research article linear approximation and asymptotic...

Research ArticleLinear Approximation and Asymptotic Expansion ofSolutions for a Nonlinear Carrier Wave Equation in an AnnularMembrane with Robin-Dirichlet Conditions

Le Thi Phuong Ngoc1 Le Huu Ky Son23 TranMinh Thuyet4 and Nguyen Thanh Long3

1University of Khanh Hoa 01 Nguyen Chanh Str Nha Trang City Vietnam2Department of Fundamental sciences Ho Chi Minh City University of Food Industry 140 Le Trong Tan StrTan Phu Dist Ho Chi Minh City Vietnam3Department of Mathematics and Computer Science University of Natural ScienceVietnam National University-Ho Chi Minh City 227 Nguyen Van Cu Str Dist 5 Ho Chi Minh City Vietnam4Department of Mathematics University of Economics of Ho Chi Minh City59C Nguyen Dinh Chieu Str Dist 3 Ho Chi Minh City Vietnam

Correspondence should be addressed to NguyenThanh Long longnt2gmailcom

Received 23 April 2016 Accepted 8 September 2016

Academic Editor Yuri Vladimirovich Mikhlin

Copyright copy 2016 LeThi Phuong Ngoc et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichletconditions Existence and uniqueness of aweak solution are proved by using the linearizationmethod for nonlinear terms combinedwith the Faedo-Galerkin method and the weak compact method Furthermore an asymptotic expansion of a weak solution of highorder in a small parameter is established

1 Introduction

In this paper we consider the following nonlinear Carrierwave equation in the annular membrane

119906119905119905 minus 120583 (119906 (119905)20) (119906119909119909 + 1119909119906119909) = 119891 (119909 119905 119906 119906119909 119906119905) 120588 lt 119909 lt 1 0 lt 119905 lt 119879 (1)

associated with Robin-Dirichlet conditions

119906 (120588 119905) = 119906119909 (1 119905) + 120577119906 (1 119905) = 0 (2)

and initial conditions

119906 (119909 0) = 0 (119909) 119906119905 (119909 0) = 1 (119909) (3)

where120583119891 0 1 are given functions 120588 120577 are given constantswith 0 lt 120588 lt 1 In (1) nonlinear term 120583(119906(119905)20) depends onintegral 119906(119905)20 = int1

1205881199091199062(119909 119905)119889119909

Equation (1) herein is the bidimensional nonlinear waveequation describing nonlinear vibrations of annular mem-brane Ω1 = (119909 119910) 1205882 lt 1199092 + 1199102 lt 1 In the vibration pro-cessing the area of the annular membrane and the tensionat various points change in time The condition on boundaryΓ1 = (119909 119910) 1199092 + 1199102 = 1 that is 119906119909(1 119905) + 120577119906(1 119905) = 0describes elastic constraints where 120577 constant has a mechan-ical signification And with the boundary condition on Γ120588 =(119909 119910) 1199092+1199102 = 1205882 requiring 119906(120588 119905) = 0 the annularmem-brane is fixed

In [1] Carrier established the equation which modelsvibrations of an elastic string when changes in tension are notsmall

120588119906119905119905 minus (1 + 1198641198601198711198790 int119871

01199062 (119910 119905) 119889119910) 119906119909119909 = 0 (4)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8031638 18 pageshttpdxdoiorg10115520168031638

2 Mathematical Problems in Engineering

where 119906(119909 119905) is 119909-derivative of the deformation 1198790 is thetension in the rest position 119864 is the Young modulus 119860 is thecross section of a string 119871 is the length of a string and 120588 is thedensity of a material Clearly if properties of a material varywith 119909 and 119905 then there is a hyperbolic equation of the type[2]

119906119905119905 minus 119861(119909 119905 int1

01199062 (119910 119905) 119889119910) 119906119909119909 = 0 (5)

The Kirchhoff-Carrier equations of form (1) receivedmuch attention We refer the reader to for example Caval-canti et al [3 4] Ebihara et al [5] Miranda and Jutuca [6]Lasiecka and Ong [7] Hosoya and Yamada [8] Larkin [2]Medeiros [9] Menzala [10] Park et al [11 12] Rabello et al[13] and Santos et al [14] for many interesting results andfurther references

Thepaper consists of four sections Preliminaries are donein Section 2 with the notations definitions list of appropriatespaces and required lemmas The main results are presentedin Sections 3 and 4

First by combining the linearization method for nonlin-ear terms the Faedo-Galerkinmethod and theweak compactmethod we prove that problem (1)ndash(3) has a unique weaksolution

Next by using Taylorrsquos expansion of given functions 120583 1205831119891 and 1198911 up to high order119873 + 1 we establish an asymptoticexpansion of solution 119906 = 119906120576 of order119873+1 in small parameter120576 for

119906119905119905 minus (120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20)) (119906119909119909 + 1119909119906119909)= 119891 (119909 119905 119906 119906119909 119906119905) + 1205761198911 (119909 119905 119906 119906119909 119906119905) (6)

120588 lt 119909 lt 1 0 lt 119905 lt 119879 associated with (1) and (2) with 120583 isin119862119873+1(R+) 1205831 isin 119862119873(R+) 120583(119911) ge 120583lowast gt 0 1205831(119911) ge 0 for all119911 isin R+119891 isin 119862119873+1([120588 1]timesR+timesR3)1198911 isin 119862119873([120588 1]timesR+timesR3)Our results can be regarded as an extension and improvementof the corresponding results of [15 16]

2 Preliminaries

First put Ω = (120588 1) 119876119879 = Ω times (0 119879) 119879 gt 0 We omit thedefinitions of the usual function spaces and denote them bynotations 119871119901 = 119871119901(Ω) 119867119898 = 119867119898(Ω) Let (sdot sdot) be a scalarproduct in 1198712 Notation sdot stands for the norm in 1198712 and wedenote sdot 119883 the norm in Banach space119883 We call1198831015840 the dualspace of 119883 We denote 119871119901(0 119879119883) 1 le 119901 le infin the Banachspace of real functions 119906 (0 119879) rarr 119883 to be measurable suchthat 119906119871119901(0119879119883) lt +infin with

119906119871119901(0119879119883) = (int119879

0119906 (119905)119901119883 119889119905)1119901 if 1 le 119901 lt infin

ess sup0lt119905lt119879

119906 (119905)119883 if 119901 = infin (7)

With 119891 isin 119862119896([120588 1] times R+ times R3) 119891 = 119891(119909 119905 1199101 1199102 1199103)we put 1198631119891 = 120597119891120597119909 1198632119891 = 120597119891120597119905 119863119894+2119891 = 120597119891120597119910119894 with

119894 = 1 3 and 119863120572119891 = 11986312057211 sdot sdot sdot 1198631205725

5 119891 120572 = (1205721 1205725) isin Z5+|120572| = 1205721 + sdot sdot sdot + 1205725 = 119896119863(00)119891 = 119891

On11986711198672 we shall use the following norms

V1198671 = (V2 + 1003817100381710038171003817V11990910038171003817100381710038172)12 (8)

V1198672 = (V2 + 1003817100381710038171003817V11990910038171003817100381710038172 + 1003817100381710038171003817V11990911990910038171003817100381710038172)12 (9)

respectivelyWe remark that 1198712 1198671 1198672 are the Hilbert spaces with

respect to the corresponding scalar products

⟨119906 V⟩ = int1

120588119909119906 (119909) V (119909) 119889119909

⟨119906 V⟩ + ⟨119906119909 V119909⟩ ⟨119906 V⟩ + ⟨119906119909 V119909⟩ + ⟨119906119909119909 V119909119909⟩ (10)

The norms in 11987121198671 and1198672 induced by the correspond-ing scalar products (10) are denoted by sdot 0 sdot 1 and sdot 2

Consider the following set119881 = V isin 1198671 V (120588) = 0 (11)

It is obviously that 119881 is a closed subspace of 1198671 and on119881 two norms V1198671 and V119909 are equivalent norms On theother hand 119881 is continuously and densely embedded in 1198712Identifying 1198712 with (1198712)1015840 (the dual of1198712) we have119881 997893rarr 1198712 997893rarr1198811015840 We note more that the notation ⟨sdot sdot⟩ is also used for thepairing between 119881 and 1198811015840

We then have the following lemmas

Lemma 1 The following inequalities are fulfilled

(i) radic120588V le V0 le V for all V isin 1198712(ii) radic120588V1198671 le V1 le V1198671 for all V isin 1198671

Proof of Lemma 1 It is easy to verify the above inequalities viathe following inequalities

120588int1

120588V2 (119909) 119889119909 le int1

120588119909V2 (119909) 119889119909 le int1

120588V2 (119909) 119889119909

forallV isin 1198712120588 int1

120588V2119909 (119909) 119889119909 le int1

120588119909V2119909 (119909) 119889119909 le int1

120588V2119909 (119909) 119889119909forallV isin 1198671

(12)

Lemma 2 Embedding 119881 997893rarr 1198620(Ω) is compact and for all V isin119881 we have(i) V1198620(Ω) le radic1 minus 120588V119909(ii) V le ((1 minus 120588)radic2)V119909(iii) V0 le ((1 minus 120588)radic2120588)V1199090(iv) V11990920 + V2(1) ge V20(v) |V(1)| le radic3V1

Mathematical Problems in Engineering 3

Proof of Lemma 2 Embedding 119881 997893rarr 1198671 is continuous andembedding 1198671 997893rarr 1198620(Ω) is compact so embedding 119881 997893rarr1198620(Ω) is compact In what follows we prove (i)ndash(v)

(i) For all V isin 119881 and 119909 isin [120588 1]|V (119909)| = 100381610038161003816100381610038161003816100381610038161003816int119909

120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1

120588

1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)

(ii) For all V isin 119881and 119909 isin [120588 1]V2 (119909) = 100381610038161003816100381610038161003816100381610038161003816int119909

120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909

120588V2119909 (119910) 119889119910

le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)

Integrating over 119909 from 120588 to 1 we obtainV2 = int1

120588V2 (119909) 119889119909 le int1

120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909

= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)

(iii) For all V isin 119881V0 le V le 1 minus 120588radic2 1003817100381710038171003817V1199091003817100381710038171003817 le 1 minus 120588radic2120588 1003817100381710038171003817V11990910038171003817100381710038170 (16)

(iv) Using integration by part it leads to

V20 = int1

120588119909V2 (119909) 119889119909

= 12 [1199092V2 (119909)]1120588 minus int1

1205881199092V (119909) V119909 (119909) 119889119909

= 12V2 (1) minus int1

1205881199092V (119909) V119909 (119909) 119889119909

le 12V2 (1) + V0 1003817100381710038171003817V11990910038171003817100381710038170le 12V2 (1) + 12 (V20 + 1003817100381710038171003817V119909100381710038171003817100381720)

(17)

for any V isin 119881 so we get (iv)(v) By V20 = (12)V2(1) minus int1

1205881199092V(119909)V119909(119909)119889119909 we have

V2 (1) = 2 V20 + 2int1

1205881199092V (119909) V119909 (119909) 119889119909

le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)

implying (v)

Lemma 2 is proved

Remark 3 On 1198712 two norms V 997891rarr V and V 997891rarr V0 areequivalent So are two norms V 997891rarr V1198671 and V 997891rarr V1 on1198671and five norms V 997891rarr V1198671 V 997891rarr V1 V 997891rarr V119909 V 997891rarr V1199090and V 997891rarr radicV11990920 + V2(1) on 119881

Now we define the following bilinear form

119886 (119906 V) = 120577119906 (1) V (1) + int1

120588119909119906119909 (119909) V119909 (119909) 119889119909

forall119906 V isin 119881 (19)

where 120577 ge 0 is a constantLemma 4 Symmetric bilinear form 119886(sdot sdot) defined by (19) iscontinuous on 119881 times 119881 and coercive on 119881 that is

(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21

for all 119906 V isin 119881 where 1198620 = (12)min1 2120588(1 minus 120588)2 and1198621 = 1 + 3120577Proof of Lemma 4 (i) Byradic1 minus 120588V119909 ge V1198620(Ω) ge |V(1)| andradic120588V119909 le V1199090 for all V isin 119881 we have

|119886 (119906 V)| le 120577 |119906 (1)| |V (1)| + int1

120588

1003816100381610038161003816119909119906119909 (119909) V119909 (119909)1003816100381610038161003816 119889119909le 3120577 1199061 V1 + 100381710038171003817100381711990611990910038171003817100381710038170 1003817100381710038171003817V11990910038171003817100381710038170le (3120577 + 1) 1199061 V1

(20)

(ii) By inequality V11990920 ge (2120588(1 minus 120588)2)V20 we get119886 (V V) = 120577V2 (1) + int1

120588119909V2119909 (119909) 119889119909 = 120577V2 (1) + 1003817100381710038171003817V119909100381710038171003817100381720

ge 1003817100381710038171003817V119909100381710038171003817100381720 = 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 1003817100381710038171003817V119909100381710038171003817100381720ge 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 2120588(1 minus 120588)2 V20ge 12 min1 2120588(1 minus 120588)2V21

(21)

Lemma 4 is proved

Lemma 5 There exists Hilbert orthonormal base 119908119895 of space1198712 consisting of eigenfunctions119908119895 corresponding to eigenvalues120582119895 such that

(i) 0 lt 1205821 le 1205822 le sdot sdot sdot le 120582119895 le 120582119895+1 le sdot sdot sdot lim119895rarr+infin120582119895 =+infin(ii) 119886(119908119895 V) = 120582119895⟨119908119895 V⟩ for all V isin 119881 119895 = 1 2 Furthermore sequence 119908119895radic120582119895 is also the Hilbert

orthonormal base of 119881 with respect to scalar product 119886(sdot sdot)


On the other hand we also have119908119895 satisfying the followingboundary value problem

119860119908119895 equiv minus(119908119895119909119909 + 1119909119908119895119909) = minus 1119909 120597120597119909 (119909119908119895119909) = 120582119895119908119895in Ω119908119895 (120588) = 119908119895119909 (1) + 120577119908119895 (1) = 0 119908119895 isin 119862infin ([120588 1])

(22)

Proof The proof of Lemma 5 can be found in [17 p 87Theorem 77] with119867 = 1198712 and 119886(sdot sdot) as defined by (19)

We also note that operator119860 119881 rarr 1198811015840 in (22) is uniquelydefined by Lax-Milgramrsquos lemma that is119886 (119906 V) = ⟨119860119906 V⟩ forall119906 V isin 119881 (23)

Lemma 6 On 119881 cap 1198672 three norms V 997891rarr V1198672 V 997891rarr V2 =radicV20 + V11990920 + V11990911990920 and V 997891rarr V2lowast = radicV11990920 + 119860V20 areequivalent

Proof of Lemma 6 (i) It is easy to see that on 119881 cap 1198672 twonorms V 997891rarr V1198672 V 997891rarr V2 = radicV20 + V11990920 + V11990911990920 areequivalent becauseradic120588 V1198672 le V2 le V1198672 forallV isin 1198672 (24)

(ii) For all 119909 isin [120588 1] and V isin 119881 cap 1198672 we have

119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(25)

(a) Proof 1199062 le const1199062lowastIt follows from (25) that1199091199062119909119909 le 119909 |119860119906 (119909)|2 + 2 10038161003816100381610038161199061199091199061199091199091003816100381610038161003816 + 11199091199062119909 (26)

Hence1003817100381710038171003817119906119909119909100381710038171003817100381720 le 11986011990620 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 11986011990620 + 1120588 (2120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1205882 1003817100381710038171003817119906119909119909100381710038171003817100381720)

+ 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= 11986011990620 + 21205882 1003817100381710038171003817119906119909100381710038171003817100381720 + 12 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720

(27)

This implies1003817100381710038171003817119906119909119909100381710038171003817100381720 le 2 11986011990620 + 61205882 1003817100381710038171003817119906119909100381710038171003817100381720le 2(1 + 31205882) (11986011990620 + 1003817100381710038171003817119906119909100381710038171003817100381720)le 2(1 + 31205882) 11990622lowast

(28)

By V0 le ((1 minus 120588)radic2120588)V1199090 for all V isin 119881 we have11990622 = 11990620 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 minus 120588)22120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 + (1 minus 120588)22120588 ) 11990622lowast + 2(1 + 31205882) 11990622lowast= ((1 minus 120588)22120588 + 3 + 61205882)11990622lowast

(29)

(b) Proof 1199062lowast le const1199062It follows from (25) that

119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(30)

Hence

119909 |119860119906 (119909)|2 le 1199091199062119909119909 + 2 10038161003816100381610038161199061199091199061199091199091003816100381610038161003816 + 11199091199062119909 (31)

Thus

11986011990620 le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 (1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720) + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= (1 + 1120588) [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 11990622

(32)

This implies

11990622lowast = 1003817100381710038171003817119906119909100381710038171003817100381720 + 11986011990620 le 11990622 + (1 + 1120588) 1120588 11990622= (1 + 1120588 + 11205882) 11990622

(33)

Lemma 6 is proved

Remark 7 The weak formulation of initial-boundary valueproblem (1)ndash(3) can be given in the following manner find119906 isin 119882 = 119906 isin 119871infin(0 119879 119881 cap 1198672) 119906119905 isin 119871infin(0 119879 119881) 119906119905119905 isin119871infin(0 119879 1198712) such that 119906 satisfies the following variationalequation ⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (34)


for all V isin 119881 ae 119905 isin (0 119879) together with the initialconditions 119906 (0) = 0119906119905 (0) = 1 (35)

where 119886(sdot sdot) is the symmetric bilinear form on 119881 defined by(19)

3 The Existence and Uniqueness Theorem

Now we shall consider problem (1)ndash(3) with constant 120577 ge 0and make the following assumptions

(1198671) 0 isin 119881 cap 1198672 1 isin 119881(1198672) 120583 isin 1198621(R+) with 120583(119911) ge 120583lowast gt 0 forall119911 isin R+

(1198673) 119891 isin 1198620(Ω times R+ times R3) such that 119891(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+ timesR and119863119894119891 isin 1198620(ΩtimesR+ timesR3) 119894 =1 3 4 5Considering 119879lowast gt 0 fixed and letting 119879 isin (0 119879lowast] and119872 gt 0 we put119872 (120583) = sup

0le119911le1198722(120583 (119911) + 100381610038161003816100381610038161205831015840 (119911)10038161003816100381610038161003816)

119870119872 (119891) = 100381710038171003817100381711989110038171003817100381710038171198620(119860lowast(119872)) + 1003817100381710038171003817119863111989110038171003817100381710038171198620(119860lowast(119872))+ sum3le119894le5

100381710038171003817100381711986311989411989110038171003817100381710038171198620(119860lowast(119872)) 100381710038171003817100381711989110038171003817100381710038171198620(119860lowast(119872))= sup 1003816100381610038161003816119891 (119909 119905 1199101 1199102 1199103)1003816100381610038161003816 (119909 119905 1199101 1199102 1199103)isin 119860lowast (119872)

(36)

where

119860lowast (119872) = (119909 119905 1199101 1199102 1199103) isin Ω times [0 119879lowast] timesR3 10038161003816100381610038161199101198941003816100381610038161003816

le radic 1 minus 120588120588 119872 119894 = 1 2 3 (37)

Also for each119872 gt 0 and 119879 isin (0 119879lowast] we set119882(119872119879) = 119906 isin 119871infin (0 119879 119881 cap 1198672) 119906119905isin 119871infin (0 119879 119881) 119906119905119905 isin 1198712 (119876119879) 119906119871infin(0119879119881cap1198672)le 119872 10038171003817100381710038171199061199051003817100381710038171003817119871infin(0119879119881) le 119872 100381710038171003817100381711990611990511990510038171003817100381710038171198712(119876119879) le 119872 1198821 (119872 119879) = 119906 isin 119882 (119872119879) 119906119905119905 isin 119871infin (0 119879 1198712)

(38)

We choose first term 1199060 equiv 0 suppose that119906119898minus1 isin 1198821 (119872 119879) (39)

and associate the following variational problemwith problem(1)ndash(3) find 119906119898 isin 1198821(119872 119879) (119898 ge 1) so that⟨119898 (119905) V⟩ + 120583119898 (119905) 119886 (119906119898 (119905) V) = ⟨119865119898 (119905) V⟩ forallV isin 119881119906119898 (0) = 0119898 (0) = 1

(40)

where120583119898 (119905) = 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720) 119865119898 (119905) = 119891 (119909 119905 119906119898minus1 (119905) nabla119906119898minus1 (119905) 119898minus1 (119905)) (41)

Then we have the following result

Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put

119906(119896)119898 (119905) = 119896sum119895=1

119888(119896)119898119895 (119905) 119908119895 (42)

where coefficients 119888(119896)119898119895 satisfy the system of linear differentialequations ⟨(119896)119898 (119905) 119908119895⟩ + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119908119895)= ⟨119865119898 (119905) 119908119895⟩ 119895 = 1 119896

119906(119896)119898 (0) = 1199060119896(119896)119898 (0) = 1199061119896(43)

with

1199060119896 = 119896sum119895=1

120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum

119895=1

120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)

The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895

(45)

in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)


Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details

Step 2 (a priori estimates) We put

119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)

Then it follows from (43) (47) and (48) that119878(119896)119898 (119905) = 119878(119896)119898 (0)+ int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)

We shall estimate terms 119868119895 on the right-hand side of (49)as follows

First Term 1198681 By the following inequalities100381610038161003816100381610038161205831015840119898 (119905)10038161003816100381610038161003816 = 2 100381610038161003816100381610038161205831015840 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720) ⟨119906119898minus1 (119905) 1199061015840119898minus1 (119905)⟩10038161003816100381610038161003816le 2 100381610038161003816100381610038161205831015840 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038170 100381710038171003817100381710038171199061015840119898minus1 (119905)100381710038171003817100381710038170le 21198722119872 (120583) 119878(119896)119898 (119905) ge 120583119898 (119905) [119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905)) + 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720]ge 120583lowast [119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905)) + 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720]

(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)

Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904

le 2radic 1 minus 12058822 119870119872 (119891)int119905

0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)

Third Term 1198683 Similarly we have

100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

radic119886 (119865119898 (119904) 119865119898 (119904))radic119886 ((119896)119898 (119904) (119896)119898 (119904))119889119904le 2int119905

0

radic119886 (119865119898 (119904) 119865119898 (119904))radic119878(119896)119898 (119904)119889119904le int119905

0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)

Note that119886 (V V) le 1198621 V21 = 1198621 (V20 + 1003817100381710038171003817V119909100381710038171003817100381720)le 1198621((1 minus 120588)22120588 1003817100381710038171003817V119909100381710038171003817100381720 + 1003817100381710038171003817V119909100381710038171003817100381720)= 1198621

(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)

We also have119865119898119909 (119905) = 1198631119891 [119906119898minus1] + 1198633119891 [119906119898minus1] nabla119906119898minus1 (119905)+ 1198634119891 [119906119898minus1] Δ119906119898minus1 (119905)+ 1198635119891 [119906119898minus1] nabla119898minus1 (119905) (56)

where119863119894119891[119906119898minus1] = 119863119894119891(119909 119905 119906119898minus1(119905) nabla119906119898minus1(119905) 119898minus1(119905)) 119894 =1 5It implies from (39) (56) that

1003817100381710038171003817119865119898119909 (119905)10038171003817100381710038170 le 119870119872 (119891)[[radic1 minus 12058822 + 1003817100381710038171003817nabla119906119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817Δ119906119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119898minus1 (119905)10038171003817100381710038170]] le (radic1 minus 12058822+ 3119872)119870119872 (119891)

(57)


Combining (53) (55) and (57) we obtain

100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)

Fourth Term 1198684 Equation (43)1 can be rewritten as follows⟨(119896)119898 (119905) 119908119895⟩ + 120583119898 (119905) ⟨119860119906(119896)119898 (119905) 119908119895⟩= ⟨119865119898 (119905) 119908119895⟩ 119895 = 1 119896 (59)

Hence it follows after replacing 119908119895 with (119896)119898 (119905) that10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720= minus120583119898 (119905) ⟨119860119906(119896)119898 (119905) (119896)119898 (119905)⟩ + ⟨119865119898 (119905) (119896)119898 (119905)⟩le [120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)100381710038171003817100381710038170 + 1003817100381710038171003817119865119898 (119905)10038171003817100381710038170] 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170le [120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)100381710038171003817100381710038170 + 1003817100381710038171003817119865119898 (119905)10038171003817100381710038170]2le 21205832119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 + 2 1003817100381710038171003817119865119898 (119905)100381710038171003817100381720le 2120583119898 (119905) 119878(119896)119898 (119905) + 2 1003817100381710038171003817119865119898 (119905)100381710038171003817100381720le 2119872 (120583) 119878(119896)119898 (119905) + 21 minus 12058822 1198702

119872 (119891)

(60)

Integrate into 119905 to get1198684 = int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)

It follows from (49) (51) (52) (58) and (61) that

119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)

By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]

le 121198722(64)

for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]

such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620

)radic119879sdot exp [119879(1 + 1120583lowast1198722119872 (120583))]sdot [411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)

Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (67)

By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)

for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)

Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that

119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)

(70)

Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete

We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section


Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate

1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)

Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])

We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem

⟨11990810158401015840119898 (119905) V⟩ + 120583119898+1 (119905) 119886 (119908119898 (119905) V)+ [120583119898+1 (119905) minus 120583119898 (119905)] ⟨119860119906119898 (119905) V⟩= ⟨119865119898+1 (119905) minus 119865119898 (119905) V⟩ forallV isin 119881119908119898 (0) = 1199081015840

119898 (0) = 0(72)

Taking V = 1199081015840119898 in (72)1 after integrating into 119905 we get119885119898 (119905)

= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)

All integrals on the right-hand side of (73) will be estimatedas below

First Integral 1198691 By (50) and (74) we have

100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)

Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)

Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)

sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840

119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840

119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1

10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904

le 2119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)


Combining (73) (75) (77) and (79) we obtain

119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + 2(1 + 1120583lowast1198722119872 (120583))

sdot int119905

0119885119898 (119904) 119889119904

(80)

Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1

10038171003817100381710038171198821(119879) forall119898 isin N (81)

which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901

100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)

It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)

Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that

119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)

(84)

We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)

Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)

On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)

Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)

Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying

⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)

for all V isin 119881 and the initial conditions119906 (0) = 01199061015840 (0) = 1 (90)

Furthermore from assumptions (1198672) (1198673) we obtainfrom (84)4 (86) (88) and (89) that

11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)

and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem

⟨11990610158401015840 (119905) V⟩ + 1205831 (119905) 119886 (119906 (119905) V)+ [1205831 (119905) minus 1205832 (119905)] ⟨1198601199062 (119905) V⟩= ⟨1198651 (119905) minus 1198652 (119905) V⟩ forallV isin 119881119906 (0) = 1199061015840 (0) = 0(92)

where 119865119894(119909 119905) = 119891(119909 119905 119906119894 nabla119906119894 1199061015840119894 ) 120583119894(119905) = 120583(119906119894(119905)20) 119894 =1 2We take 119908 = 1199061015840 in (92)1 and integrate in 119905 to get119885 (119905) = int119905

012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)

with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast

119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that

119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)

Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved


4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter

In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions

(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+

(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is

a small parameter and |120576| le 1119906119905119905 + 120583120576 [119906] 119860119906 = 119865120576 [119906] (119909 119905) 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906 (120588 119905) = 119906119909 (1 119905) + 120577119906 (1 119905) = 0119906 (119909 0) = 0 (119909) 119906119905 (119909 0) = 1 (119909) (119875120576)

with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)

First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840

3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)

We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)

Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z

119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)

We need the following lemma

Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen

( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)

where coefficients 119875(119898)119896

[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873

119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)

Proof The proof of Lemma 10 is easy hence we omit thedetails

Now we assume that

(119867(119873)2 ) 120583 isin 119862119873+1(R+) 1205831 isin 119862119873(R+) with 120583(119911) ge 120583lowast gt0 1205831(119911) ge 0 forall119911 isin R+

(119867(119873)3 ) 119891 isin 119862119873+1(ΩtimesR+ timesR3) 1198911 isin 119862119873(ΩtimesR+ timesR3) such

that 119891(120588 119905 0 1199102 0) = 1198911(120588 119905 0 1199102 0) = 0 forall(119905 1199102) isinR+ timesR

Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)

Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)

where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)

119865119896 = Φ119896 [119873 119891 1199060 ] + Φ119896minus1 [119873 minus 1 1198911 1199060 ]minus (1205831 [1199060] + Φ1 [119873 120583 1199060 ]) 119860119906119896minus1


minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

(Φ119895+1 [119873 120583 1199060 ]+ Φ119895 [119873 minus 1 1205831 1199060 ]) 119860119906119894minus1 2 le 119896 le 119873

(99b)

with Φ119896[119873 119891 1199060 ] Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 are definedby the following formulas

Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816

= (1199061 119906119873) 1015840 = (11990610158401 1199061015840119873) nabla = (nabla1199061 nabla119906119873) (120574119873) = (1198961 1198962 1198963) isin Z3+ 120574119894 le 119896119894 le 119873120574119894 forall119894 = 1 2 3 120574 = (1205741 1205742 1205743) isin Z

3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)

Then we have the following theorem

Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)

3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows

1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)

where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873

In order to prove Theorem 11 we need the followinglemmas

Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873

119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)

with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873

Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873

119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+


ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain

119891 [ℎ] = 119891 (119909 119905 1199060 + ℎ1 11990610158400 + ℎ10158401 nabla1199060 + nablaℎ1)= 119891 [1199060] + sum

1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873

sdot 119863120574119891 (119909 119905 1199060 + 120579ℎ1 11990610158400 + 120579ℎ10158401 nabla1199060 + 120579nablaℎ1)sdot ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 119889120579 equiv |120576|119873+1 119877(1)119873 [119891 1199060 ℎ1 120576]

(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743

5 119891119863120574119891[1199060] = 119863120574119891(119909 119905 1199060 11990610158400 nabla1199060)By formula (97) we get

ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)

where = (1199061 119906119873)Similarly with (ℎ10158401)1205742 (nablaℎ1)1205743 we also have(ℎ10158401)1205742 = ( 119873sum

119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)

where 1015840 = (11990610158401 1199061015840119873) nabla = (nabla1199061 nabla119906119873)Hence we deduce from (106)-(107) that

ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)

where Ψ119896[120574119873 ] 1 le 119896 le 119873|120574| are defined by (100)We deduce from (104) (108) that119891 [ℎ] = 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)

where Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 are defined by (100) and

|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)

By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved

Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873

119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum

119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)


Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2


Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)

= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)

On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873

120590119896120576119896 (114)

where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that

120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)

where = (1205901 1205902119873)We deduce from (112) (115) that

120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)

By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved

Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576

Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873

119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem

V10158401015840 + 120583120576 [V + ℎ]119860V = 119865120576 [V + ℎ] minus 119865120576 [ℎ]minus (120583120576 [V + ℎ] minus 120583120576 [ℎ]) 119860ℎ+ 119864120576 (119909 119905) 120588 lt 119909 lt 1 0 lt 119905 lt 119879V (120588 119905) = V119909 (1 119905) + 120577V (1 119905) = 0V (119909 0) = V1015840 (119909 0) = 0

(118)

where

119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum

119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)

3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576

1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)

where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873


Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum

119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)

where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873

By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum

119896=2

Φ119896minus1 [119873 minus 1 1198911 1199060 ] 120576119896+ 120576 |120576|119873 119877119873minus1 [1198911 1199060 120576] (122)

Hence we deduce from (103) and (122) that119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ] = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576+ 119873sum

119896=2

(Φ119896 [119873 119891 1199060 ] + Φ119896minus1 [119873 minus 1 1198911 1199060 ])sdot 120576119896 + |120576|119873+1 119877119873 [119891 1198911 1199060 120576]

(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1

[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873

On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that

minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum

119896=2

(Φ119896 [119873 120583 1199060 ] + Φ119896minus1 [119873 minus 1 1205831 1199060 ])sdot 120576119896] minus 119860ℎ (|120576|119873+1 119873 [120583 1199060 120576] + 120576 |120576|119873sdot 119873minus1 [1205831 1199060 120576]) = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum

119896=2

(Φ119896 [119873 120583 1199060 ] + Φ119896minus1 [119873 minus 1 1205831 1199060 ])sdot 120576119896] + |120576|119873+1 119873 [120583 1205831 1199060 120576] equiv minus( 119873sum

119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1

119860119906119894minus1120576119894)1205781minus 1205781119860119906119873120576119873+1 minus (119873+1sum

119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1

sdot 119873 [120583 1205831 1199060 120576] = minus12057811198601199060120576 minus ( 119873sum119896=2

119860119906119896minus1120576119896)sdot 1205781 minus 2119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus12057811198601199060120576 minus ( 119873sum

119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus12057811198601199060120576 minus 119873sum

119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)


where|120576|119873+1 119873 [120583 1205831 1199060 120576]= minus119860ℎ (|120576|119873+1 119873 [120583 1199060 120576]+ 120576 |120576|119873 119873minus1 [1205831 1199060 120576]) |120576|119873+1 (1)

119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)

Combining (99a)-(99b) (119) (123) and (125) leads to

119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ] minus (120583 [ℎ] minus 120583 [1199060]+ 1205761205831 [ℎ]) 119860ℎ minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2

(Φ119896 [119873 119891 1199060 ]+ Φ119896minus1 [119873 minus 1 1198911 1199060 ]) 120576119896 + |120576|119873+1 119877119873 [119891 1198911 1199060 120576] minus 12057811198601199060120576 minus 119873sum

119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2

(Φ119896 [119873 119891 1199060 ]+ Φ119896minus1 [119873 minus 1 1198911 1199060 ] minus 1205781119860119906119896minus1minus sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)

By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576

1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)

where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete

Proof of Theorem 11 Consider sequence V119898 defined by

V0 equiv 0V10158401015840119898 + 120583120576 [V119898minus1 + ℎ]119860V119898= 119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ]minus (120583120576 [V119898minus1 + ℎ] minus 120583120576 [ℎ]) 119860ℎ + 119864120576 (119909 119905) 120588 lt 119909 lt 1 0 lt 119905 lt 119879V119898 (120588 119905) = V119898119909 (1 119905) + 120577V119898 (1 119905) = 0V119898 (119909 0) = V1015840119898 (119909 0) = 0 119898 ge 1

(128)

By multiplying two sides of (128)1 with V1015840119898 and afterintegrating in 119905 we have119885119898 (119905)

= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905

0(120583120576 [V119898minus1 + ℎ] minus 120583120576 [ℎ]) ⟨119860ℎ V1015840119898 (119904)⟩ 119889119904

(129)

where120583119898 (119905) = 120583120576 [V119898minus1 + ℎ] (119905)= 120583 (1003817100381710038171003817V119898minus1 (119905) + ℎ (119905)100381710038171003817100381720)+ 1205761205831 (1003817100381710038171003817V119898minus1 (119905) + ℎ (119905)100381710038171003817100381720) 119885119898 (119905) = 10038171003817100381710038171003817V1015840119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (V119898 (119905) V119898 (119905)) (130)

Note that100381610038161003816100381610038161205831015840119898 (119905)10038161003816100381610038161003816 = 2 10038161003816100381610038161003816[1205831015840 (1003817100381710038171003817V119898minus1 (119905) + ℎ (119905)100381710038171003817100381720)+ 12057612058310158401 (1003817100381710038171003817V119898minus1 (119905) + ℎ (119905)100381710038171003817100381720)] ⟨V119898minus1 (119905)+ ℎ (119905) V1015840119898minus1 (119905) + ℎ1015840 (119905)⟩10038161003816100381610038161003816 le 2 [1198722(120583)

+ 1198722(1205831)] 1003817100381710038171003817V119898minus1 (119905) + ℎ (119905)10038171003817100381710038170 10038171003817100381710038171003817V1015840119898minus1 (119905) + ℎ1015840 (119905)100381710038171003817100381710038170le 2 [1198722

(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)


where1198722 = (119873 + 2)119872 and119885119898 (119905) ge 10038171003817100381710038171003817V1015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (V119898 (119905) V119898 (119905))ge 10038171003817100381710038171003817V1015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)

Using Lemma 15 (129) gives

119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0

1003817100381710038171003817119891 [V119898minus1 + ℎ] minus 119891 [ℎ]10038171003817100381710038170 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 + 2 |120576|sdot int119905

0

10038171003817100381710038171198911 [V119898minus1 + ℎ] minus 1198911 [ℎ]10038171003817100381710038170 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904+ 2int119905

0

1003816100381610038161003816120583120576 [V119898minus1 + ℎ] minus 120583120576 [ℎ]1003816100381610038161003816 119860ℎ0 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904le 1198791198622

lowast |120576|2119873+2 + (1 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0

10038171003817100381710038171198911 [V119898minus1 + ℎ] minus 1198911 [ℎ]1003817100381710038171003817 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904+ 2 (119873 + 1)sdot 119872int119905

0

1003816100381610038161003816120583 [V119898minus1 + ℎ] minus 120583 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 + 2 |120576|sdot (119873 + 1)sdot 119872int119905

0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)

We estimate the integrals on the right-hand side of (133)as follows

Estimating 1198691We note that1003817100381710038171003817119891 [V119898minus1 + ℎ] minus 119891 [ℎ]10038171003817100381710038170 le 1198701198722(119891)(1 + 1 minus 120588radic2120588 )sdot [1003817100381710038171003817nablaV119898minus1 (119905)10038171003817100381710038170 + 10038171003817100381710038171003817V1015840119898minus1 (119905)100381710038171003817100381710038170] le 1198701198722

(119891)sdot (1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817V119898minus1

10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872

It follows from (134) that

1198691 = 2int119905

0

1003817100381710038171003817119891 [V119898minus1 + ℎ] minus 119891 [ℎ]10038171003817100381710038170 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904le 1198791198702

1198722(119891)(1 + 1 minus 120588radic2120588 )2 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)

Estimating 1198692 Similarly

1198692 = 2int119905

0


1198722(1198911) (1 + 1 minus 120588radic2120588 )2 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)

Estimating 1198693We have1003816100381610038161003816120583 [V119898minus1 + ℎ] minus 120583 [ℎ]1003816100381610038161003816le 1198722(120583) 100381610038161003816100381610038161003817100381710038171003817V119898minus1 (119905) + ℎ (119905)100381710038171003817100381720 minus ℎ (119905)2010038161003816100381610038161003816le (2119873 + 3)1198721198722

(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722

(120583) 1 minus 120588radic2120588 1003817100381710038171003817V119898minus110038171003817100381710038171198821(119879)

(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0

1003816100381610038161003816120583 [V119898minus1 + ℎ] minus 120583 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904le 2 (119873 + 1) (2119873 + 3)11987221198722(120583)

sdot 1 minus 120588radic2120588 1003817100381710038171003817V119898minus110038171003817100381710038171198821(119879) int119905

0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)

Estimating 1198694 Similarly1198694 = 2 (119873 + 1)sdot 119872int119905

0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904le (119873 + 1)2 (2119873 + 3)211987241198791198722(1205831)

sdot (1 minus 120588)2120588 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)


Combining (133) (135) (136) (138) (139) it leads to

119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2

1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1

10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)

where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)

We require the following lemma and its proof is imme-diate so we omit the details

Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)

where 0 le 120590 lt 1 120575 ge 0 are the given constants Then

120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)

Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that

1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)

where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined

by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get

V1198821(119879) le 119862119879 |120576|119873+1 (146)

This implies (102) The proof of Theorem 11 is complete

Competing Interests

The authors declare that they have no competing interests

Authorsrsquo Contributions

All authors contributed equally in this article They read andapproved the final manuscript

References

[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945

[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002

[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001

[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004

[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986

[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999

[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999

[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991

[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994

[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980

[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002

[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002

[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003

[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003

[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010


[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010

[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994

[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969

[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where 119906(119909 119905) is 119909-derivative of the deformation 1198790 is thetension in the rest position 119864 is the Young modulus 119860 is thecross section of a string 119871 is the length of a string and 120588 is thedensity of a material Clearly if properties of a material varywith 119909 and 119905 then there is a hyperbolic equation of the type[2]

119906119905119905 minus 119861(119909 119905 int1

01199062 (119910 119905) 119889119910) 119906119909119909 = 0 (5)

The Kirchhoff-Carrier equations of form (1) receivedmuch attention We refer the reader to for example Caval-canti et al [3 4] Ebihara et al [5] Miranda and Jutuca [6]Lasiecka and Ong [7] Hosoya and Yamada [8] Larkin [2]Medeiros [9] Menzala [10] Park et al [11 12] Rabello et al[13] and Santos et al [14] for many interesting results andfurther references

Thepaper consists of four sections Preliminaries are donein Section 2 with the notations definitions list of appropriatespaces and required lemmas The main results are presentedin Sections 3 and 4

First by combining the linearization method for nonlin-ear terms the Faedo-Galerkinmethod and theweak compactmethod we prove that problem (1)ndash(3) has a unique weaksolution

Next by using Taylorrsquos expansion of given functions 120583 1205831119891 and 1198911 up to high order119873 + 1 we establish an asymptoticexpansion of solution 119906 = 119906120576 of order119873+1 in small parameter120576 for

119906119905119905 minus (120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20)) (119906119909119909 + 1119909119906119909)= 119891 (119909 119905 119906 119906119909 119906119905) + 1205761198911 (119909 119905 119906 119906119909 119906119905) (6)

120588 lt 119909 lt 1 0 lt 119905 lt 119879 associated with (1) and (2) with 120583 isin119862119873+1(R+) 1205831 isin 119862119873(R+) 120583(119911) ge 120583lowast gt 0 1205831(119911) ge 0 for all119911 isin R+119891 isin 119862119873+1([120588 1]timesR+timesR3)1198911 isin 119862119873([120588 1]timesR+timesR3)Our results can be regarded as an extension and improvementof the corresponding results of [15 16]

2 Preliminaries

First put Ω = (120588 1) 119876119879 = Ω times (0 119879) 119879 gt 0 We omit thedefinitions of the usual function spaces and denote them bynotations 119871119901 = 119871119901(Ω) 119867119898 = 119867119898(Ω) Let (sdot sdot) be a scalarproduct in 1198712 Notation sdot stands for the norm in 1198712 and wedenote sdot 119883 the norm in Banach space119883 We call1198831015840 the dualspace of 119883 We denote 119871119901(0 119879119883) 1 le 119901 le infin the Banachspace of real functions 119906 (0 119879) rarr 119883 to be measurable suchthat 119906119871119901(0119879119883) lt +infin with

119906119871119901(0119879119883) = (int119879

0119906 (119905)119901119883 119889119905)1119901 if 1 le 119901 lt infin

ess sup0lt119905lt119879

119906 (119905)119883 if 119901 = infin (7)

With 119891 isin 119862119896([120588 1] times R+ times R3) 119891 = 119891(119909 119905 1199101 1199102 1199103)we put 1198631119891 = 120597119891120597119909 1198632119891 = 120597119891120597119905 119863119894+2119891 = 120597119891120597119910119894 with

119894 = 1 3 and 119863120572119891 = 11986312057211 sdot sdot sdot 1198631205725

5 119891 120572 = (1205721 1205725) isin Z5+|120572| = 1205721 + sdot sdot sdot + 1205725 = 119896119863(00)119891 = 119891

On11986711198672 we shall use the following norms

V1198671 = (V2 + 1003817100381710038171003817V11990910038171003817100381710038172)12 (8)

V1198672 = (V2 + 1003817100381710038171003817V11990910038171003817100381710038172 + 1003817100381710038171003817V11990911990910038171003817100381710038172)12 (9)

respectivelyWe remark that 1198712 1198671 1198672 are the Hilbert spaces with

respect to the corresponding scalar products

⟨119906 V⟩ = int1

120588119909119906 (119909) V (119909) 119889119909

⟨119906 V⟩ + ⟨119906119909 V119909⟩ ⟨119906 V⟩ + ⟨119906119909 V119909⟩ + ⟨119906119909119909 V119909119909⟩ (10)

The norms in 11987121198671 and1198672 induced by the correspond-ing scalar products (10) are denoted by sdot 0 sdot 1 and sdot 2

Consider the following set119881 = V isin 1198671 V (120588) = 0 (11)

It is obviously that 119881 is a closed subspace of 1198671 and on119881 two norms V1198671 and V119909 are equivalent norms On theother hand 119881 is continuously and densely embedded in 1198712Identifying 1198712 with (1198712)1015840 (the dual of1198712) we have119881 997893rarr 1198712 997893rarr1198811015840 We note more that the notation ⟨sdot sdot⟩ is also used for thepairing between 119881 and 1198811015840

We then have the following lemmas

Lemma 1 The following inequalities are fulfilled

(i) radic120588V le V0 le V for all V isin 1198712(ii) radic120588V1198671 le V1 le V1198671 for all V isin 1198671

Proof of Lemma 1 It is easy to verify the above inequalities viathe following inequalities

120588int1

120588V2 (119909) 119889119909 le int1

120588119909V2 (119909) 119889119909 le int1

120588V2 (119909) 119889119909

forallV isin 1198712120588 int1

120588V2119909 (119909) 119889119909 le int1

120588119909V2119909 (119909) 119889119909 le int1

120588V2119909 (119909) 119889119909forallV isin 1198671

(12)

Lemma 2 Embedding 119881 997893rarr 1198620(Ω) is compact and for all V isin119881 we have(i) V1198620(Ω) le radic1 minus 120588V119909(ii) V le ((1 minus 120588)radic2)V119909(iii) V0 le ((1 minus 120588)radic2120588)V1199090(iv) V11990920 + V2(1) ge V20(v) |V(1)| le radic3V1




120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1

120588

1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)


120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909

120588V2119909 (119910) 119889119910

le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)


120588V2 (119909) 119889119909 le int1

120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909

= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)



V20 = int1

120588119909V2 (119909) 119889119909

= 12 [1199092V2 (119909)]1120588 minus int1

1205881199092V (119909) V119909 (119909) 119889119909


1205881199092V (119909) V119909 (119909) 119889119909

le 12V2 (1) + V0 1003817100381710038171003817V11990910038171003817100381710038170le 12V2 (1) + 12 (V20 + 1003817100381710038171003817V119909100381710038171003817100381720)

(17)


1205881199092V(119909)V119909(119909)119889119909 we have

V2 (1) = 2 V20 + 2int1

1205881199092V (119909) V119909 (119909) 119889119909

le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)

implying (v)

Lemma 2 is proved



119886 (119906 V) = 120577119906 (1) V (1) + int1

120588119909119906119909 (119909) V119909 (119909) 119889119909

forall119906 V isin 119881 (19)


(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21


|119886 (119906 V)| le 120577 |119906 (1)| |V (1)| + int1

120588

1003816100381610038161003816119909119906119909 (119909) V119909 (119909)1003816100381610038161003816 119889119909le 3120577 1199061 V1 + 100381710038171003817100381711990611990910038171003817100381710038170 1003817100381710038171003817V11990910038171003817100381710038170le (3120577 + 1) 1199061 V1

(20)


120588119909V2119909 (119909) 119889119909 = 120577V2 (1) + 1003817100381710038171003817V119909100381710038171003817100381720


(21)

Lemma 4 is proved







(22)






119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(25)


Hence1003817100381710038171003817119906119909119909100381710038171003817100381720 le 11986011990620 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 11986011990620 + 1120588 (2120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1205882 1003817100381710038171003817119906119909119909100381710038171003817100381720)

+ 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= 11986011990620 + 21205882 1003817100381710038171003817119906119909100381710038171003817100381720 + 12 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720

(27)


(28)


(29)


119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(30)

Hence

119909 |119860119906 (119909)|2 le 1199091199062119909119909 + 2 10038161003816100381610038161199061199091199061199091199091003816100381610038161003816 + 11199091199062119909 (31)

Thus

11986011990620 le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 (1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720) + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= (1 + 1120588) [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 11990622

(32)

This implies

11990622lowast = 1003817100381710038171003817119906119909100381710038171003817100381720 + 11986011990620 le 11990622 + (1 + 1120588) 1120588 11990622= (1 + 1120588 + 11205882) 11990622

(33)

Lemma 6 is proved









0le119911le1198722(120583 (119911) + 100381610038161003816100381610038161205831015840 (119911)10038161003816100381610038161003816)



(36)

where


le radic 1 minus 120588120588 119872 119894 = 1 2 3 (37)


(38)



(40)




119906(119896)119898 (119905) = 119896sum119895=1

119888(119896)119898119895 (119905) 119908119895 (42)


119906(119896)119898 (0) = 1199060119896(119896)119898 (0) = 1199061119896(43)

with

1199060119896 = 119896sum119895=1


119895=1

120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)


(45)

in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)




119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)


01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)



(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)


0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904


0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)


100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0


0


0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)


(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)




(57)



100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1

120588

1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)


120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909

120588V2119909 (119910) 119889119910

le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)


120588V2 (119909) 119889119909 le int1

120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909

= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)



V20 = int1

120588119909V2 (119909) 119889119909

= 12 [1199092V2 (119909)]1120588 minus int1

1205881199092V (119909) V119909 (119909) 119889119909


1205881199092V (119909) V119909 (119909) 119889119909

le 12V2 (1) + V0 1003817100381710038171003817V11990910038171003817100381710038170le 12V2 (1) + 12 (V20 + 1003817100381710038171003817V119909100381710038171003817100381720)

(17)


1205881199092V(119909)V119909(119909)119889119909 we have

V2 (1) = 2 V20 + 2int1

1205881199092V (119909) V119909 (119909) 119889119909

le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)

implying (v)

Lemma 2 is proved



119886 (119906 V) = 120577119906 (1) V (1) + int1

120588119909119906119909 (119909) V119909 (119909) 119889119909

forall119906 V isin 119881 (19)


(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21


|119886 (119906 V)| le 120577 |119906 (1)| |V (1)| + int1

120588

1003816100381610038161003816119909119906119909 (119909) V119909 (119909)1003816100381610038161003816 119889119909le 3120577 1199061 V1 + 100381710038171003817100381711990611990910038171003817100381710038170 1003817100381710038171003817V11990910038171003817100381710038170le (3120577 + 1) 1199061 V1

(20)


120588119909V2119909 (119909) 119889119909 = 120577V2 (1) + 1003817100381710038171003817V119909100381710038171003817100381720


(21)

Lemma 4 is proved







(22)






119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(25)


Hence1003817100381710038171003817119906119909119909100381710038171003817100381720 le 11986011990620 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 11986011990620 + 1120588 (2120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1205882 1003817100381710038171003817119906119909119909100381710038171003817100381720)

+ 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= 11986011990620 + 21205882 1003817100381710038171003817119906119909100381710038171003817100381720 + 12 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720

(27)


(28)


(29)


119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(30)

Hence

119909 |119860119906 (119909)|2 le 1199091199062119909119909 + 2 10038161003816100381610038161199061199091199061199091199091003816100381610038161003816 + 11199091199062119909 (31)

Thus

11986011990620 le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 (1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720) + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= (1 + 1120588) [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 11990622

(32)

This implies

11990622lowast = 1003817100381710038171003817119906119909100381710038171003817100381720 + 11986011990620 le 11990622 + (1 + 1120588) 1120588 11990622= (1 + 1120588 + 11205882) 11990622

(33)

Lemma 6 is proved









0le119911le1198722(120583 (119911) + 100381610038161003816100381610038161205831015840 (119911)10038161003816100381610038161003816)



(36)

where


le radic 1 minus 120588120588 119872 119894 = 1 2 3 (37)


(38)



(40)




119906(119896)119898 (119905) = 119896sum119895=1

119888(119896)119898119895 (119905) 119908119895 (42)


119906(119896)119898 (0) = 1199060119896(119896)119898 (0) = 1199061119896(43)

with

1199060119896 = 119896sum119895=1


119895=1

120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)


(45)

in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)




119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)


01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)



(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)


0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904


0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)


100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0


0


0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)


(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)




(57)



100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(22)






119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(25)


Hence1003817100381710038171003817119906119909119909100381710038171003817100381720 le 11986011990620 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 11986011990620 + 1120588 (2120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1205882 1003817100381710038171003817119906119909119909100381710038171003817100381720)

+ 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= 11986011990620 + 21205882 1003817100381710038171003817119906119909100381710038171003817100381720 + 12 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720

(27)


(28)


(29)


119909 |119860119906 (119909)|2 = 119909 11199092 [ 120597120597119909 (119909119906119909)]2= 1199091199062119909119909 + 2119906119909119906119909119909 + 11199091199062119909

(30)

Hence

119909 |119860119906 (119909)|2 le 1199091199062119909119909 + 2 10038161003816100381610038161199061199091199061199091199091003816100381610038161003816 + 11199091199062119909 (31)

Thus

11986011990620 le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 2120588 100381710038171003817100381711990611990910038171003817100381710038170 100381710038171003817100381711990611990911990910038171003817100381710038170 + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720le 1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 (1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720) + 11205882 1003817100381710038171003817119906119909100381710038171003817100381720= (1 + 1120588) [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1120588 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 [1003817100381710038171003817119906119909119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720]le (1 + 1120588) 1120588 11990622

(32)

This implies

11990622lowast = 1003817100381710038171003817119906119909100381710038171003817100381720 + 11986011990620 le 11990622 + (1 + 1120588) 1120588 11990622= (1 + 1120588 + 11205882) 11990622

(33)

Lemma 6 is proved









0le119911le1198722(120583 (119911) + 100381610038161003816100381610038161205831015840 (119911)10038161003816100381610038161003816)



(36)

where


le radic 1 minus 120588120588 119872 119894 = 1 2 3 (37)


(38)



(40)




119906(119896)119898 (119905) = 119896sum119895=1

119888(119896)119898119895 (119905) 119908119895 (42)


119906(119896)119898 (0) = 1199060119896(119896)119898 (0) = 1199061119896(43)

with

1199060119896 = 119896sum119895=1


119895=1

120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)


(45)

in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)




119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)


01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)



(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)


0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904


0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)


100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0


0


0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)


(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)




(57)



100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















0le119911le1198722(120583 (119911) + 100381610038161003816100381610038161205831015840 (119911)10038161003816100381610038161003816)



(36)

where


le radic 1 minus 120588120588 119872 119894 = 1 2 3 (37)


(38)



(40)




119906(119896)119898 (119905) = 119896sum119895=1

119888(119896)119898119895 (119905) 119908119895 (42)


119906(119896)119898 (0) = 1199060119896(119896)119898 (0) = 1199061119896(43)

with

1199060119896 = 119896sum119895=1


119895=1

120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)


(45)

in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)




119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)


01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)



(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)


0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904


0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)


100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0


0


0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)


(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)




(57)



100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119878(119896)119898 (119905) = 119883(119896)119898 (119905) + 119884(119896)

119898 (119905) + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904 (47)

where119883(119896)119898 (119905) = 10038171003817100381710038171003817(119896)119898 (119905)1003817100381710038171003817100381720 + 120583119898 (119905) 119886 (119906(119896)119898 (119905) 119906(119896)119898 (119905))

119884(119896)119898 (119905) = 119886 ((119896)119898 (119905) (119896)119898 (119905)) + 120583119898 (119905) 10038171003817100381710038171003817119860119906(119896)119898 (119905)1003817100381710038171003817100381720 (48)


01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

+ 2int119905

0⟨119865119898 (119904) (119896)119898 (119904)⟩ 119889119904

+ 2int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 119889119904 + int119905

0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904equiv 119878(119896)119898 (0) + 4sum

119895=1

119868119895

(49)



(50)

we have

1198681 = int119905

01205831015840119898 (119904) [119886 (119906(119896)119898 (119904) 119906(119896)119898 (119904)) + 10038171003817100381710038171003817119860119906(119896)119898 (119904)1003817100381710038171003817100381720] 119889119904

le 2120583lowast1198722119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (51)


0⟨119865119898 (119904) (119896)119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0

1003817100381710038171003817119865119898 (119904)10038171003817100381710038170 10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904


0

10038171003817100381710038171003817(119896)119898 (119904)100381710038171003817100381710038170 119889119904le 1 minus 12058822 1198791198702

119872 (119891) + int119905

0119878(119896)119898 (119904) 119889119904

(52)


100381610038161003816100381611986831003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905

0119886 (119865119898 (119904) (119896)119898 (119904)) 11988911990410038161003816100381610038161003816100381610038161003816

le 2int119905

0


0


0119886 (119865119898 (119904) 119865119898 (119904)) 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

(53)


(1 + 1205882)2120588 1003817100381710038171003817V119909100381710038171003817100381720 forallV isin 119881(54)

so

119886 (119865119898 (119904) 119865119898 (119904)) le 1198621

(1 + 1205882)2120588 1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 (55)




(57)



100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











100381610038161003816100381611986831003816100381610038161003816 le 1198621

(1 + 1205882)2120588 int119905

0

1003817100381710038171003817119865119898119909 (119904)100381710038171003817100381720 119889119904 + int119905

0119878(119896)119898 (119904) 119889119904

le 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2 1198791198702119872 (119891)

+ int119905

0119878(119896)119898 (119904) 119889119904

(58)



119872 (119891)

(60)


0

10038171003817100381710038171003817(119896)119898 (119904)1003817100381710038171003817100381720 119889119904le (1 minus 1205882) 1198791198702

119872 (119891) + 2119872 (120583) int119905

0119878(119896)119898 (119904) 119889119904 (61)


119878(119896)119898 (119905) le 119878(119896)119898 (0) + 1198791198631 (119872) + 1198632 (119872)int119905

0119878(119896)119898 (119904) 119889119904 (62)

where

1198631 (119872) = [[[32 (1 minus 1205882)

+ 1198621

(1 + 1205882)2120588 (radic1 minus 12058822 + 3119872)2]]]1198702119872 (119891)

1198632 (119872) = 2 [1 + (1 + 1120583lowast1198722) 119872 (120583)] (63)


le 121198722(64)


such that

(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)

119896119879 = (1 + 1radic120583lowast1198620


119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]12 lt 1

(66)


0119878(119896)119898 (119904) 119889119904 (67)





(70)









119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119898 (0) = 0(72)


= int119905

01205831015840119898+1 (119904) 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904

minus 2int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904+ 2int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 119889119904equiv 1198691 + 1198692 + 1198693

(73)

where

119885119898 (119905) = 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583119898+1 (119905) 119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast119886 (119908119898 (119905) 119908119898 (119905))

ge 100381710038171003817100381710038171199081015840119898 (119905)1003817100381710038171003817100381720 + 120583lowast1198620

1003817100381710038171003817119908119898 (119905)100381710038171003817100381721 (74)



100381610038161003816100381611986911003816100381610038161003816 le int119905

0

100381610038161003816100381610038161205831015840119898+1 (119904)10038161003816100381610038161003816 119886 (119908119898 (119904) 119908119898 (119904)) 119889119904le 1120583lowast 21198722119872 (120583) int119905

0119885119898 (119904) 119889119904 (75)


(76)

Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 4119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)

sdot int119905

0

1003817100381710038171003817119860119906119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 41198722119872 (120583)

sdot 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879) int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 411987911987242

119872 (120583)sdot 1003817100381710038171003817119908119898minus1

100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(77)





10038171003817100381710038171198821(119879)

(78)

Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905

0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840

119898 (119904)⟩ 11988911990410038161003816100381610038161003816100381610038161003816le 2int119905

0

1003817100381710038171003817119865119898+1 (119904) minus 119865119898 (119904)10038171003817100381710038170 100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904


sdot int119905

0

100381710038171003817100381710038171199081015840119898 (119904)100381710038171003817100381710038170 119889119904 le 1198791198702

119872 (119891)(1 + 1 minus 120588radic2120588 )2

sdot 1003817100381710038171003817119908119898minus1100381710038171003817100381721198821(119879) + int119905

0119885119898 (119904) 119889119904

(79)



119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119885119898 (119905) le 119879[411987242

119872 (120583) + 1198702119872 (119891)(1 + 1 minus 120588radic2120588 )2]


sdot int119905

0119885119898 (119904) 119889119904

(80)


10038171003817100381710038171198821(119879) forall119898 isin N (81)






(84)






⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (89)



11990610158401015840 = minus120583 (119906 (119905)20)119860119906 (119905) + 119891 (119909 119905 119906 119906119909 119906119905)isin 119871infin (0 119879 1198712) (91)




012058310158401 (119904) 119886 (119906 (119904) 119906 (119904)) 119889119904

+ 2int119905

0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904

minus 2int119905

0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904

(93)



119885 (119905) le 119870lowast119872int119905

0119885 (119904) 119889119904 forall119905 isin [0 119879] (94)








with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















with 119860119906 = minus1119909 120597120597119909 (119909119906119909) = minus (119906119909119909 + 1119909119906119909) 120583120576 [119906] = 120583120576 (119906 (119905)20)= 120583 (119906 (119905)20) + 1205761205831 (119906 (119905)20) 119865120576 [119906] (119909 119905) = 119891 [119906] (119909 119905) + 1205761198911 [119906] (119909 119905) 119891 [119906] (119909 119905) = 119891 (119909 119905 119906 119906119905 119906119909) 1198911 [119906] (119909 119905) = 1198911 (119909 119905 119906 119906119905 119906119909)

(95)








( 119873sum119894=1

119909119894120576119894)119898 = 119898119873sum119896=119898

119875(119898)119896 [119873 119909] 120576119896 (97)



119875(119898)119896 [119873 119909] = sum

120572isin119860(119898)119896

(119873)

119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)

119896 (119873) = 120572 isin Z119873+ |120572| = 119898 119873sum

119894=1

119894120572119894 = 119896 (98)


Now we assume that









minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


(99b)


Φ119896 [119873 119891 1199060 ] = sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ] 1 le 119896 le 119873Ψ119896 [120574119873 ] = sum

(1198961 1198962 1198963)isin(120574119873)1198961+1198962+1198963=119896

119875(1205741)

1198961[119873 ] 119875(1205742)

1198962[119873 1015840] 119875(1205743)

1198963[119873 nabla] 1 le 119896 le 119873 10038161003816100381610038161205741003816100381610038161003816


3+ 1 le 10038161003816100381610038161205741003816100381610038161003816 le 119873

(100)

Φ119896 [119873 120583 1199060 ] = 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ] 1 le 119896 le 119873

= (1205901 1205902119873) 120590119896 =

2⟨1199060 (119905) 1199061 (119905)⟩ 119896 = 12 ⟨1199060 (119905) 119906119896 (119905)⟩ + sum

119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896

⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)


Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0

11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)




119896=0 119906119896120576119896 then we have

119891 [ℎ] = 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576] (103)







1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ 119877119873 [119891 1199060 ℎ1]

(104)

where

119877119873 [119891 1199060 ℎ1] = sum|120574|=119873+1

119873 + 1120574 int1

0(1 minus 120579)119873


(105)

120574 = (1205741 1205742 1205743) isin Z3+ |120574| = 1205741 + 1205742 + 1205743 120574 = 120574112057421205743 119863120574119891 =1198631205741

3 11986312057424 1198631205743


ℎ12057411 = ( 119873sum119896=1

119906119896120576119896)1205741 = 1198731205741sum119896=1205741

119875(1205741)

119896 [119873 ] 120576119896 (106)


119896=1

1199061015840119896120576119896)1205742 = 1198731205742sum119896=1205742

119875(1205742)

119896[119873 1015840] 120576119896

(nablaℎ1)1205743 = ( 119873sum119896=1

nabla119906119896120576119896)1205743 = 1198731205743sum119896=1205743

119875(1205743)

119896 [119873 nabla] 120576119896 (107)


ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743 = 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ 119873|120574|sum

119896=119873+1

Ψ119896 [120574119873 ] 120576119896 (108)


+ sum1le|120574|le119873

1120574119863120574119891 [1199060] ℎ12057411 (ℎ10158401)1205742 (nablaℎ1)1205743+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]

+ sum1le|120574|le119873

1120574119863120574119891 [1199060] 119873sum119896=|120574|

Ψ119896 [120574119873 ] 120576119896+ sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576]= 119891 [1199060]+ 119873sum

119896=1

( sum1le|120574|le119896

1120574119863120574119891 [1199060] Ψ119896 [120574119873 ]) 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

= 119891 [1199060] + 119873sum119896=1

Φ119896 [119873 119891 1199060 ] 120576119896+ |120576|119873+1 119877119873 [119891 1199060 120576]

(109)


|120576|119873+1 119877119873 [119891 1199060 120576]= sum

1le|120574|le119873

1120574119863120574119891 [1199060] 119873|120574|sum119896=119873+1

Ψ119896 [120574119873 ] 120576119896+ |120576|119873+1 119877(1)

119873 [119891 1199060 ℎ1 120576] (110)




119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576] (111)





= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ 119877119873 [120583 1199060 120585]

(112)

where119877119873 [120583 1199060 120585]= 1119873 int1

0(1 minus 120579)119873 120583(119873+1) (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120579120585) 120585119873+1119889120579

= |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

120583(119898) [1199060] = 119889119898120583119889119911119898 [1199060] = 119889119898120583119889119911119898 (10038171003817100381710038171199060 (119905)100381710038171003817100381720) (113)


120590119896120576119896 (114)


120585119898 = ( sum1le119896le2119873

120590119896120576119896)119898 = 2119898119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

= 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896 + 2119898119873sum

119896=119873+1

119875(119898)119896 [2119873 ] 120576119896 (115)


120583 [ℎ] = 120583 [1199060] + 119873sum119898=1

1119898120583(119898) [1199060] 120585119898+ |120576|119873+1 119877(1)

119873 [120583 1199060 120585 120576]= 120583 [1199060] + 119873sum

119898=1

1119898120583(119898) [1199060] 119873sum119896=119898

119875(119898)119896 [2119873 ] 120576119896

+ 119873sum119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]= 120583 [1199060]

+ 119873sum119896=1

( 119896sum119898=1

1119898120583(119898) [1199060] 119875(119898)119896 [2119873 ]) 120576119896

+ |120576|119873+1 119873 [120583 1199060 120576]

= 120583 [1199060] + 119873sum119896=1

Φ119896 [119873 120583 1199060 ] 120576119896+ |120576|119873+1 119873 [120583 1199060 120576]

(116)


|120576|119873+1 119873 [120583 1199060 120576]= 119873sum

119898=1

1119898120583(119898) [1199060] 2119898119873sum119896=119873+1

119875(119898)119896 [2119873 ] 120576119896

+ |120576|119873+1 119877(1)119873 [120583 1199060 120585 120576]

(117)






(118)

where


119896=1

119865119896120576119896 (119)

Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)




119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896=1

Φ119896 [119873 minus 1 1198911 1199060 ] 120576119896+ |120576|119873 119877119873minus1 [1198911 1199060 120576] (121)



119896=2



119896=2


(123)

where 119877119873[119891 1198911 1199060 120576] = 119877119873[119891 1199060 120576] + (120576|120576|)119877119873minus1




119896=2


119896=2


119894=0

119860119906119894120576119894)

sdot (1205781120576 + 119873sum119895=2

120578119895120576119895) + |120576|119873+1 119873 [120583 1205831 1199060 120576]= minus( 119873sum

119894=0

119860119906119894120576119894)1205781120576 minus ( 119873sum119894=0

119860119906119894120576119894+1)( 119873sum119895=2

120578119895120576119895minus1)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119894=1


119894=1

119860119906119894minus1120576119894)(119873minus1sum119895=1

120578119895+1120576119895)+ |120576|119873+1 119873 [120583 1205831 1199060 120576] = minus( 119873sum

119896=1

119860119906119896minus1120576119896)1205781minus 119873+1sum

119894=1

119873minus1sum119895=1

120578119895+1119860119906iminus1120576119894+119895 minus 1205781119860119906119873120576119873+1 + |120576|119873+1



119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

1205781119860119906119896minus1120576119896)minus 119873sum

119896=2

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896

minus 2119873sum119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1

sdot (1)

119873 [120583 1205831 1199060 120576] (124)



119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873 [120583 1205831 1199060 120576]= minus 2119873sum

119896=119873+1

( sum119894+119895=119896

1le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896minus 1205781119860119906119873120576119873+1 + |120576|119873+1 119873 [120583 1205831 1199060 120576]

(125)



119896=1

119865119896120576119896 = (1198911 [1199060]+ Φ1 [119873 119891 1199060 ]) 120576 + 119873sum

119896=2


119896=2

(1205781119860119906119896minus1+ sum

119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1)120576119896 + |120576|119873+1 (1)

119873 [1205831205831 1199060 120576] minus 119873sum

119896=1

119865119896120576119896 = (1198911 [1199060] + Φ1 [119873 119891 1199060 ]minus 12057811198601199060 minus 1198651) 120576 + 119873sum

119896=2


119894+119895=1198961le119894le119873+11le119895le119873minus1

120578119895+1119860119906119894minus1 minus 119865119896)120576119896 + |120576|119873+1

sdot (119877119873 [119891 1198911 1199060 120576] + (1)

119873 [120583 1205831 1199060 120576])= |120576|119873+1 (119877119873 [119891 1198911 1199060 120576]+ (1)

119873 [120583 1205831 1199060 120576])

(126)


1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)




(128)


= int119905

01205831015840119898 (119904) 119886 (V119898 (119904) V119898 (119904)) 119889119904

+ 2int119905

0⟨119864120576 (119904) V1015840119898 (119904)⟩ 119889119904

+ 2int119905

0⟨119865120576 [V119898minus1 + ℎ] minus 119865120576 [ℎ] V1015840119898 (119904)⟩ 119889119904

minus 2int119905


(129)




(120583) + 1198722(1205831)]1198722

2 equiv 1205902 (119872) (131)



1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817V119898 (119905)100381710038171003817100381721 (132)


119885119898 (119905) le 1198791198622lowast |120576|2119873+2 + int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904 + 1120583lowast 120590119898 (119872)sdot int119905

0119885119898 (119904) 119889119904

+ 2int119905

0


0


0



0119885119898 (119904) 119889119904

+ 2int119905

0


0


0


0

10038161003816100381610038161205831 [V119898minus1 + ℎ] minus 1205831 [ℎ]1003816100381610038161003816 10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904= 1198791198622


0119885119898 (119904) 119889119904

+ 4sum119894=1

119869119894

(133)




10038171003817100381710038171198821(119879) (134)

with1198722 = (119873 + 2)119872


1198691 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(135)


1198692 = 2int119905

0



100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(136)


(120583) 1003817100381710038171003817V119898minus1 (119905)10038171003817100381710038170le (2119873 + 3)1198721198722


(137)

Hence1198693 = 2 (119873 + 1)sdot 119872int119905

0



0

10038171003817100381710038171003817V1015840119898 (119904)100381710038171003817100381710038170 119889119904 le (119873 + 1)2sdot (2119873 + 3)211987241198792

1198722(120583) (1 minus 120588)2120588 1003817100381710038171003817V119898minus1

100381710038171003817100381721198821(119879)+ int119905

0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904

(138)


0



0

10038171003817100381710038171003817V1015840119898 (119904)1003817100381710038171003817100381720 119889119904(139)



119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119885119898 (119905) le 119879120588119872 1003817100381710038171003817V119898minus1100381710038171003817100381721198821(119879) + 1198791198622

lowast |120576|2119873+2

+ (5 + 1120583lowast 1205902 (119872))int119905

0119885119898 (119904) 119889119904 (140)

where 120588119872 = [11987021198722(119891)+1198702

1198722(1198911)](1+(1minus120588)radic2120588)2+[2

1198722(120583)+2


10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)







1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)



V1198821(119879) le 119862119879 |120576|119873+1 (146)


Competing Interests




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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