 120582nabla119891120572) is 120577-averaged for each120582 isin (0 2(120572 + 119860

2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)

Indeed it is easy to see that nabla119891 = 119860lowast(119868 minus 119875

119876)119860 is 1119860

2-ism that is

⟨nabla119891 (119909) minus nabla119891 (119910) 119909 minus 119910⟩ ge1

1198602

1003817100381710038171003817nabla119891 (119909) minus nabla119891 (119910)1003817100381710038171003817

2 (39)

Observe that

(120572 + 1198602) ⟨nabla119891120572 (119909) minus nabla119891

120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

+ ⟨nabla119891 (119909) minus nabla119891 (119910) 119909 minus 119910⟩ ]

= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2+ 120572 ⟨nabla119891 (119909) minus nabla119891 (119910) 119909 minus 119910⟩

+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 1198602⟨nabla119891 (119909) minus nabla119891 (119910) 119909 minus 119910⟩

ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+1003817100381710038171003817nabla119891 (119909) minus nabla119891 (119910)

1003817100381710038171003817

2

=1003817100381710038171003817120572 (119909 minus 119910) + nabla119891 (119909) minus nabla119891 (119910)

1003817100381710038171003817

2

=1003817100381710038171003817nabla119891120572 (119909) minus nabla119891

120572(119910)

1003817100381710038171003817

2

(40)

Hence it follows that nabla119891120572

= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according

to Proposition 12(ii) By Proposition 12(iii) the complement


119868 minus 120582nabla119891120572is 120582(120572 + 119860

2)2-averaged Therefore noting that

119875119862is 12-averaged and utilizing Proposition 13(iv) we know

that for each 120582 isin (0 2(120572+1198602)) 119875119862(119868minus120582nabla119891

120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)

This shows that 119875119862(119868 minus 120582nabla119891120572) is nonexpansive Furthermore

for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602

= lim119899rarrinfin

1

120572119899+ 119860

2 (42)

Without loss of generality we may assume that

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)

Consequently it follows that for each integer 119899 ge 0 119875119862(119868 minus

120582119899nabla119891120572119899

) is 120577119899-averaged with

120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)

This immediately implies that 119875119862(119868 minus 120582119899nabla119891120572119899

) is nonex-pansive for all 119899 ge 0

Next we divide the remainder of the proof into severalsteps

Step 1 119909119899 is boundedIndeed take 119901 isin Fix(119878) cap Ξ cap Γ arbitrarily Then 119878119901 = 119901

119875119862(119868 minus 120582nabla119891)119901 = 119901 for 120582 isin (0 2119860

2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)

From (16) it follows that

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899

) 119901 minus 119875119862 (119868 minus 120582119899nabla119891) 11990110038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899

) 119901 minus (119868 minus 120582119899nabla119891) 119901

10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)

Utilizing Lemma 19 we also have

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899

) 119909119899 minus 119875119862 (119868 minus 120582119899nabla119891120572119899

) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)

For simplicity we write 119902 = 119875119862(119901 minus 120583

21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)

119906119899= 119875119862[119875119862(119911119899minus 12058321198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)

for each 119899 ge 0Then 119910119899= 120590119899119909119899+120591119899119906119899+(1minus120590

119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H

1is 120573119894-inverse strongly monotone

and 0 lt 120583119894lt 2120573119894for 119894 = 1 2 we know that for all 119899 ge 0

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)

minus 12058311198611119875119862 (119911119899 minus 12058321198612119911119899)] minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus 119875119862 [119875119862 (119901 minus 12058321198612119901) minus 12058311198611119875119862 (119901 minus 12058321198612119901)]1003817100381710038171003817

2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]

minus 1205831[1198611119875119862(119911119899minus 12058321198612119911119899) minus 1198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus 1205832(21205732minus 1205832)10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)

Furthermore by Proposition 8(ii) we have

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2

minus10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

minus 2 ⟨119909119899 minus 119911119899 119911119899 minus 119906119899⟩ minus1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)

Further by Proposition 8(i) we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)

So from (46) we obtain

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)

Hence it follows from (46) (49) and (52) that

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901) + (1 minus 120590119899 minus 120591119899) (119906119899 minus 119901)

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)

Since (120574119899 + 120575119899)119896 le 120574119899 for all 119899 ge 0 utilizing Lemma 18 we

obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin

119899=0120572119899lt infin it is clear that suminfin

119899=02119887119901120572

119899lt infin Thus

by Corollary 17 we conclude that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 exists for each 119901 isin Fix (119878) cap Ξ cap Γ (55)

and the sequence 119909119899 is bounded Taking into account that

119875119862 nabla119891120572119899

1198611and 119861

2are Lipschitz continuous we can easily

see that 119911119899 119906119899 119906119899 119910119899 and

119899 are bounded where

119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0

Step 2 Consider lim119899rarrinfin

1198612119911119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119899minus

1198611119902 = 0 and lim

119899rarrinfin119909119899minus119911119899 = 0 where 119902 = 119875

119862(119901minus12058321198612119901)

Indeed utilizing Lemma 18 and the convexity of sdot 2 we

obtain from (16) and (47)ndash(52) that

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

minus (1 minus 120590119899 minus 120591119899)

times [1205832(21205732minus 1205832)10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899

rarr 0 lim119899rarrinfin

119909119899minus119901 exists lim inf

119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+

120591119899) lt 1 it follows that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899

(119911119899)) minus (119909119899 minus 120582119899nabla119891120572119899

(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)

This together with 119911119899

minus 119909119899 rarr 0 implies that

lim119899rarrinfin

119906119899minus 119911119899 = 0 and hence lim

119899rarrinfin119906119899minus 119909119899 = 0 By

firm nonexpansiveness of 119875119862 we have

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119911119899 minus 12058321198612119911119899) minus 119875119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

le ⟨(119911119899minus 12058321198612119911119899) minus (119901 minus 120583

21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus

119899) minus 1205832(1198612119911119899minus 1198612119901) minus (119901 minus 119902)

1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119899 minus (119901 minus 119902)1003817100381710038171003817

2

+ 21205832⟨119911119899minus 119899minus (119901 minus 119902) 119861

2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2

+212058321003817100381710038171003817119911119899 minus 119899 minus (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119899minus (119901 minus 119902)

1003817100381710038171003817

2

+ 212058321003817100381710038171003817119911119899 minus 119899 minus (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)

Moreover using the argument technique similar to theprevious one we derive

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le ⟨(119899 minus 12058311198611119899) minus (119902 minus 12058311198611119902) 119906119899 minus 119901⟩

=1

2[1003817100381710038171003817119899 minus 119902 minus 1205831 (1198611119899 minus 1198611119902)

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817(119899 minus 119906119899) minus 1205831 (1198611119899 minus 1198611119902) + (119901 minus 119902)

1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)

Utilizing (47) (52) (61) and (63) we have

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)

+ (1 minus 120590119899minus 120591119899) (119906119899minus 119901)

1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

minus (1 minus 120590119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)

Thus utilizing Lemma 14 from (16) and (64) it follows that

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

minus (1 minus 120573119899) (1 minus 120590

119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)

which hence implies that

(1 minus 120573119899) (1 minus 120590119899 minus 120591119899)

times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)

Since 0 lt lim inf119899rarrinfin

120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861

1119899minus 1198611119902 rarr 0 and lim

119899rarrinfin119909119899minus 119901 exists

it follows from the boundedness of 119906119899 119911119899 and

119899 that

lim119899rarrinfin

119911119899minus 119899minus (119901 minus 119902) = 0

lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)

Consequently it immediately follows that

lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)

Also note that1003817100381710038171003817119910119899 minus 119906

119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 + (1 minus 120590119899minus 120591119899)1003817100381710038171003817119906119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(69)

This together with 119909119899 minus 119906119899 rarr 0 implies that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911

119899 converge weakly to an element 119909 isin

Fix(119878) cap Ξ cap Γ

Indeed since 119909119899 is bounded there exists a subsequence

119909119899119894

of 119909119899 that converges weakly to some 119909 isin 119862 We obtain

that119909 isin Fix(119878)capΞcapΓ Taking into account that 119909119899minus119910119899 rarr 0

and 119909119899minus 119911119899 rarr 0 as 119899 rarr infin we deduce that 119910

119899119894

rarr 119909

weakly and 119911119899119894

rarr 119909 weakly First it is clear from Lemma 15and 119878119910

119899minus 119910119899 rarr 0 that 119909 isin Fix(119878) Now let us show that

119909 isin Ξ Note that

1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875

119862[119875119862(119911119899minus 12058321198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)

as 119899 rarr infin where 119866 119862 rarr 119862 is defined as that in Lemma 1According to Lemma 15 we get 119909 isin Ξ Further let us showthat 119909 isin Γ As a matter of fact define

119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)

where 119873119862119907 = 119908 isin H1 ⟨119907 minus 119906 119908⟩ ge 0 forall119906 isin 119862 Then 119879 ismaximal monotone and 0 isin 119879119907 if and only if 119907 isin VI(119862 nabla119891)see [40] for more details Let (119907 119908) isin Gph(119879) Then we have

119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have

⟨119907 minus 119906 119908 minus nabla119891 (119907)⟩ ge 0 forall119906 isin 119862 (77)

On the other hand from

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907) minus nabla119891 (119911119899119894

)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get

⟨119907 minus 119909 119908⟩ ge 0 as 119894 997888rarr infin (83)

Since 119879 is maximal monotone we have 119909 isin 119879minus1

0 and hence119909 isin VI(119862 nabla119891) Thus it is clear that 119909 isin Γ Therefore 119909 isin

Fix(119878) cap Ξ cap ΓLet 119909

119899119895

be another subsequence of 119909119899 such that 119909

119899119895

converges weakly to 119909 isin 119862 Then 119909 isin Fix(119878) cap Ξ cap Γ Let usshow that119909 = 119909 Assume that119909 = 119909 From theOpial condition[41] we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

= lim inf119894rarrinfin

10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817lt lim inf119894rarrinfin

10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817 = lim inf119895rarrinfin

100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817

lt lim inf119895rarrinfin

100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817= lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)

This leads to a contradiction Consequently we have 119909 = 119909This implies that 119909

119899 converges weakly to 119909 isin Fix(119878) cap Ξ cap Γ

Further from 119909119899minus 119910119899 rarr 0 and 119909

119899minus 119911119899 rarr 0 it follows

that both 119910119899 and 119911

119899 converge weakly to 119909 This completes

the proof

Corollary 21 Let 119862 be a nonempty closed convex subset of areal Hilbert space H

1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr

H1be 120573119894-inverse strongly monotone for 119894 = 1 2 Let 119878

119862 rarr 119862 be a 119896-strictly pseudocontractive mapping such thatFix(119878)capΞcapΓ = 0 For given 119909

0isin 119862 arbitrarily let the sequences

119909119899 119910119899 119911119899 be generated iteratively by

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0

(iii) 0 lt lim inf119899rarrinfin120591119899 le lim sup119899rarrinfin

120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0

(v) 0 lt lim inf119899rarrinfin120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2


element 119909 isin Fix(119878) capΞcapΓ Furthermore (119909 119910) is a solution ofthe GSVI (3) where 119910 = 119875

119862(119909 minus 120583

21198612119909)

Proof In Theorem 20 put 120590119899

= 0 for all 119899 ge 0 Then in thiscase Theorem 20 reduces to Corollary 21

Next utilizing Corollary 21 we give the following result

Corollary 22 Let 119862 be a nonempty closed convex subset ofa real Hilbert space H

1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr

119862 be a nonexpansive mapping such that Fix(119878) cap Γ = 0 Forgiven 119909

0isin 119862 arbitrarily let the sequences 119909

119899 119910119899 119911119899 be

generated iteratively by

119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)

where 120572119899 sub (0infin) 120582119899 sub (0 1119860

2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin

(ii) 0 lt lim inf119899rarrinfin

120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1

(iv) 0 lt lim inf119899rarrinfin120582119899 le lim sup

119899rarrinfin120582119899 lt 1119860

2

Then all the sequences 119909119899 119910119899 119911119899 converge weakly to anelement 119909 isin Fix(119878) cap Γ


Proof In Corollary 21 put 1198611

= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573

119899+ 120575119899= 1 for all 119899 ge 0 and the iterative scheme (85)

is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)

This is equivalent to (86) Since 119878 is a nonexpansive mapping119878must be a 119896-strictly pseudocontractive mapping with 119896 = 0In this case it is easy to see that all the conditions (i)ndash(v) inCorollary 21 are satisfied Therefore in terms of Corollary 21we obtain the desired result

Now we are in a position to prove the weak convergenceof the sequences generated by the Mann-type extragradientiterative algorithm (17) with regularization

Theorem 23 Let 119862 be a nonempty closed convex subset ofa real Hilbert space H

1 Let 119860 isin 119861(H

1H2) and 119861

119894

119862 rarr H1be 120573119894-inverse strongly monotone for 119894 = 1 2

Let 119878 119862 rarr 119862 be a 119896-strictly pseudocontractive mappingsuch that Fix(119878) cap Ξ cap Γ = 0 For given 119909

0isin 119862 arbitrarily

let 119909119899 119906119899 119899 be the sequences generated by the Mann-

type extragradient iterative algorithm (17) with regularizationwhere 120583119894 isin (0 2120573119894) for 119894 = 1 2 120572119899 sub (0infin) 120582119899 sub

(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0

(iii) lim sup119899rarrinfin

120590119899lt 1

(iv) 0 lt lim inf119899rarrinfin120573119899 le lim sup119899rarrinfin

120573119899 lt 1 andlim inf119899rarrinfin120575119899 gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2

Then the sequences 119909119899 119906119899 119899 converge weakly to anelement 119909 isin Fix(119878) capΞcapΓ Furthermore (119909 119910) is a solution ofthe GSVI (3) where 119910 = 119875119862(119909 minus 12058321198612119909)

Proof First taking into account 0 lt lim inf119899rarrinfin120582119899 le


120582119899 lt 11198602 without loss of generality we may


2) Repeat-

ing the same argument as that in the proof of Theorem 20we can show that 119875

119862(119868 minus 120582nabla119891

120572) is 120577-averaged for each

120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4

Further repeating the same argument as that in the proof ofTheorem 20 we can also show that for each integer 119899 ge 0119875119862(119868minus120582119899nabla119891120572119899

) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin

(0 1)Next we divide the remainder of the proof into several

steps

Step 1 119909119899 is bounded

Indeed take 119901 isin Fix(119878) cap Ξ cap Γ arbitrarily Then 119878119901 = 119901119875119862(119868 minus 120582nabla119891)119901 = 119901 for 120582 isin (0 2119860

2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)

For simplicity we write

119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0

Utilizing the arguments similar to those of (46) and (47) inthe proof of Theorem 20 from (17) we can obtain

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894

119862 rarr H1is 120573119894-inverse strongly monotone and

0 lt 120583119894lt 2120573119894for 119894 = 1 2 utilizing the argument similar to

that of (49) in the proof of Theorem 20 we can obtain thatfor all 119899 ge 0

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)

Utilizing the argument similar to that of (52) in the proof ofTheorem 20 from (90) we can obtain

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)

Hence it follows from (92) and (93) that1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + (1 minus 120590119899) (119906119899 minus 119901)

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574

119899for all 119899 ge 0 by Lemma 18 we can readily

see from (94) that1003817100381710038171003817119909119899+1 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



and the sequence 119909119899 is bounded Since 119875

119862 nabla119891120572119899

1198611and 119861

2

are Lipschitz continuous it is easy to see that 119906119899 119899 119906119899

119910119899 and 119909

119899 are bounded where 119909

119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)


obtain from (17) (92) and (93) that

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2

minus1205831 (21205731minus1205831)10038171003817100381710038171198611119909119899minus1198611119902

1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899

rarr 0 lim sup119899rarrinfin

120590119899

lt 1 lim119899rarrinfin

119909119899minus 119901 exists

lim inf119899rarrinfin

(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin

(0 11198602) it follows from the boundedness of 119899 that

lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0

Indeed utilizing the Lipschitz continuity of nabla119891120572119899

we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817

(100)This together with

119899minus 119906119899 rarr 0 implies that lim


119899 = 0 and hence lim

119899rarrinfin119906119899

minus 119906119899 = 0 Utilizing the

arguments similar to those of (61) and (63) in the proof ofTheorem 20 we get1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)

Utilizing (91) and (101) we have1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2

+ 212058321003817100381710038171003817119909119899minus119909119899 minus (119901minus119902)

1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus (1 minus 120573

119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909119899 minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901

exists it follows from the boundedness of 119909119899 119909119899 119906119899 119899

and 119906119899 that lim

119899rarrinfin119909119899minus 119909119899minus (119901 minus 119902) = 0

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)

This together with 119910119899 minus 119906119899 le 120590119899119909119899 minus 119906119899 rarr 0 implies that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)

we havelim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and


Fix(119878) cap Ξ cap ΓIndeed since 119909

119899 is bounded there exists a subsequence

119909119899119894



and 119899minus 119906119899 rarr 0 and 119909

119899minus 119910119899 rarr 0 we deduce that

119910119899119894

rarr 119909 weakly and 119899119894

rarr 119909 weaklyFirst it is clear from Lemma 15 and 119878119910

119899minus 119910119899 rarr 0 that

119909 isin Fix(119878) Now let us show that 119909 isin Ξ Note that1003817100381710038171003817119909119899 minus 119866 (119909

119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H

1 ⟨119907minus119906 119908⟩ ge 0 forall119906 isin 119862 Utilizing the

argument similar to that of Step 4 in the proof ofTheorem 20from the relation

119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)

we can easily conclude that

⟨119907 minus 119909 119908⟩ ge 0 (113)

It is easy to see that119909 isin ΓTherefore119909 isin Fix(119878)capΞcapΓ Finallyutilizing theOpial condition [41] we infer that 119909119899 convergesweakly to 119909 isin Fix(119878)capΞcapΓ Further from 119909119899minus119906119899 rarr 0 and119909119899 minus 119899 rarr 0 it follows that both 119906119899 and 119899 convergeweakly to 119909 This completes the proof

Corollary 24 Let 119862 be a nonempty closed convex subset of areal Hilbert spaceH1 Let 119860 isin 119861(H1H2) and 119861119894 119862 rarr H1be 120573119894-inverse strongly monotone for 119894 = 1 2 Let 119878 119862 rarr 119862 bea 119896-strictly pseudocontractive mapping such that Fix(119878) cap Ξ cap

Γ = 0 For given 1199090isin 119862 arbitrarily let the sequences 119909

119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2

Then the sequences 119909119899 119906119899 119899 converge weakly to an

element 119909 isin Fix(119878) capΞcapΓ Furthermore (119909 119910) is a solution ofGSVI (3) where 119910 = 119875119862(119909 minus 12058321198612119909)

Next utilizing Corollary 24 we derive the followingresult

Corollary 25 Let 119862 be a nonempty closed convex subset of areal Hilbert space H1 Let 119860 isin 119861(H1H2) and 119878 119862 rarr 119862

be a nonexpansive mapping such that Fix(119878) cap Γ = 0 For given1199090

isin 119862 arbitrarily let the sequences 119909119899 119899 be generated

iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2

Then both the sequences 119909119899 and

119899 converge weakly to an

element 119909 isin Fix(119878) cap Γ


= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573

119899+120575119899= 1 for all 119899 ge 0 and the iterative scheme (114)

is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)

This is equivalent to (115) Since 119878 is a nonexpansivemapping119878must be a 119896-strictly pseudocontractive mapping with 119896 = 0In this case it is easy to see that all the conditions (i)ndash(iv) inCorollary 24 are satisfiedTherefore in terms of Corollary 24we obtain the desired result

Remark 26 Compared with the Ceng and Yao [31 Theorem31] our Corollary 25 coincides essentially with [31 Theorem31] This shows that our Theorem 23 includes [31 Theorem31] as a special case

Remark 27 Our Theorems 20 and 23 improve extend anddevelop [20 Theorem 57] [31 Theorem 31] [7 Theorem32] and [14 Theorem 31] in the following aspects

(i) Compared with the relaxed extragradient iterativealgorithm in [7 Theorem 32] our Mann-typeextragradient iterative algorithms with regularizationremove the requirement of boundedness for thedomain 119862 in which various mappings are defined

(ii) Because [31 Theorem 31] is the supplementationimprovement and extension of [20Theorem 57] andourTheorem 23 includes [31Theorem 31] as a specialcase beyond question our results are very interestingand quite valuable

(iii) The problem of finding an element of Fix(119878) cap Ξ cap Γ

in our Theorems 20 and 23 is more general than thecorresponding problems in [20Theorem 57] and [31Theorem 31] respectively

(iv) The hybrid extragradient method for finding anelement of Fix(119878) cap Ξ cap VI(119862 119860) in [14 Theorem 31]is extended to develop our Mann-type extragradientiterative algorithms (16) and (17) with regularizationfor finding an element of Fix(119878) cap Ξ cap Γ

(v) The proof of our results are very different from thatof [14 Theorem 31] because our argument techniquedepends on the Opial condition the restriction onthe regularization parameter sequence 120572

119899 and the

properties of the averaged mappings 119875119862(119868 minus 120582

119899nabla119891120572119899

)

to a great extent


(vi) Because our iterative algorithms (16) and (17) involvetwo inverse strongly monotone mappings 119861

1and 119861

2

a 119896-strictly pseudocontractive self-mapping 119878 andseveral parameter sequences they are more flexibleand more subtle than the corresponding ones in [20Theorem 57] and [31 Theorem 31] respectively

Acknowledgments

In this research the first author was partially supported bythe National Science Foundation of China (11071169) andPhD ProgramFoundation ofMinistry of Education ofChina(20123127110002)The third authorwas partially supported byGrant NSC 101-2115-M-037-001

References

[1] A BnouhachemM AslamNoor and Z Hao ldquoSome new extra-gradient iterative methods for variational inequalitiesrdquo Nonlin-ear Analysis Theory Methods amp Applications vol 70 no 3 pp1321ndash1329 2009

[2] L-C Ceng Q H Ansari and J-C Yao ldquoViscosity approxima-tion methods for generalized equilibrium problems and fixedpoint problemsrdquo Journal of Global Optimization vol 43 no 4pp 487ndash502 2009

[3] L-C Ceng and S Huang ldquoModified extragradient methodsfor strict pseudo-contractions and monotone mappingsrdquo Tai-wanese Journal ofMathematics vol 13 no 4 pp 1197ndash1211 2009

[4] L-C Ceng C-Y Wang and J-C Yao ldquoStrong convergencetheorems by a relaxed extragradient method for a generalsystem of variational inequalitiesrdquo Mathematical Methods ofOperations Research vol 67 no 3 pp 375ndash390 2008

[5] L-C Ceng and J-C Yao ldquoAn extragradient-like approximationmethod for variational inequality problems and fixed pointproblemsrdquo Applied Mathematics and Computation vol 190 no1 pp 205ndash215 2007

[6] L-C Ceng and J-C Yao ldquoRelaxed viscosity approximationmethods for fixed point problems and variational inequalityproblemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 69 no 10 pp 3299ndash3309 2008

[7] Y Yao Y-C Liou and S M Kang ldquoApproach to common ele-ments of variational inequality problems and fixed point pro-blems via a relaxed extragradient methodrdquo Computers ampMath-ematics with Applications vol 59 no 11 pp 3472ndash3480 2010

[8] L-C Zeng and J-C Yao ldquoStrong convergence theorem by anextragradient method for fixed point problems and variationalinequality problemsrdquo Taiwanese Journal of Mathematics vol 10no 5 pp 1293ndash1303 2006

[9] N Nadezhkina and W Takahashi ldquoStrong convergence theo-rem by a hybrid method for nonexpansive mappings and Lip-schitz-continuousmonotonemappingsrdquo SIAM Journal onOpti-mization vol 16 no 4 pp 1230ndash1241 2006

[10] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003

[11] G M Korpelevic ldquoAn extragradient method for finding saddlepoints and for other problemsrdquo Ekonomika i MatematicheskieMetody vol 12 no 4 pp 747ndash756 1976

[12] Y Yao and J-C Yao ldquoOn modified iterative method fornonexpansive mappings and monotone mappingsrdquo Applied

Mathematics and Computation vol 186 no 2 pp 1551ndash15582007

[13] R U Verma ldquoOn a new system of nonlinear variational ineq-ualities and associated iterative algorithmsrdquo Mathematical Sci-ences Research Hot-Line vol 3 no 8 pp 65ndash68 1999

[14] L-C Ceng S-M Guu and J-C Yao ldquoFinding common sol-utions of a variational inequality a general system of variationalinequalities and a fixed-point problem via a hybrid extragra-dient methodrdquo Fixed Point Theory and Applications vol 2011Article ID 626159 22 pages 2011

[15] Y Censor and T Elfving ldquoA multiprojection algorithm usingBregman projections in a product spacerdquoNumerical Algorithmsvol 8 no 2ndash4 pp 221ndash239 1994

[16] C Byrne ldquoIterative oblique projection onto convex sets and thesplit feasibility problemrdquo Inverse Problems vol 18 no 2 pp 441ndash453 2002

[17] Y Censor T Bortfeld B Martin and A Trofimov ldquoA unifiedapproach for inversion problems in intensity-modulated radia-tion therapyrdquo Physics in Medicine amp Biology vol 51 no 10 pp2353ndash2365 2006

[18] Y Censor T Elfving N Kopf and T Bortfeld ldquoThe multiple-sets split feasibility problem and its applications for inverseproblemsrdquo Inverse Problems vol 21 no 6 pp 2071ndash2084 2005

[19] Y Censor A Motova and A Segal ldquoPerturbed projectionsand subgradient projections for themultiple-sets split feasibilityproblemrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 1244ndash1256 2007

[20] H-K Xu ldquoIterative methods for the split feasibility problem ininfinite-dimensional Hilbert spacesrdquo Inverse Problems vol 26no 10 Article ID 105018 17 pages 2010

[21] C Byrne ldquoA unified treatment of some iterative algorithms insignal processing and image reconstructionrdquo Inverse Problemsvol 20 no 1 pp 103ndash120 2004

[22] B Qu and N Xiu ldquoA note on the CQ algorithm for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1655ndash1665 2005

[23] H-K Xu ldquoA variable Krasnoselrsquoskii-Mann algorithm and themultiple-set split feasibility problemrdquo Inverse Problems vol 22no 6 pp 2021ndash2034 2006

[24] Q Yang ldquoThe relaxed CQ algorithm solving the split feasibilityproblemrdquo Inverse Problems vol 20 no 4 pp 1261ndash1266 2004

[25] J Zhao and Q Yang ldquoSeveral solution methods for the splitfeasibility problemrdquo Inverse Problems vol 21 no 5 pp 1791ndash1799 2005

[26] M I Sezan and H Stark ldquoApplications of convex projectiontheory to image recovery in tomography and related areasrdquo inImage Recovery Theory and Applications H Stark Ed pp 415ndash462 Academic Press Orlando Fla USA 1987

[27] B Eicke ldquoIteration methods for convexly constrained ill-posedproblems in Hilbert spacerdquo Numerical Functional Analysis andOptimization vol 13 no 5-6 pp 413ndash429 1992

[28] L Landweber ldquoAn iteration formula for Fredholm integralequations of the first kindrdquo American Journal of Mathematicsvol 73 pp 615ndash624 1951

[29] L C Potter and K S Arun ldquoA dual approach to linear inverseproblems with convex constraintsrdquo SIAM Journal on Controland Optimization vol 31 no 4 pp 1080ndash1092 1993

[30] P L Combettes and V R Wajs ldquoSignal recovery by proximalforward-backward splittingrdquoMultiscaleModelingamp Simulationvol 4 no 4 pp 1168ndash1200 2005


[31] L-C Ceng Q H Ansari and J-C Yao ldquoAn extragradientmethod for solving split feasibility and fixed point problemsrdquoComputers amp Mathematics with Applications vol 64 no 4 pp633ndash642 2012

[32] N Nadezhkina andW Takahashi ldquoWeak convergence theoremby an extragradient method for nonexpansive mappings andmonotone mappingsrdquo Journal of Optimization Theory andApplications vol 128 no 1 pp 191ndash201 2006

[33] D P Bertsekas and E M Gafni ldquoProjection methods for vari-ational inequalities with application to the traffic assignmentproblemrdquoMathematical Programming Study vol 17 pp 139ndash1591982

[34] D Han and H K Lo ldquoSolving non-additive traffic assignmentproblems a descentmethod for co-coercive variational inequal-itiesrdquo European Journal of Operational Research vol 159 no 3pp 529ndash544 2004

[35] P L Combettes ldquoSolving monotone inclusions via composi-tions of nonexpansive averaged operatorsrdquo Optimization vol53 no 5-6 pp 475ndash504 2004

[36] K Goebel and W A Kirk Topics in Metric Fixed Point The-ory vol 28 of Cambridge Studies in Advanced MathematicsCambridge University Press 1990

[37] G Marino and H-K Xu ldquoWeak and strong convergence the-orems for strict pseudo-contractions in Hilbert spacesrdquo Journalof Mathematical Analysis and Applications vol 329 no 1 pp336ndash346 2007

[38] M O Osilike S C Aniagbosor and B G Akuchu ldquoFixedpoints of asymptotically demicontractive mappings in arbitraryBanach spacesrdquo Panamerican Mathematical Journal vol 12 no2 pp 77ndash88 2002

[39] K-K Tan and H K Xu ldquoApproximating fixed points of non-expansive mappings by the Ishikawa iteration processrdquo Journalof Mathematical Analysis and Applications vol 178 no 2 pp301ndash308 1993

[40] R T Rockafellar ldquoOn the maximality of sums of nonlinearmonotone operatorsrdquo Transactions of the American Mathemat-ical Society vol 149 pp 75ndash88 1970

[41] Z Opial ldquoWeak convergence of the sequence of successiveapproximations for nonexpansive mappingsrdquo Bulletin of theAmerican Mathematical Society vol 73 pp 591ndash597 1967

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


extended in [12] to develop a new iterativemethod for findingelements in Fix(119878) cap VI(119862 119860)

Let 1198611 1198612

119862 rarr H be two mappings Recently Cenget al [4] introduced and considered the following problem offinding (119909

lowast 119910lowast) isin 119862 times 119862 such that





(3)

which is called a general system of variational inequalities(GSVI) where 120583

1gt 0 and 120583

2gt 0 are two constants The

set of solutions of problem (3) is denoted by GSVI(119862 1198611 1198612)

In particular if 1198611

= 1198612 then problem (3) reduces to the

new system of variational inequalities (NSVI) introducedand studied byVerma [13] Further if 119909lowast = 119910

lowast then theNSVIreduces to VIP (1)

Recently Ceng et al [4] transformed problem (3) into afixed point problem in the following way

Lemma 1 (see [4]) For given 119909 119910 isin 119862 (119909 119910) is a solutionof problem (3) if and only if 119909 is a fixed point of the mapping119866 119862 rarr 119862 defined by

119866 (119909) = 119875119862[119875119862(119909 minus 120583

21198612119909) minus 120583

11198611119875119862(119909 minus 120583

21198612119909)]

forall119909 isin 119862

(4)

where 119910 = 119875119862(119909 minus 120583

21198612119909)

In particular if the mapping 119861119894 119862 rarr H is 120573

119894-inverse

strongly monotone for 119894 = 1 2 then the mapping 119866 isnonexpansive provided 120583

119894isin (0 2120573

119894) for 119894 = 1 2

Utilizing Lemma 1 they introduced and studied a relaxedextragradient method for solving GSVI (3)

Throughout this paper unless otherwise specified the setof fixed points of themapping119866 is denoted byΞ Based on therelaxed extragradient method and viscosity approximationmethod Yao et al [7] proposed and analyzed an iterativealgorithm for finding a common solution of GSVI (3) andfixed point problem of a strictly pseudocontractive mapping119878 119862 rarr 119862 where 119862 is a nonempty bounded closed convexsubset of a real Hilbert spaceH

Subsequently Ceng at al [14] further presented andanalyzed an iterative scheme for finding a common elementof the solution set of VIP (1) the solution set of GSVI (3)and fixed point set of a strictly pseudocontractive mapping119878 119862 rarr 119862

Theorem 2 (see [14 Theorem 31]) Let 119862 be a nonemptyclosed convex subset of a real Hilbert spaceH Let119860 119862 rarr Hbe 120572-inverse strongly monotone and let 119861

119894 119862 rarr H be

120573119894-inverse strongly monotone for 119894 = 1 2 Let 119878 119862 rarr 119862


Ξ cap VI(119862 119860) = 0 Let 119876 119862 rarr 119862 be a 120588-contraction with

120588 isin [0 12) For given 1199090

isin 119862 arbitrarily let the sequences119909119899 119910119899 and 119911


119911119899 = 119875119862 (119909119899 minus 120582119899119860119909119899)

119910119899= 120572119899119876119909119899

+ (1 minus 120572119899) 119875119862 [119875119862 (119911119899 minus 12058321198612119911119899)

minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(5)

where 120583119894 isin (0 2120573119894) for 119894 = 1 2 120582119899 sub (0 2120572] and 120572119899 120573119899120574119899 120575119899 sub [0 1] such that

(i) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0

(ii) lim119899rarrinfin

120572119899= 0 and sum

infin

119899=0120572119899= infin


120573119899


120573119899

lt 1 andlim inf119899rarrinfin120575119899 gt 0

(iv) lim119899rarrinfin(120574119899+1(1 minus 120573119899+1) minus 120574119899(1 minus 120573119899)) = 0(v) 0 lt lim inf

119899rarrinfin120582119899


120582119899

lt 2120572 andlim119899rarrinfin

|120582119899+1

minus 120582119899| = 0

Then the sequence 119909119899 generated by (5) converges strongly to

119909 = 119875Fix(119878)capΞcapVI(119862119860)119876119909 and (119909 119910) is a solution of GSVI (3)where 119910 = 119875

119862(119909 minus 120583

21198612119909)

On the other hand let 119862 and 119876 be nonempty closed con-vex subsets of real Hilbert spaces H

1and H

2 respectively

The split feasibility problem (SFP) is to find a point 119909lowast withthe following property

119909lowast

isin 119862 119860119909lowast

isin 119876 (6)

where 119860 isin 119861(H1H2) and 119861(H

1H2) denotes the family of

all bounded linear operators fromH1toH2

In 1994 the SFP was first introduced by Censor andElfving [15] in finite-dimensional Hilbert spaces for mod-eling inverse problems which arise from phase retrievalsand in medical image reconstruction A number of imagereconstruction problems can be formulated as the SFP seefor example [16] and the references therein Recently it isfound that the SFP can also be applied to study intensity-modulated radiation therapy see for example [17ndash19] andthe references therein In the recent past a wide varietyof iterative methods have been used in signal processingand image reconstruction and for solving the SFP see forexample [16ndash26] and the references therein A special caseof the SFP is the following convex constrained linear inverseproblem [27] of finding an element 119909 such that

119909 isin 119862 119860119909 = 119887 (7)

It has been extensively investigated in the literature using theprojected Landweber iterative method [28] Comparativelythe SFP has received much less attention so far due tothe complexity resulting from the set 119876 Therefore whethervarious versions of the projected Landweber iterativemethod[28] can be extended to solve the SFP remains an interesting







119862andor 119875

119876






min119909isin119862

119891 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2 (8)


min119909isin119862

119891120572 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2+

1

21205721199092 (9)






119875119862(119868 minus 120582nabla119891) 119909



119876)119860 and 119860











(i) the gradient

nabla119891120572= nabla119891 + 120572119868 = 119860


119876) 119860 + 120572119868 (13)



119862(119868 minus 120582nabla119891


ficient



2120572120582)

(14)

where 0 lt 120582 le 120572(1198602+ 120572)2


119909120572exists




119899 and


gradient algorithm


119910119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1 = 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119910119899)) forall119899 ge 0

(15)

where suminfin

119899=0120572119899lt infin 120582

119899 sub [119886 119887] for some 119886 119887 isin (0 1119860

2)


119909119899 and 119910




isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899 120574119899 120575119899 sub [0 1] such

that 120590119899+ 120591119899

le 1 and 120573119899+ 120574119899+ 120575119899



given 1199090



119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120590119899119909119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))


minus 12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(16)




119862(119909 minus 120583

21198612119909)


isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590119899 120573119899 120574119899 120575119899 sub [0 1] such that






119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572

119899

(119906119899))

119910119899= 120590119899119909119899+ (1 minus 120590

119899) 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(17)




119862(119909 minus 120583

21198612119909)


2 Preliminaries





1003817100381710038171003817119909 minus 1198751198701199091003817100381710038171003817 = inf119910isin119870

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 119889 (119909 119870) (18)




(ii) 119911 = 119875119870119909 hArr 119909 minus 119911

2le 119909 minus 119910

2minus 119910 minus 119911

2 for all119910 isin 119870

(iii) ⟨119875119870119909 minus 119875

119870119910 119909 minus 119910⟩ ge 119875

119870119909 minus 119875

1198701199102 for all 119910 isin


monotone



1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin H (19)



1003817100381710038171003817

2 forall119909 119910 isin H (20)


119879 =1

2(119868 + 119878) (21)







1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (23)



1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (24)



119870is 1-ism



119879 equiv (1 minus 120572) 119868 + 120572119878 (25)




is 12-ism








119894119873

119894=1is


(v) If the mappings 119879119894119873



119873

⋂

119894=1

Fix (119879119894) = Fix (119879

1sdot sdot sdot 119879119873) (26)




⟨119878119909 minus 119878119910 119909 minus 119910⟩

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2minus

1 minus 119896

2


2

forall119909 119910 isin 119862

(27)




1003817100381710038171003817120582119909 + (1 minus 120582)1199101003817100381710038171003817

2= 120582119909

2+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817

2

(28)


be a mapping



1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 le

1 + 119896

1 minus 119896



119899 is a



119878)119909119899





119899=0 119887119899infin

119899=0 and 120575

119899infin

119899=0be


119886119899+1

le (1 + 120575119899) 119886119899 + 119887

119899 forall119899 ge 0 (30)

If suminfin119899=0

120575119899


119899=0119887119899


119886119899exists If




119886119899= 0


119899=0and 119887

119899infin

119899=0be two


119886119899+1

le 119886119899+ 119887119899 forall119899 ge 0 (31)

If suminfin119899=0





1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)


(32)



1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)













Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)


3 Main Results



1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1




119899 119910119899 119911119899 be the


119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin


119899+ 120591119899le 1 for all 119899 ge 0


120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)


120582119899

le


120582119899



2)


2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)


119876)119860 is 1119860

2-ism that is


1198602


2 (39)

Observe that


120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2


= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817



1003817100381710038171003817

2


1003817100381710038171003817

2


120572(119910)

1003817100381710038171003817

2

(40)


= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according



119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119862andor 119875

119876






min119909isin119862

119891 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2 (8)


min119909isin119862

119891120572 (119909) =1

2

1003817100381710038171003817119860119909 minus 119875119876119860119909

1003817100381710038171003817

2+

1

21205721199092 (9)






119875119862(119868 minus 120582nabla119891) 119909



119876)119860 and 119860











(i) the gradient

nabla119891120572= nabla119891 + 120572119868 = 119860


119876) 119860 + 120572119868 (13)



119862(119868 minus 120582nabla119891


ficient



2120572120582)

(14)

where 0 lt 120582 le 120572(1198602+ 120572)2


119909120572exists




119899 and


gradient algorithm


119910119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1 = 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119910119899)) forall119899 ge 0

(15)

where suminfin

119899=0120572119899lt infin 120582

119899 sub [119886 119887] for some 119886 119887 isin (0 1119860

2)


119909119899 and 119910




isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899 120574119899 120575119899 sub [0 1] such

that 120590119899+ 120591119899

le 1 and 120573119899+ 120574119899+ 120575119899



given 1199090



119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120590119899119909119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))


minus 12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(16)




119862(119909 minus 120583

21198612119909)


isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590119899 120573119899 120574119899 120575119899 sub [0 1] such that






119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572

119899

(119906119899))

119910119899= 120590119899119909119899+ (1 minus 120590

119899) 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(17)




119862(119909 minus 120583

21198612119909)


2 Preliminaries





1003817100381710038171003817119909 minus 1198751198701199091003817100381710038171003817 = inf119910isin119870

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 119889 (119909 119870) (18)




(ii) 119911 = 119875119870119909 hArr 119909 minus 119911

2le 119909 minus 119910

2minus 119910 minus 119911

2 for all119910 isin 119870

(iii) ⟨119875119870119909 minus 119875

119870119910 119909 minus 119910⟩ ge 119875

119870119909 minus 119875

1198701199102 for all 119910 isin


monotone



1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin H (19)



1003817100381710038171003817

2 forall119909 119910 isin H (20)


119879 =1

2(119868 + 119878) (21)







1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (23)



1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (24)



119870is 1-ism



119879 equiv (1 minus 120572) 119868 + 120572119878 (25)




is 12-ism








119894119873

119894=1is


(v) If the mappings 119879119894119873



119873

⋂

119894=1

Fix (119879119894) = Fix (119879

1sdot sdot sdot 119879119873) (26)




⟨119878119909 minus 119878119910 119909 minus 119910⟩

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2minus

1 minus 119896

2


2

forall119909 119910 isin 119862

(27)




1003817100381710038171003817120582119909 + (1 minus 120582)1199101003817100381710038171003817

2= 120582119909

2+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817

2

(28)


be a mapping



1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 le

1 + 119896

1 minus 119896



119899 is a



119878)119909119899





119899=0 119887119899infin

119899=0 and 120575

119899infin

119899=0be


119886119899+1

le (1 + 120575119899) 119886119899 + 119887

119899 forall119899 ge 0 (30)

If suminfin119899=0

120575119899


119899=0119887119899


119886119899exists If




119886119899= 0


119899=0and 119887

119899infin

119899=0be two


119886119899+1

le 119886119899+ 119887119899 forall119899 ge 0 (31)

If suminfin119899=0





1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)


(32)



1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)













Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)


3 Main Results



1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1




119899 119910119899 119911119899 be the


119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin


119899+ 120591119899le 1 for all 119899 ge 0


120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)


120582119899

le


120582119899



2)


2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)


119876)119860 is 1119860

2-ism that is


1198602


2 (39)

Observe that


120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2


= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817



1003817100381710038171003817

2


1003817100381710038171003817

2


120572(119910)

1003817100381710038171003817

2

(40)


= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according



119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










given 1199090



119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120590119899119909119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))


minus 12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(16)




119862(119909 minus 120583

21198612119909)


isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin)

120582119899 sub (0 1119860

2) and 120590119899 120573119899 120574119899 120575119899 sub [0 1] such that






119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572

119899

(119906119899))

119910119899= 120590119899119909119899+ (1 minus 120590

119899) 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(17)




119862(119909 minus 120583

21198612119909)


2 Preliminaries





1003817100381710038171003817119909 minus 1198751198701199091003817100381710038171003817 = inf119910isin119870

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 = 119889 (119909 119870) (18)




(ii) 119911 = 119875119870119909 hArr 119909 minus 119911

2le 119909 minus 119910

2minus 119910 minus 119911

2 for all119910 isin 119870

(iii) ⟨119875119870119909 minus 119875

119870119910 119909 minus 119910⟩ ge 119875

119870119909 minus 119875

1198701199102 for all 119910 isin


monotone



1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 forall119909 119910 isin H (19)



1003817100381710038171003817

2 forall119909 119910 isin H (20)


119879 =1

2(119868 + 119878) (21)







1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (23)



1003817100381710038171003817

2 forall119909 119910 isin 119863 (119879) (24)



119870is 1-ism



119879 equiv (1 minus 120572) 119868 + 120572119878 (25)




is 12-ism








119894119873

119894=1is


(v) If the mappings 119879119894119873



119873

⋂

119894=1

Fix (119879119894) = Fix (119879

1sdot sdot sdot 119879119873) (26)




⟨119878119909 minus 119878119910 119909 minus 119910⟩

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2minus

1 minus 119896

2


2

forall119909 119910 isin 119862

(27)




1003817100381710038171003817120582119909 + (1 minus 120582)1199101003817100381710038171003817

2= 120582119909

2+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817

2

(28)


be a mapping



1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 le

1 + 119896

1 minus 119896



119899 is a



119878)119909119899





119899=0 119887119899infin

119899=0 and 120575

119899infin

119899=0be


119886119899+1

le (1 + 120575119899) 119886119899 + 119887

119899 forall119899 ge 0 (30)

If suminfin119899=0

120575119899


119899=0119887119899


119886119899exists If




119886119899= 0


119899=0and 119887

119899infin

119899=0be two


119886119899+1

le 119886119899+ 119887119899 forall119899 ge 0 (31)

If suminfin119899=0





1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)


(32)



1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)













Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)


3 Main Results



1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1




119899 119910119899 119911119899 be the


119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin


119899+ 120591119899le 1 for all 119899 ge 0


120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)


120582119899

le


120582119899



2)


2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)


119876)119860 is 1119860

2-ism that is


1198602


2 (39)

Observe that


120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2


= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817



1003817100381710038171003817

2


1003817100381710038171003817

2


120572(119910)

1003817100381710038171003817

2

(40)


= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according



119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870is 1-ism



119879 equiv (1 minus 120572) 119868 + 120572119878 (25)




is 12-ism








119894119873

119894=1is


(v) If the mappings 119879119894119873



119873

⋂

119894=1

Fix (119879119894) = Fix (119879

1sdot sdot sdot 119879119873) (26)




⟨119878119909 minus 119878119910 119909 minus 119910⟩

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2minus

1 minus 119896

2


2

forall119909 119910 isin 119862

(27)




1003817100381710038171003817120582119909 + (1 minus 120582)1199101003817100381710038171003817

2= 120582119909

2+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2


1003817100381710038171003817

2

(28)


be a mapping



1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 le

1 + 119896

1 minus 119896



119899 is a



119878)119909119899





119899=0 119887119899infin

119899=0 and 120575

119899infin

119899=0be


119886119899+1

le (1 + 120575119899) 119886119899 + 119887

119899 forall119899 ge 0 (30)

If suminfin119899=0

120575119899


119899=0119887119899


119886119899exists If




119886119899= 0


119899=0and 119887

119899infin

119899=0be two


119886119899+1

le 119886119899+ 119887119899 forall119899 ge 0 (31)

If suminfin119899=0





1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)


(32)



1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)













Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)


3 Main Results



1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1




119899 119910119899 119911119899 be the


119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin


119899+ 120591119899le 1 for all 119899 ge 0


120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)


120582119899

le


120582119899



2)


2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)


119876)119860 is 1119860

2-ism that is


1198602


2 (39)

Observe that


120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2


= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817



1003817100381710038171003817

2


1003817100381710038171003817

2


120572(119910)

1003817100381710038171003817

2

(40)


= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according



119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817120574 (119909 minus 119910) + 120575 (119878119909 minus 119878119910)1003817100381710038171003817 le (120574 + 120575)


(32)



1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 119909

2+ 2 ⟨119910 119909 + 119910⟩ forall119909 119910 isin H (33)













Define

119879119907 = 119865119907 + 119873

119870119907 if 119907 isin 119870

0 if 119907 notin 119870(37)


3 Main Results



1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr H

1




119899 119910119899 119911119899 be the


119894isin (0 2120573

119894) for 119894 =

1 2 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120590

119899 120591119899 120573119899

120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin


119899+ 120591119899le 1 for all 119899 ge 0


120591119899le lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)


120582119899

le


120582119899



2)


2)) where

120577 =

2 + 120582 (120572 + 1198602)

4isin (0 1) (38)


119876)119860 is 1119860

2-ism that is


1198602


2 (39)

Observe that


120572(119910) 119909 minus 119910⟩

= (120572 + 1198602) [120572

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2


= 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817


+ 12057211986021003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


ge 12057221003817100381710038171003817119909 minus 119910

1003817100381710038171003817



1003817100381710038171003817

2


1003817100381710038171003817

2


120572(119910)

1003817100381710038171003817

2

(40)


= 120572119868 + 119860lowast(119868 minus 119875

119876)119860 is

1(120572+ 1198602)-ismThus 120582nabla119891

120572is 1120582(120572+ 119860

2)-ism according



119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119868 minus 120582nabla119891120572is 120582(120572 + 119860




120572) is 120577-averaged

with

120577 =1

2+

120582 (120572 + 1198602)

2minus

1

2sdot

120582 (120572 + 1198602)

2

=

2 + 120582 (120572 + 1198602)

4isin (0 1)

(41)


for 120582119899 sub [119886 119887] with 119886 119887 isin (0 1119860

2) we have

119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

1198602


1

120572119899+ 119860

2 (42)


119886 le inf119899ge0

120582119899le sup119899ge0

120582119899le 119887 lt

1

120572119899 + 1198602 forall119899 ge 0 (43)


120582119899nabla119891120572119899


120577119899 =

1

2+

120582119899(120572119899+ 119860

2)

2minus

1

2sdot

120582119899(120572119899+ 119860

2)

2

=

2 + 120582119899 (120572119899 + 119860

2)

4isin (0 1)

(44)






2) and

119901 = 119875119862[119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)] (45)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

le10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891120572119899

) 11990110038171003817100381710038171003817

+10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 +10038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(46)


1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

=10038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119909119899minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

=10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 119901

+ 119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

2

le10038171003817100381710038171003817119875119862 (119868 minus 120582119899nabla119891120572

119899


) 11990110038171003817100381710038171003817

2

+ 2 ⟨119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901

minus 119875119862(119868 minus 120582

119899nabla119891) 119901 119911

119899minus 119901⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817119875119862(119868 minus 120582

119899nabla119891120572119899

) 119901 minus 119875119862(119868 minus 120582

119899nabla119891) 119901

10038171003817100381710038171003817

times1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 210038171003817100381710038171003817(119868 minus 120582

119899nabla119891120572119899


10038171003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(47)


21198612119901) 119899

= 119875119862(119911119899minus

12058321198612119911119899)


119906119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899))

(48)


119899minus120591119899)119906119899for each

119899 ge 0 Since 119861119894 119862 rarr H



1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899)


1003817100381710038171003817

2

=1003817100381710038171003817119875119862 [119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]


2

le1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 12058311198611119875119862(119911119899minus 12058321198612119911119899)]

minus [119875119862(119901 minus 120583

21198612119901) minus 120583

11198611119875119862(119901 minus 120583

21198612119901)]

1003817100381710038171003817

2

=1003817100381710038171003817[119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875119862(119901 minus 120583

21198612119901)]


21198612119901)]

1003817100381710038171003817

2


le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le1003817100381710038171003817119875119862 (119911119899 minus 120583

21198612119911119899) minus 119875

119862 (119901 minus 12058321198612119901)

1003817100381710038171003817

2

minus 1205831(21205731minus 1205831)10038171003817100381710038171198611119875119862 (119911

119899minus 12058321198612119911119899)

minus 1198611119875119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817(119911119899 minus 120583

21198612119911119899) minus (119901 minus 120583

21198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

=1003817100381710038171003817(119911119899 minus 119901) minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2


11199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

2

(49)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le10038171003817100381710038171003817119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119901

10038171003817100381710038171003817

2


(119911119899) minus 119906119899

10038171003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2+ 2120582119899⟨nabla119891120572119899

(119911119899) 119901 minus 119906

119899⟩

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119901) 119901 minus 119911119899⟩

+ ⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩

+⟨nabla119891120572119899

(119911119899) 119911119899minus 119906119899⟩)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899(⟨nabla119891120572119899

(119901) 119901 minus 119911119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[ ⟨(120572119899119868 + nabla119891) 119901 119901minus119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus119906119899⟩ ]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2


1003817100381710038171003817

2

+ 2120582119899[120572119899⟨119901 119901 minus 119911

119899⟩ + ⟨nabla119891

120572119899

(119911119899) 119911119899minus 119906119899⟩]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

(50)


⟨119909119899 minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899 minus 119911119899⟩

= ⟨119909119899minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119906119899minus 119911119899⟩

+ ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899minus 119911119899⟩

le ⟨120582119899nabla119891120572119899

(119909119899) minus 120582119899nabla119891120572119899

(119911119899) 119906119899 minus 119911119899⟩

le 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119909119899) minus nabla119891

120572119899

(119911119899)10038171003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

(51)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 2 ⟨119909119899minus 120582119899nabla119891120572119899

(119911119899) minus 119911119899 119906119899minus 119911119899⟩

+ 2120582119899120572119899⟨119901 119901 minus 119911

119899⟩

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 1199111198991003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 1199061198991003817100381710038171003817

2

+ 2120582119899 (120572119899 + 1198602)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199111198991003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2minus

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817

2

+ 1205822

119899(120572119899+ 119860

2)21003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 4120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 4120582

2

1198991205722

119899

10038171003817100381710038171199011003817100381710038171003817

2

= (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(52)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899minus 120591119899)1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899) (

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + [2120591119899+ (1 minus 120590

119899minus 120591119899)] 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(53)


obtain from (53)

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817


1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 +

1003817100381710038171003817120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899 + 120575119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(54)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin



119875119862 nabla119891120572119899

1198611and 119861


see that 119911119899 119906119899 119906119899 119910119899 and


119899= 119875119862(119911119899minus 12058321198612119911119899) for all 119899 ge 0


1198612119911119899minus 1198612119901 = 0 lim


1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times

10038171003817100381710038171003817100381710038171003817

1

120574119899+ 120575119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120591119899[1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2]

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2

minus 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899+ 120575119899)

times (1 minus 120590119899) 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 120591119899(1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2



21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus (120574119899+ 120575119899) 120591119899(1 minus 120582

2

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119909119899 minus 119911

119899

1003817100381710038171003817

2

+ (1 minus 120590119899 minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

(56)


Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Therefore

(120574119899 + 120575119899) 120591119899 (1 minus 120582

2

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817

2

+ (1 minus 120590119899minus 120591119899)


21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119899 minus 119861

11199021003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

(57)

Since120572119899



119899rarrinfin(120574119899+120575119899) gt

0 120582119899 sub [119886 119887] and 0 lt lim inf

119899rarrinfin120591119899


(120590119899+


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119911119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 = 0

(58)


119878119910119899minus 119910119899 = 0

Indeed observe that

1003817100381710038171003817119906119899 minus 119911119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899)) minus 119875

119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

10038171003817100381710038171003817

le10038171003817100381710038171003817(119909119899 minus 120582119899nabla119891120572

119899


(119909119899))10038171003817100381710038171003817

= 120582119899

10038171003817100381710038171003817nabla119891120572119899

(119911119899) minus nabla119891

120572119899

(119909119899)10038171003817100381710038171003817

le 120582119899(120572119899+ 119860

2)1003817100381710038171003817119911119899 minus 119909

119899

1003817100381710038171003817

(59)



lim119899rarrinfin




1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


1003817100381710038171003817

2


21198612119901) 119899minus 119902⟩

=1

2[1003817100381710038171003817119911119899 minus 119901 minus 120583

2(1198612119911119899minus 1198612119901)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2


2(1198612119911119899minus 1198612119901) minus (

119899minus 119902)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817(119911119899 minus


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2minus


2


2119911119899minus 1198612119901⟩

minus1205832

2

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

2]

le1

2[1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817 ]

(60)

that is

1003817100381710038171003817119899 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

(61)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817119875119862 (119899 minus 120583

11198611119899) minus 119875119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


=1


1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1(1198611119899minus 1198611119902) minus (119906

119899minus 119901)

1003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2]

=1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831⟨119899minus 119906119899+ (119901 minus 119902) 119861

1119899minus 1198611119902⟩

minus 1205832

1

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

2]

le1

2[1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2+

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817]

(62)


that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










that is

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(63)


1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120590119899 (119909119899 minus 119901) + 120591119899 (119906119899 minus 119901)


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 120591119899

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2


1003817100381710038171003817

2

le 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times [1003817100381710038171003817119899 minus 119902

1003817100381710038171003817

2

minus1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817 ]

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119911119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817


1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ 120591119899(1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817)

+ (1 minus 120590119899minus 120591119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

minus1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+ 21205832


1003817100381710038171003817

times10038171003817100381710038171198612119911119899 minus 119861

21199011003817100381710038171003817 minus

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

(64)


1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119909119899 minus 119901) + 120574

119899(119910119899minus 119901) + 120575

119899(119878119910119899minus 119901)

1003817100381710038171003817

2

=

10038171003817100381710038171003817100381710038171003817

120573119899(119909119899minus 119901) + (1 minus 120573

119899) sdot

1

1 minus 120573119899

times [120574119899(119910119899minus 119901)+120575

119899(119878119910119899minus119901)]

10038171003817100381710038171003817100381710038171003817

2

= 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119910119899 minus 119901) + 120575119899 (119878119910119899 minus 119901)]

10038171003817100381710038171003817100381710038171003817

2

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817



1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)

minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817


119899minus 120591119899) (


1003817100381710038171003817

2

+1003817100381710038171003817119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2)


minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(65)



times (1003817100381710038171003817119911119899 minus

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119911119899 minus 1199011003817100381710038171003817

+ 21205832


10038171003817100381710038171198612119911119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119899 minus 11986111199021003817100381710038171003817

(66)


120573119899


120573119899

lt 1lim sup

119899rarrinfin(120590119899+ 120591119899) lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119911119899minus

1198612119901 rarr 0 119861




119899 that

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 = 0

(67)


lim119899rarrinfin

1003817100381710038171003817119911119899 minus 119906119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119906119899

1003817100381710038171003817 = 0 (68)


119899

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 119906119899


119899

1003817100381710038171003817 997888rarr 0

(69)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (70)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(71)

we have

lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199091198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119878119910119899 minus 1199101198991003817100381710038171003817 = 0 (72)

Step 4 119909119899 119910119899 and 119911




119909119899119894




119899119894

rarr 119909

weakly and 119911119899119894




1003817100381710038171003817119911119899 minus 119866 (119911119899)1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119875


1003817100381710038171003817

=1003817100381710038171003817119911119899 minus 119906

119899

1003817100381710038171003817 997888rarr 0

(73)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(74)


119908 isin 119879119907 = nabla119891 (119907) + 119873119862119907 (75)

and hence

119908 minus nabla119891 (119907) isin 119873119862119907 (76)

So we have



119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899)) 119907 isin 119862 (78)

we have

⟨119909119899 minus 120582119899nabla119891120572119899

(119909119899) minus 119911119899 119911119899 minus 119907⟩ ge 0 (79)

and hence

⟨119907 minus 119911119899119911119899minus 119909119899

120582119899

+ nabla119891120572119899

(119909119899)⟩ ge 0 (80)

Therefore from

119908 minus nabla119891 (119907) isin 119873119862119907 119911

119899119894

isin 119862 (81)


we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










we have

⟨119907 minus 119911119899119894

119908⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891120572119899119894

(119909119899119894

)⟩

= ⟨119907 minus 119911119899119894

nabla119891 (119907)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

+ nabla119891 (119909119899119894

)⟩

minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

= ⟨119907 minus 119911119899119894


)⟩

+ ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

ge ⟨119907 minus 119911119899119894

nabla119891 (119911119899119894

) minus nabla119891 (119909119899119894

)⟩

minus ⟨119907 minus 119911119899119894

119911119899119894

minus 119909119899119894

120582119899119894

⟩ minus 120572119899119894

⟨119907 minus 119911119899119894

119909119899119894

⟩

(82)

Hence we get





119899119895


119899119895


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817


10038171003817100381710038171003817119909119899119894


10038171003817100381710038171003817119909119899119894

minus 11990910038171003817100381710038171003817



100381710038171003817100381710038171003817119909119899119895

minus 119909100381710038171003817100381710038171003817


100381710038171003817100381710038171003817119909119899119895


1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817

(84)





that both 119910119899 and 119911


the proof


1 Let 119860 isin 119861(H

1H2) and 119861

119894 119862 rarr





119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899= 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

+ (1 minus 120591119899) 119875119862[119875119862(119911119899minus 12058321198612119911119899)

minus12058311198611119875119862 (119911119899 minus 12058321198612119911119899)]

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(85)

where 120583119894isin (0 2120573

119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub (0 1

1198602) and 120591119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120591119899 lt 1


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


119899rarrinfin120582119899lt 1119860

2



119862(119909 minus 120583

21198612119909)





1 Let 119860 isin 119861(H

1H2) and 119878 119862 rarr



119899 119910119899 119911119899 be


119911119899 = 119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119909119899))

119910119899= (1 minus 120591

119899) 119911119899+ 120591119899119875119862(119909119899minus 120582119899nabla119891120572119899

(119911119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119910119899 forall119899 ge 0

(86)


2) and 120591119899 120573119899 sub [0 1]

such that

(i) suminfin

119899=0120572119899lt infin


120591119899le lim sup

119899rarrinfin120591119899lt 1


120573119899 lt 1


119899rarrinfin120582119899 lt 1119860

2




= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119911119899= 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119910119899 = 120591119899119875119862 (119909119899 minus 120582119899nabla119891120572119899

(119911119899)) + (1 minus 120591119899) 119911119899

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(87)




1 Let 119860 isin 119861(H

1H2) and 119861

119894






(0 11198602) and 120590

119899 120573119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899 lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120590119899lt 1




120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2






2) Repeat-


119862(119868 minus 120582nabla119891


120582 isin (0 1(120572 + 1198602)) where 120577 = (2 + 120582(120572 + 119860

2))4


) is 120577119899-averagedwith 120577

119899= (2+120582

119899(120572119899+1198602))4 isin


steps



2) and

119901 = 119875119862 [119875119862 (119901 minus 120583

21198612119901) minus 120583

11198611119875119862 (119901 minus 120583

21198612119901)] (88)


119902 = 119875119862 (119901 minus 12058321198612119901)

119909119899= 119875119862(119909119899minus 12058321198612119909119899)

119906119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

(89)

for each 119899 ge 0 Then 119910119899

= 120590119899119909119899+ (1 minus 120590

119899)119906119899for each 119899 ge 0


1003817100381710038171003817119899 minus 1199011003817100381710038171003817 le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 + 120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817 (90)

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 (91)

Since 119861119894




1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


21199011003817100381710038171003817

2


1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

(92)


1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899+ 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le (1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)2

(93)


1003817100381710038171003817


1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

le 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817)

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

(94)

Since (120574119899+120575119899)119896 le 120574



1003817100381710038171003817

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + (120574

119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + 21198871003817100381710038171003817119901

1003817100381710038171003817 120572119899

(95)

Since suminfin


119899=02119887119901120572

119899lt infin Thus


lim119899rarrinfin




119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862 nabla119891120572119899

1198611and 119861

2


119910119899 and 119909


119899= 119875119862(119909119899minus 12058321198612119909119899) for

all 119899 ge 0


1198612119909119899minus 1198612119901 = 0 lim

119899rarrinfin1198611119909119899minus

1198611119902 = 0 and lim


119862(119901minus12058321198612119901)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (120574119899 + 120575119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)

times [1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (120574119899+ 120575119899)

times 1205901198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120590119899)

times [1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus 1205832 (21205732 minus 1205832)

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

2


1003817100381710038171003817

2+2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899minus1199011003817100381710038171003817

+ (1205822

119899(120572119899 + 119860

2)2

minus 1)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2]

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 minus (120574

119899+ 120575119899) (1 minus 120590

119899)

times 1205832

(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831 (21205731 minus 1205831)10038171003817100381710038171198611119909119899 minus 1198611119902

1003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899 + 119860

2)2

)1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

2

(97)

Therefore

(120574119899+ 120575119899) (1 minus 120590

119899) 1205832(21205732minus 1205832)10038171003817100381710038171198612119909119899 minus 119861

21199011003817100381710038171003817

2

+ 1205831(21205731minus 1205831)10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817

2

+ (1 minus 1205822

119899(120572119899+ 119860

2)2

)1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

(98)

Since 120572119899


120590119899


119909119899minus 119901 exists


(120574119899+ 120575119899) gt 0 and 120582

119899 sub [119886 119887] for some 119886 119887 isin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817 = 0

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817 = 0

(99)


119878119910119899minus 119910119899 = 0


we have1003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

=10038171003817100381710038171003817119875119862(119906119899minus 120582119899nabla119891120572119899

(119899)) minus 119875

119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

10038171003817100381710038171003817

le 120582119899 (120572119899 + 119860

2)1003817100381710038171003817119899 minus 119906119899

1003817100381710038171003817





119899rarrinfin119906119899



1003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

2le

1003817100381710038171003817119909119899 minus 1199021003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

(101)


1003817100381710038171003817

2

=1003817100381710038171003817119899 minus 119901 + 119906119899 minus 119899

1003817100381710038171003817

2

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2 ⟨119906

119899minus 119899 119906119899minus 119901⟩

le1003817100381710038171003817119899 minus 119901

1003817100381710038171003817

2+ 2

1003817100381710038171003817119906119899 minus 1198991003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119902

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus


1003817100381710038171003817

2

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(102)



1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)1003817100381710038171003817119910119899 minus 119901

1003817100381710038171003817

2minus 120573119899(1 minus 120573

119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 1205731198991003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2+ (1 minus 120573119899)

times [120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120590

119899)1003817100381710038171003817119906119899 minus 119901

1003817100381710038171003817

2]

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le 120573119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2+ (1 minus 120573

119899)

times 120590119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120590119899) [

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2minus

1003817100381710038171003817119909119899 minus 119909119899

minus (119901 minus 119902)1003817100381710038171003817

2


1003817100381710038171003817

10038171003817100381710038171198612119909119899minus11986121199011003817100381710038171003817

minus1003817100381710038171003817119909119899 minus 119906

119899+ (119901 minus 119902)

1003817100381710038171003817

2

+ 212058311003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)

1003817100381710038171003817

times10038171003817100381710038171198611119909119899 minus 119861

11199021003817100381710038171003817 + 2120582

119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 ]

minus 120573119899 (1 minus 120573119899)

times

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817


119899) (1 minus 120590

119899)

times (1003817100381710038171003817119909119899 minus 119909

119899minus (119901 minus 119902)

1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

2)

+ 21205832


10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899+ (119901 minus 119902)

1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817 + 2

1003817100381710038171003817119906119899 minus 119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

minus 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

(103)


(1 minus 120573119899) (1 minus 120590

119899)


1003817100381710038171003817

2+

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

2)

+ 120573119899(1 minus 120573

119899)

10038171003817100381710038171003817100381710038171003817

1

1 minus 120573119899

[120574119899(119909119899minus 119910119899) + 120575119899(119909119899minus 119878119910119899)]

10038171003817100381710038171003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2minus

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2+ 2120582119899120572119899

10038171003817100381710038171199011003817100381710038171003817

1003817100381710038171003817119899 minus 1199011003817100381710038171003817

+ 21205832


1003817100381710038171003817

10038171003817100381710038171198612119909119899 minus 11986121199011003817100381710038171003817

+ 21205831

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817

10038171003817100381710038171198611119909119899 minus 11986111199021003817100381710038171003817

+ 21003817100381710038171003817119906119899 minus

119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817

(104)


120573119899


120573119899

lt 1lim sup

119899rarrinfin120590119899lt 1 120582

119899 sub [119886 119887] 120572

119899rarr 0 119861

2119909119899minus 1198612119901 rarr

0 1198611119909119899minus 1198611119902 rarr 0 119906


119899rarrinfin119909119899minus 119901




lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899 + (119901 minus 119902)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817120574119899 (119909119899 minus 119910119899) + 120575119899 (119909119899 minus 119878119910119899)1003817100381710038171003817 = 0

(105)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 lim

119899rarrinfin

1003817100381710038171003817119909119899 minus 1199061198991003817100381710038171003817 = 0 (106)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (107)

Since1003817100381710038171003817120575119899 (119878119910119899 minus 119909

119899)1003817100381710038171003817


1003817100381710038171003817

le1003817100381710038171003817120574119899 (119909119899 minus 119910

119899) + 120575119899(119909119899minus 119878119910119899)1003817100381710038171003817 + 120574119899

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817

(108)


1003817100381710038171003817119878119910119899 minus 119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119878119910119899 minus 119910119899

1003817100381710038171003817 = 0 (109)

Step 4 119909119899 119906119899 and




119909119899119894





119910119899119894

rarr 119909 weakly and 119899119894


119899minus 119910119899 rarr 0 that


119899)1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119875119862 [119875119862 (119909119899 minus 12058321198612119909119899)

minus 12058311198611119875119862(119909119899minus 12058321198612119909119899)]

1003817100381710038171003817

=1003817100381710038171003817119909119899 minus 119906119899

1003817100381710038171003817 997888rarr 0

(110)


119879119907 = nabla119891 (119907) + 119873

119862119907 if 119907 isin 119862

0 if 119907 notin 119862(111)


where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where119873119862119907 = 119908 isin H



119899 = 119875119862 (119906119899 minus 120582119899nabla119891120572119899

(119906119899)) 119907 isin 119862 (112)


⟨119907 minus 119909 119908⟩ ge 0 (113)




119899 119906119899



119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120574119899119910119899+ 120575119899119878119910119899 forall119899 ge 0

(114)

where 120583119894

isin (0 2120573119894) for 119894 = 1 2 120572

119899 sub (0infin) 120582

119899 sub

(0 11198602) and 120573

119899 120574119899 120575119899 sub [0 1] such that

(i) suminfin

119899=0120572119899lt infin

(ii) 120573119899+ 120574119899+ 120575119899= 1 and (120574

119899+ 120575119899)119896 le 120574

119899for all 119899 ge 0


120573119899


120573119899

lt 1 andlim inf

119899rarrinfin120575119899gt 0


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2







iteratively by

119899 = 119875119862(119909119899minus 120582119899nabla119891120572119899

(119909119899))

119909119899+1

= 120573119899119909119899+ (1 minus 120573

119899) 119878119875119862(119909119899minus 120582119899nabla119891120572119899

(119899))

forall119899 ge 0

(115)

where 120572119899 sub (0infin) 120582

119899 sub (0 1119860

2) and 120573

119899 sub [0 1] such

that

(i) suminfin

119899=0120572119899lt infin


120573119899le lim sup

119899rarrinfin120573119899lt 1


120582119899le lim sup

119899rarrinfin120582119899lt 1119860

2





= 1198612

= 0 and 120574119899

= 0 ThenΞ = 119862 120573


is equivalent to

119906119899= 119909119899

119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119906119899))

119910119899= 119875119862(119906119899minus 120582119899nabla119891120572119899

(119899))

119909119899+1

= 120573119899119909119899+ 120575119899119878119910119899 forall119899 ge 0

(116)










119899 and the


119899nabla119891120572119899

)

to a great extent



1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1and 119861

2


Acknowledgments


References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the mann-type extragradient iterative...

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