yii - hindawi publishing corporationdownloads.hindawi.com/journals/ijmms/1997/230875.pdfinternat. j....

Internat. J. Math. & Math. Sci.VOL. 20 NO. 3 (1997) 517-520

517

CONVERGENCE OF ISHIKAWA ITERATES OFGENERALIZED NONEXPANSIVE MAPPINGS

M. K. GHOSHDeprtment ofMathematics

Narajole Raj College, Midnapore, 721211West Bengal, INDIA

and

L. DEBNATHDepment ofMathematicsUniversity ofCentral Florida

Orlando, Florida, 32816, U.S.A.

(Received Hay 15, 1995 and in revised form September 30, 1996)

ABSTRACT. This paper is concerned with the convergence of Ishikawa iterates of generalized

nonexpansive mappings in both uniformly convex and strictly convex Banach spaces. Several fixed pointtheorems are discussed.

KEY WORDS AND PHRASES. Fixed point theorems, quasi-nonexpansive mappings, uniformlyconvex Banach spaces, and strictly convex Banach spaces.1991 AMS SUBJECT CLASSIFICATION CODES. 47H10, 54H25.

1. INTRODUCTIONRecently, Hardy and Rogers (1973) discussed the convergence of Ishikawa (1974) iterates of

generalized nonexpansive mappings in a metric space These mappings have also been discussed in a

Banach space by Goebel et al. (1973), Wong (1976) and Shimi (1978).The purpose of this paper is to discuss the convergence of the Ishikawa iterates of generalized

nonexpansive mappings in a uniformly convex space and in a strictly convex anach space It is shown

that results proved by Ghosh (1990) for nonexpansive mappings are also true for generalized

nonexpansive mappings. Furthermore, it is shown that a result of Wong (1976) for the case of Manniterates (1953) is also true for the case ofIshikawa iterates.

2. GENERALIZED NON-EXPANSIVE MAPPINGS AND ISHIKAWA ITERATIVE PROCESSSuppose/3 is a Banach space and D is a convex subset of/3. A mapping T" D --, D is said to be

generalized nonexpansive if

IITz Tyll <_ ailx YII + b{llx Tall + II TylI} + c{llx Tyll + Ily Txll} (2.1)

for all x, y 6 D, where a, b, c k 0 and

(a + 25 + 2c) < 1 (2.2)

In this connection we recall a fixed point theorem due to Goebel et al. (1973), an analogue ofwhich

has been established by Browder (1965) and Kirk (1965).

518 M K. GHOSH AND L. DEBNATH

THEOREM 2.1 (Goebel et al. (1973)) If D is a closed bounded convex subset of a uniformlyconvex Banach space/3, and T D D is a cominuous mapping that satisfies (2.1) and (2.2), then Thas a fixed point.

REMARK 1. If b > 0, then the fixed point is unique. For, if possible, suppose p and q are twofixed points of T. Then it is easy to derive from (2. l) that

II- qll -< ( + 2c)ll,- qll < (= + 2b + 2c)11,- qll (2.3)

which implies that p q.

REMARK 2. A generalized nonexpansiv mapping T is also quasi-nonexpansive if it has a fixedpoint. Ifp is a fixed poim ofT, then

IIT il <_ (a /-b-cb/ c_)IIz ’11 <- IIz ’11 (2.)

because of the fact that (1 b c) > (a + b + c). Thus T is quasi-nonexpansive. This was introducedby Bose and Mukherjee (.1981).

REMARK 3. IfD is a dosed bounded convex subset of a uniformly convex Banach space B andT D D is a continuous mapping satisfying (2.1) with (2.2), then T is asymptotically regular. Theproofofthis result may be established exactly in the same way as it has been done by Ghosh (1990).

Suppose D is a convex subset of a Banach space B and T is a self-mapping of D. For an z0 D,we define a sequence {z,,},,=l such that

x, T.,x0, T,, (1 ,X)I + AT[(1 #)I + gTl. (2 5)

The interativ scheme (2.5) was introduced by Ishikawa (1974). Or, in earlier notation,

T,. (1 A)I + ATTu, (2.6)

where A (0,1) and/ q [0,1), so that T.g T when/ 0.

3. CONVERGENCE OF I$ltIKAWA ITEITE$

First, we observe that F(T) F(T,,) where F(T) denotes the fixed point set of the generalizednonexpansiv mapping T. It is obvious that

F(T) C F(T,x.t,) (3.1)

If p . F(T,.,), that is, T,,t,p p, which implies that TT,p p, since T is generalized nonexpansiveWe then have

[[Tp p[[ [[Tp TT,p[[ < a[[p T,p[[ + b[[[p Tp[[ + lip- Tup[[ + c[[Tp- Tup[[

The above remit gives

(1 b c)llTp- Pll < (a + b + c)l]T,p Pll #(a + b + c)l]Tp Pl]

This implies that

[[Tp p[] _<\ -_- - [[Tp p[[ (3.2)

whence, we get Tp p, since 1-b-c / < 1. Then

F(Tx,u C F(T) (3.3)

Thus (3.1) and (3.3) yield F(T) F(T,x,,).We now state the following variants of theorems due to Ghosh (1990) and they can be proved by

following Ghosh’s paper.

CONVERGENCE OF ISHIKAWA ITERATES OF GENERALIZED NON’EXPANSIVE MAPPINGS 519

THEOREM 3.1. If D is a bounded closed convex subset of a uniformly convex Banach space Band T D D be a cominuous mapping satisfying (2.1) with (2.2), then Tx,u is asymptotically regular.

It may be noted that the case, when/ 0 in the above theorem, has been proved by Wong (1976).THEOREM 3.2. Suppose D is a bounded closed convex subset of a uniformly convex Banach

space B, and T D --, D is a continuous mapping which satisfies (2.1) with (2.2). IfT satisfies any one

ofthe following conditions:

(a) (I TTu) maps closed sets in D imo closed sets in B,(b) TT, is demicompact at 0,then, for an z0 e D, the sequence {zn} with zn T.,z0 converges to a fixed poim ofT in D.

THEOREM 3.3. If D is a bounded closed convex subset of a uniformly convex Banach space 13and T D D be a continuous mapping satisfying (2.1) with (2.2). Ifthere exists a number k > 0 such

that, for each z D,

Ii(I TT.)xll >_ k d(x, F) (3.4)

then, for an xo D, the sequence {z.) with x. T,.z0 converges to a fixed point ofT in D.

DEFINITION 3.1. A Banach space is said to be sctly convex provided that if

IIz + /[I [Izll + II/ll and z # 0, # 0, then /= tz, > 0

e now prove a result which is close in spirit with the one proved by Wong [4] in connection with theconvergence ofMann iterates.

THEOREM 3.4. Suppose D is a compact convex subset of a strictly convex Banach space B andT D D is a continuous mapping satisfying (2.1) with (2.2). Then, for an z0 6 D, the sequence {z.}with z. T,z0 converges to a fixed point ofT in D.

PROOF. First, we observe that F(T) is nonempty, because the existence of a fixed point is ensuredby the Schauder-Tychonoff theorem. Then, obviously, F(T) F(T,u # . Since D is compact, thesequence {z.} has a subsequence {z., } which converges to a point z, say, in D. We show that z is a

fixed point of T. The quasi-nonexpansiveness ofT follows from 10]. If T is quasi-nonexpansive, then

Tx,. is also quasi-nonexpansive, as seen from (3.6) below. From the quasi-nonexpansiveness of T,. we

observe that the sequence { IIz. Pll } is non-increasing, where p is a fixed point ofT. In fact,

-< -IIz= pll. (3.5)

Now, if F(T) is a singleton, then p z and the theorem is proved. We now. assume otherwise. Thenfrom the continuity ofthd norm, I[-It and Tx,u we obtain

lira IlTx,.z=, P]I IIT,.z pll. (3 6)

Therefore, because ofthe quasi-nonexpansiveness ofT, and (3 6), we have

[]T..z p]] }Ix Pl[ (3.7)

520 M.K. GHOSH AND L. DEBNATH

Indeed,

IIT,.,z Pl] I](1 A)x + ATTvzI1(1 A)(x p) + A(TT,x

< (x )llz t, / ,XlITT.z_< (1 A)liz ;’11 / AIIT.z

(I A)]]z p]] + A]](I p)z + #rz

If we take imo account (3.?) and (3.8), it follows that inequalities involved in (3.8) reduce to

equalities. Thus we have the following results:

II(I A)( v) + A(TT v)ll (I A)II= II + AIIT, il 0.9)

fITS,= vll II= vll. (3 0)

Since the space B is strictly convex, then for some > 0, we have

TT=- t(z v). (.

From (3. 0) we observe that t I, which implies that TTvz z. Thus z is a fixed point of TA,v andhence of T. But we have already observed that {]]z, z[[} is nonincreasing and hence {z} convergesto z. This completes the proof.

REMARK 4. The present analysis can be extended to a more general mapping T which satisfies

1 1T= Tvll < ma{ll -vll, [II= =II + fly- vlll, [II vll + fly- T=II]), (3.)

for all a:,b, ( D. This mapping includes nonexpansive and generalized nonexpansive mappings (seeRhoades (I 977)). It is easy to verify that T is quasi-nonexpansive.

ACKNOWLEDGEMENT. The first author is indebtexl to Prof. M. Maiti for his suggestion in thepreparation ofthis paper.

REFERENCESBOSE, R. K. and MUKHERJEE, R. N., Approximating fixed points of some mappings, Proc. Amer.

Math. Soc. $2 (I98 I), 603-606.

BROWDER, F. E., Nonexpansiv nonlinear operations in Banach space, Proc. Nat. ActwL ScL U.S.A. $4(196s), i0a-i0.

GHOSH, M. K., On the convergence of Ishikawa iterates to fixed points, Ph.D. thesis, JadavpurUniversity, India, 1990.

GOEBEL, K., KIRK, W. A. and SHIM1, T. N., A fixed point theorem in uniformly convex Banachspaces, Bull. U.M.I. 7 (1973), 67-75.

HARDY, G. F. and ROGERS, T. D., A generalization of a fixed point theorem of Reich, Cand. Math.Bull. 1 (I 973), 20.1-206.

ISHIKAWA, S., Fixed points by a new iteration mhod, Proc. Amer. Math. Soc. 44 (1974), 147-150KIRK, W A., A fixed point theorem for mappings which do not decrease distances, Amer. Math.

Monthly 72 (196S), 004-006.

MANN, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510RHOADES, B. E., A comparison of various definitions of comractiv mappings, Trans. Amer. Soc. 22(i

(977), 2s7-290.

SHIM1, T N., Approximation offixed poims of certain nonlinear mappings, J. Math. Anal. andAppl. (iS(97s),

WONG, C. S., Approximation of fixed points ofgeneralized nonexpansive mappings, Proc. Amer. Math.Soc. 4 (I 976), 93-97.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

yii - hindawi publishing corporationdownloads.hindawi.com/journals/ijmms/1997/230875.pdfinternat. j....

Documents