Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings

Volume 2010, Article ID 476913,8pages doi:10.1155/2010/476913

Research Article

Viscosity Approximation to Common Fixed

Points of Families of Nonexpansive Mappings with Weakly Contractive Mappings

A. Razani

and S. Homaeipour

1Department of Mathematics, Faculty of Science, Imam Khomeini International University, P.O. Box 34149-16818, Qazvin, Iran

2School of Mathematics, Institute for Research in Fundamental Sciences, P.O. Box 19395-5746, Tehran, Iran

Correspondence should be addressed to S. Homaeipour,s [email protected] Received 5 June 2010; Accepted 26 July 2010

Academic Editor: Brailey Sims

Copyrightq2010 A. Razani and S. Homaeipour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let X be a reflexive Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation sequencexnλnfxn1−λnTnxn, whereλn∈0, 1,{Tn}is a uniformly asymptotically regular sequence, and f is a weakly contractive mapping. Strong convergence of the sequence{xn}is proved.

1. Introduction

LetCbe a nonempty closed convex subset of a Banach spaceX. Recall that a self-mapping T :C → Cis nonexpansive if

Tx−T

y≤x−y ∀x, y∈C. 1.1 Alber and Guerre-Delabriere1defined the weakly contractive maps in Hilbert spaces, and Rhoades2showed that the result of1is also valid in the complete metric spaces as follows.

Definition 1.1. LetX, dbe a complete metric space. A mappingT :X → Xis called weakly contractive if

d

Tx, Ty

≤d x, y

−ψ d

x, y

, 1.2

wherex, y ∈ X andψ : 0,∞ → 0,∞is a continuous and nondecreasing function such thatψt 0 if and only ift0 and lim_{t→ ∞}ψt ∞.

Theorem 1.2. LetT :X → Xbe a weakly contractive mapping, whereX, dis a complete metric space, thenThas a unique fixed point.

In 2007, Song and Chen3considered the iterative sequence

xnλnfxn 1−λnTnxn, n∈ {1,2, . . .}. 1.3

They proved the strong convergence of the iterative sequence{xn}, wheref is a contraction mapping and{Tn}is a uniformly asymptotically regular sequence of nonexpansive mappings in a reflexive Banach spaceX, as follows.

Theorem 1.3see 3, Theorem 3.1. LetX be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJfromXtoX^∗. Suppose thatCis a nonempty closed convex subset ofXand{Tn}, n∈ {1,2, . . .},is a uniformly asymptotically regular sequence of nonexpansive mappings fromCinto itself such that

F:^∞

n1

FixTn/∅, 1.4

where FixTn :{x ∈C :x Tnx}, n ∈ {1,2, . . .}. Let{xn}be defined by1.3andλn ∈0,1, such that lim_{n→ ∞}λn0. Then asn → ∞, the sequence{xn}converges strongly top, such thatpis the unique solution, inF, to the variational inequality:

f p

−p, J y−p

≤0, ∀y∈F. 1.5

In this paper, inspired by the above results, strong convergence of sequence1.3is proved, wherefis a weakly contractive mapping.

2. Preliminaries

A Banach spaceXis called strictly convex if

x y1, x /y implies xy

2 <1. 2.1

A Banach spaceXis called uniformly convex, if for allε∈0,2,there existδε>0 such that

x y1 withx−y≥εimplies that xy

2 <1−δε. 2.2 The following results are well known which can be founded in4.

1A uniformly convex Banach spaceXis reflexive and strictly convex.

2IfCis a nonempty convex subset of a strictly convex Banach spaceXandT :C → C is a nonexpansive mapping, then the fixed point setFTofT is a closed convex subset ofC.

By a gauge function we mean a continuous strictly increasing functionϕdefined on0,∞ such thatϕ0 0 and lim_{r→ ∞}ϕr ∞. The mappingJϕ:X → 2^X∗defined by

Jϕx

x^∗∈X^∗:x, x^∗ x x^∗ , x^∗ ϕ x , for eachx∈X, 2.3

is called the duality mapping with gauge functionϕ. In the case whereϕt t,thenJϕ J which is the normalized duality mapping.

Proposition 2.1see5. (1)JIif and only ifXis a Hilbert space.

(2)Jis surjective if and only ifXis reflexive.

(3)Jϕλx signλϕ|λ| · x / x Jxfor allx∈X\ {0}, λ∈R; in particularJ−x

−Jx, for allx∈X.

We say that a Banach spaceXhas a weakly sequentially continuous duality mapping if there exists a gauge function ϕ such that the duality mapping Jϕ is single-valued and continuous from the weak topology to the weak^∗topology ofX.

We recall 6 that a Banach space X is said to satisfy Opial’s condition, if for any sequence{xn}inX, which converges weakly tox∈X, we have

lim sup

n→ ∞ xn−x <lim sup

n→ ∞

xn−y ∀y∈X, y /x. 2.4

It is known7that any separable Banach space can be equivalently renormed such that it satisfies Opial’s condition. A space with a weakly sequentially continuous duality mapping is easily seen to satisfy Opial’s condition8.

Lemma 2.2 see9, Lemma 4. LetX be a Banach space satisfying Opial’s condition andC a nonempty, closed, and convex subset ofX. Suppose thatT :C → Cis a nonexpansive mapping. Then I−T is demiclosed at zero, that is, if{xn}is a sequence inCwhich converges weakly toxand if the sequencexn−Txnconverges strongly to zero, thenx−Tx0.

Definition 2.3see3. LetCbe a nonempty closed convex subset of a Banach spaceXand Tn : C → C, where n ∈ {1,2, . . .}. Then the mapping sequence{Tn}is called uniformly asymptotically regular onC, if for allm∈ {1,2, . . .}and any bounded subsetKofCwe have

n→lim∞sup

x∈K TmTnx−Tnx 0. 2.5

3. Main Result

In this section, we prove a new version ofTheorem 1.3.

Theorem 3.1. Let X be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJfromXtoX^∗. Suppose thatCis a nonempty closed convex subset ofXandTm : C → C, m ∈ {1,2, . . .},is a uniformly asymptotically regular sequence of nonexpansive mappings such that

F: ^∞

m1

FixTm/∅. 3.1

Letf:C → Cbe a weakly contractive mapping. Suppose that{tm}is a sequence of positive numbers in0,1satisfying lim_{m→ ∞}tm0. Assume that{xm}is defined by the following iterative process:

xmtmfxm 1−tmTmxm, m∈ {1,2, . . .}. 3.2

Then the above sequence{xm}converges strongly to a common fixed pointpof{Tm}, m∈ {1,2, . . .}

such thatpis the unique solution, inF, to the variational inequality f

p

−p, J y−p

≤0, ∀y∈F. 3.3

Proof.

Step 1. We prove the uniqueness of the solution to the variational inequality3.3. Suppose thatp, q∈Fare distinct solutions to3.3. Then

f p

−p, J q−p

≤0, f

q

−q, J p−q

≤0.

3.4

By adding up the above relations, we get

0≥ p−f

p

− q−f

q , J

p−q

≥ p−q, J p−q

− f

p

−f q

, J p−q

≥p−q²−f p

−f qJ

p−q

≥p−q²−p−q²ψp−qp−q.

3.5

Thusψ p−q p−q ≤0, hencepq. We denote bypthe unique solution, inF, to3.3.

Step 2. We show that the sequence{xm}is bounded. Letq∈F; from3.2we get then that xm−q²

tm

fxm−q

1−tm

Tmxm−q , J

xm−q tm

fxm−f q

f

q

−q , J

xm−q 1−tm

Tmxm−Tmq, J

xm−q

≤tmfxm−f qJ

xm−qtm

f q

−q, J

xm−q 1−tmTmxm−TmqJ

xm−q

≤tmxm−q−ψxm−qxm−q f

q

−q, J

xm−q 1−tmTmxm−TmqJ

xm−q

≤tm

xm−q²−ψxm−qxm−q f

q

−q, J

xm−q 1−tmxm−q²

≤xm−q²−tmxm−qψxm−qtmf q

−qxm−q.

3.6

Thus

xm−qψxm−q≤f q

−qxm−q, 3.7

or

ψxm−q≤f q

−q. 3.8

Therefore{xm}is bounded.

Step 3. We prove that lim_m→∞ xm−Tnxm 0, for alln∈ {1,2, . . .}. Since the sequence{xm} is bounded, so{fxm}and{Tmxm}are bounded. Hence lim_{m→ ∞}tm Tmxm−fxm 0, thus limm→ ∞ xm−Tmxm 0. Let K be a bounded subset ofCwhich contains{xm}. Since the sequence{Tm}is uniformly asymptotically regular, we can obtain

mlim→ ∞ TnTmxm−Tmxm ≤ lim

m→ ∞sup

x∈K TnTmx−Tmx 0. 3.9

Letm → ∞, then

xm−Tnxm ≤ xm−Tmxm Tmxm−TnTmxm TnTmxm−Tnxm

≤2 xm−Tmxm Tmxm−TnTmxm −→0. 3.10

Hence lim_{m→ ∞} xm−Tnxm 0, for alln∈ {1,2, . . .}.

Step 4. We show that the sequence{xm}is sequentially compact. SinceXis reflexive and{xm} is bounded, there exists a subsequence{xmk}of{xm}such that{xmk}is weakly convergent toq∈Cask → ∞. Since limk→ ∞ xmk−Tnxmk 0 for alln∈ {1,2, . . .}, byLemma 2.2, we haveqTnqfor alln∈ {1,2, . . .}. Thusq∈F.

Step 2implies that

xmk −q²≤tmkxmk−q−ψxmk−qxmk−q f

q

−q, J

xmk−q 1−tmkxmk−q².

3.11

Hence

tmkxmk−qψxmk−q≤tmk

f q

−q, J

xmk−q

. 3.12

SinceJis single valued and weakly sequentially continuous fromXtoX^∗, we have lim sup

k→ ∞

xmk−qψxmk−q≤ lim

k→ ∞

f q

−q, J

xmk −q

0. 3.13

Thus lim_{k→ ∞}xmk q. Hence the sequence{xm}is sequentially compact.

Step 5. We now prove thatq∈Fis a solution to the variational inequality3.3. Suppose that y∈F, then

xm−y²tm

fxm−xm

xm−y , J

xm−y 1−tm

Tmxm−Tmy, J

xm−y

≤tm

fxm−xm

, J

xm−y

xm−y².

3.14

Hence

fxm−xm

, J y−xm

≤0 for each m∈ {1,2, . . .}. 3.15

Since{xmk} → qask → ∞, we have xmk−fxmk

− q−f

q−→0 ask−→ ∞, xmk−fxmk

, J

xmk−y

− q−f

q , J

q−y xmk−fxmk

− q−f

q , J

xmk−y

q−f q

, J

xmk−y

−J

q−y

≤xmk−fxmk

− q−f

qxmk−y q−f

q , J

xmk−y

−J

q−y−→0,

3.16

ask → ∞. Hence f

q

−q, J y−q

lim

k→ ∞

fxmk−xmk, J y−xmk

≤0. 3.17

Thusq ∈ Fis a solution to the variational inequality3.3. By uniqueness,q p. Since the sequence{xm}is sequentially compact and each cluster point of it is equal top, then{xm} → pasm → ∞. The proof is completed.

It is known that10, Example 2in a uniformly convex Banach spaceE, the Ces`aro meansTn 1/n_n−1

j0T^jfor nonexpansive mappingT is uniformly asymptotically regular.

So we have the following corollary, which is a new version of10, Theorem 3.2.

Corollary 3.2. LetXbe a real uniformly convex Banach space which admits a weakly sequentially continuous duality mappingJ fromX toX^∗andCa nonempty closed convex subset ofX. Suppose thatT : C → C is a nonexpansive mapping,FT/∅ and f : C → C is a weakly contractive mapping. Let{zm}be defined by

zmtmfzm 1−tm 1

m1Σ^m_j0T^jzm, m≥0, 3.18 wheretm ∈0,1and lim_{m→ ∞}tm0. Then asm → ∞,{zm}converges strongly to a fixed pointp ofT, wherepis the unique solution inFTto the following variational inequality:

f p

−p, j u−p

≤0 ∀u∈FT. 3.19

Acknowledgment

A. Razani would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran for supporting this paperGrant no.89470126.

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