Strong Convergence of an Implicit Iteration Algorithm for a Finite Family of Pseudocontractive Mappings

Volume 2008, Article ID 280908,10pages doi:10.1155/2008/280908

Research Article

Strong Convergence of an Implicit Iteration

Algorithm for a Finite Family of Pseudocontractive

Mappings

Yonghong Yao1 _{and Yeong-Cheng Liou}2

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China} 2_{Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan}

Correspondence should be addressed to Yonghong Yao,[email protected]

Received 2 December 2007; Accepted 2 January 2008

Recommended by Ram Verma

Strong convergence theorems for approximation of common fixed points of a finite family of pseudocontractive mappings are proven in Banach spaces using an implicit iteration scheme. The results presented in this paper improve and extend the corresponding results of Osilike, Xu and Ori, Chidume and Shahzad, and others.

Copyrightq2008 Y. Yao and Y.-C. Liou. 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.

1. Introduction

LetEbe a real Banach space and letJdenote the normalized duality mapping fromEinto 2E∗ given by

Jx x∗∈E∗:x, x∗x2x∗2, 1.1

whereE∗denotes the dual space ofEand·,·denotes the generalized duality pairing. IfE∗is strictly convex, thenJ is single valued. In the sequel, we will denote the single-value duality mapping byj.

LetCbe a nonempty closed convex subset ofE. Recall that a self-mappingf :C→Cis said to be a contraction if there exists a constantδ∈0,1such that

_{fx−fy≤δx−y, ∀x, y∈C. 1.2}

A mapping T with domain DTand RTin E is called pseudocontractive if, for all x, y ∈DT, there existsjx−y∈Jx−ysuch that

Tx−Ty, jx−y≤ x−y2. 1.3 We use FixTto denote the fixed point set ofT, that is, FixT {x∈C:Txx}.

Recently, Xu and Ori1have introduced an implicit iteration process below for a finite family of nonexpansive mappings. Let T1, T2, . . . , TN be N self-mappings of E and suppose

thatN_i1FixTi_/∅, the set of common fixed points ofTi, i 1,2, . . . , N.An implicit iteration process for a finite family of nonexpansive mappings is defined as follows with {tn} a real sequence in0,1,x0∈E:

x1 t1x0

1−t1 T1x1,

x2 t2x1

1−t2 T2x2,

.. .

xN tNxN−1

1−tN TNxN,

xN1 tN1xN

1−tN1 T1xN1,

.. .

1.4

which can be written in the following compact form:

xntnxn−1

1−tn Tnxn, n≥1, 1.5 whereTn TnmodN.

Xu and Ori proved the weak convergence of the above iterative process 1.5 to a common fixed point of a finite family of nonexpansive mappings{Tn}N_n1in a Hilbert space.

They further remarked that it is yet unclear what assumptions on the mapping and/or the parameters{t_n}are sufficient to guarantee the strong convergence of the sequence{x_n}.

Very recently, Osilike 2first extended Xu and Ori 1from the class of nonexpansive mappings to the more general class of strictly pseudocontractive mappings in a Hilbert space. He proved the following two convergence theorems.

Theorem O1. LetH be a real Hilbert space and let Cbe a nonempty closed convex subset ofH. Let

{Ti}N

i1beN strictly pseudocontractive self-mappings ofCsuch that _N

i1FixTi/∅. Letx0 ∈Cand let{αn}∞_n1be a sequence in0,1such thatlimn→∞αn0. Then the sequence{xn}∞n1defined by

xnαnxn−1

1−αn Tnxn, n≥1, 1.6

whereTnTnmodN, converges weakly to a common fixed point of the mappings{Ti}Ni1.

Theorem O2. LetE be a real Banach space and let Cbe a nonempty closed convex subset ofE. Let

{Ti}N

i1beN strictly pseudocontractive self-mappings ofCsuch that _N

i1FixTi/∅, and let{αn}∞n1 be a real sequence satisfying the conditions0 < αn <1,∞_n11−αn ∞and∞_n11−αn2 < ∞. Letx0∈Cand let{xn}∞_n1be defined by

xnαnxn−1

1−αn Tnxn, n≥1, 1.7

Remark 1.1. We note that Theorem O1 has only weak convergence even in a Hilbert space and Theorem O2 has strong convergence, but imposed condition lim infn→∞dxn, F 0.

In 2005, Chidume and Shahzad 3also proved the strong convergence of the implicit iteration process1.5to a common fixed point for a finite family of nonexpansive mappings. They gave the following theorem.

Theorem CS. LetEbe a uniformly convex Banach space, letCbe a nonempty closed convex subset

of E. Let {Ti}N_i1 be N nonexpansive self-mappings of C with N_i1FixTi_/∅. Suppose that one of the mappings in {Ti}N_i1 is semicompact. Let {tn} ⊂ δ,1−δfor some δ ∈ 0,1. From arbitrary x0∈C, define the sequence{xn}by1.5. Then{xn}converges strongly to a common fixed point of the mappings{Ti}N_i1.

Remark 1.2. Chidume and Shahzad gave an affirmative response to the question raised by Xu and Ori1, but they imposed compactness condition on some mapping of{Ti}N_i1.

In this paper, we will consider a process for a finite family of pseudocontractive mappings which include the nonexpansive mappings as special cases. Let f : C → C be a contraction. Let{αn},{βn}, and{γn}be three real sequences in0,1and an initial pointx0∈C.

Let the sequence{xn}be defined by

x1α1f

x0 β1x0γ1T1x1,

x2α2f

x1 β2x1γ2T2x2,

.. .

xN αNfxN−1 βNxN−1γNTNxN,

xN1αN1f

xN βN1xNγN1T1xN1,

.. .

1.8

which can be written in the following compact form:

xnαnfxn−1 βnxn−1γnTnxn, n≥1, 1.9

whereTn TnmodN.

Motivated by the works in 1–6, our purpose in this paper is to study the implicit

iteration process1.9in the general setting of a uniformly smooth Banach space and prove the strong convergence of the iterative process1.9to a common fixed point of a finite family of pseudocontractive mappings{Ti}N_i1. The results presented in this paper generalize and extend

the corresponding results of Chidume and Shahzad3, Osilike2, Xu and Ori1, and others.

2. Preliminaries

LetEbe a Banach space. Recall the norm ofEis said to be Gateaux differentiableandEis said to be smoothif

lim t→0

xty − x

exists for eachx, yin its unit sphereU {x∈E: x1}. It is said to be uniformly Frechet differentiableandEis said to be uniformly smoothif the limit in2.1is attained uniformly forx, y∈U×U.It is well known that a Banach spaceEis uniformly smooth if and only if the duality mapJis single valued and norm-to-norm uniformly continuous on bounded sets ofE. Recall that ifCandDare nonempty subsets of a Banach spaceEsuch thatCis nonempty closed convex andD⊂C, then a mapQ:C→Dis called a retraction fromContoDprovided Qx xfor allx∈D. A retractionQ : C→ Dis sunny providedQxtx−Qx Qx for allx ∈ C andt ≥ 0 wheneverxtx−Qx ∈ C. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive.

We need the following lemmas for proof of our main results.

Lemma 2.1 see 7. Let E be a uniformly smooth Banach space, C a closed convex subset of E,

T :C→Ca nonexpansive with FixT_/∅. For eachf ∈ΠCand everyt∈0,1, then{xt}defined by

xttfxt 1−tTxt 2.2

converges strongly ast→ 0to a fixed point ofT.

In particular, if f u ∈ C is a constant, then 2.2 is reduced to the sunny nonexpansive retraction of Reich fromConto FixT,

Qu−u, JQu−p ≤0, p∈FixT. 2.3

Lemma 2.2see8. LetEbe a real uniformly smooth Banach space, then there exists a nondecreasing

continuous functionb:0,∞→0,∞satisfying

ibct≤cbtfor allc≥1; iilim_t→0bt 0;

iiixy2 _{≤ x}2_{2y, jxmax{x,1}yby, for allx, y ∈E.}

The inequalityiiiis called Reich’s inequality.

Lemma 2.3see9. Let{an}∞_n0be a sequences of nonegative real numbers satisfying the property

an1≤1−γnanγnσn, n≥0,where{γn}∞_n0⊂0,1and{σn}∞n0are such that

i∞_n0γn∞;

iieitherlim sup_n→∞σn≤0or∞_n0|γnσn|<∞.

Then{an}∞_n0converges to0.

3. Main results

Theorem 3.1. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset

ofE. Let{T_i}N_i1beNpseudocontractive self-mappings ofCsuch that_iN₁FixT_i/∅. Let{α_n},{β_n}, and{γn}be three real sequences in0,1satisfying the following conditions:

iαnβnγn1;

iilimn→∞βn 0andlimn→∞αn/βn 0;

iii∞

Forf ∈ΠC and givenx0 ∈Carbitrarily, let the sequence{xn}be defined by1.9. Then{xn} converges strongly to a common fixed pointpof the mappings{Ti}N_i1, wherep Qfis the unique solution of the following variational inequality:

f−IQf, jz−Qf ≤0 ∀z∈ N

i1

FixTi . 3.1

Proof. First, we observe that{xn}is bounded. Indeed, if we take a fixed point pof T, noting that

xn−p

1−γn αn 1−γnf

xn−1 βn

1−γnxn−1

γnTnxn−p

1−γn αn 1−γn

fxn−1 −fp αn

fp−p 1−γn

βnxn−1−p

1−γn

γnTnxn−p .

3.2

It follows that _xn−p2

1−γn αn 1−γn

fxn−1 −fp

αnfp−p 1−γn

βnxn−1−p

1−γn , j

xn−p

γnTnxn−p, jxn−p

≤1−γn ₁αn₋ γn

fxn−1 −fp

αnfp−p 1−γn

βnxn−1−p

1−γn

xn−pγnxn−p2, 3.3

which implies that

_xn−p≤αnfxn−1 −fp

1−γn

αnfp−p 1−γn

βnxn−1−p

1−γn

≤ αn

1−γnfp−p

δαnβn

1−γn xn−1−p

1−δαn 1−γn ×

_fp−p 1−δ

1−1−δαn 1−γn

xn−1−p

≤maxfp−p

1−δ ,xn−1−p

.

3.4

Now, an induction yields

_xn−p≤maxfp−p

1−δ ,x0−p

. 3.5

Observe that

_{xn−Tnxn≤αnf}

xn−1 −Tnxnβnxn−1−Tnxn−→0. 3.6

SetAn 2I−Tn−1for alln 1,2, . . . , N, it is well known that{An}N_n1are all nonexpansive

mappings and FixAn FixTnas a consequence of10, Theorem 6. Then we have

_xn−AnxnAnA−1

n xn−Anxn≤xn−Tnxn. 3.7

It also follows from3.6that lim_n→∞xn−Anxn0. Next, we claim that

lim sup n→∞

fp−p, jxn−p ≤0 p∈ N

i1

FixTi , 3.8

where p Qf limt→0zt with zt being the fixed point of z → tfz 1− tAnz see Lemma 2.1.

Indeed,ztsolves the fixed point equation

zttfzt 1−tAnzt. 3.9

Then we have

zt−xn 1−tAnzt−xn tfzt −xn . 3.10

Thus we obtain

_zt−xn2_≤1−t2

Anzt−Anxnxn−Anxn2

2tfzt −zt, jzt−xn 2tzt−xn2.

3.11

Noting that

fzt −zt, jzt−xn fzt −fp, jzt−xn fp−zt, jzt−xn

≤δzt−pzt−xnfp−zt, jzt−xn . 3.12

Thus3.11gives

_zt−xn2_≤1−t2

zt−xnxn−Anxn22δtzt−pzt−xn

2tfp−zt, jzt−xn 2tzt−xn2

≤1−t2

zt−xn2ant 2δtzt−pzt−xn

2tfp−zt, jzt−xn 2tzt−xn2,

3.13

where

It follows that

zt−fp, jzt−xn ≤ t

2zt−xn

2 1

2tant δzt−pzt−xn. 3.15

Lettingn→ ∞in3.15and noting3.14yields

lim sup n→∞

zt−fp, jzt−xn ≤ t

2MδMzt−p, 3.16

whereM >0 is a constant.

For 3.9, since zt strongly converges to p, then {zt} is bounded. Hence we obtain immediately that the set {zt − xn} is bounded. At the same time, we note that the duality mapjis norm-to-norm uniformly continuous on bounded sets ofE. By lettingt→0 in3.16,

it is not hard to find that the two limits can be interchanged and3.8is thus proven. Finally, we show thatxn→pstrongly.

Indeed, usingLemma 2.2and noting that3.4, we obtain

_xn−p2

≤ αn 1−γn

fxn−1 −p βn

1−γn

xn−1−p 2

≤

βn 1−γn

2

_xn−1−p2

2 αnβn 1−γn 2

fxn−1 −p, j

xn−1−p

max βn 1−γn

xn−1−p ,1

αn 1−γnf

xn−1 −pb

αn 1−γnf

xn−1 −p

βn 1−γn

2

_xn−1−p2

2 αnβn 1−γn 2

fp−p, jxn−1−p

2 αnβn 1−γn 2

fxn−1 −fp, j

xn−1−p

max βn 1−γn

xn−1−p ,1

_{fxn−1 −pαn} 1−γn b

_{fxn−1 −pαn} 1−γn

≤

1− αn

αn2βn

αnβn 2

xn−1−p2 2αnβn

αnβn 2

δxn−1−p2

2αnβn αnβn 2

fp−p, jxn−1−p

max βn 1−γn

xn−1−p ,1

_{fxn−1 −pαn} αnβn b

_{fxn−1 −pαn} αnβn

1− αn

αn21−δβn

αnβn 2

_xn−1−p2 αn

αn21−δβn

αnβn 2

×

2βn αn21−δβn

fp−p, jxn−1−p

max βn 1−γn

xn−1−p ,1

_{fxn−1 −pαnβn} αn21−δβn b

_{fxn−1 −pαn} αnβn

1−λn xn−1−p2λnσn,

3.17 whereλnαnαn21−δβn/αnβn2and

σn 2βn αn21−δβn

fp−p, jxn−1−p max

βnxn−1−p

1−γn ,1

×f

xn−1 −pαnβn

αn21−δβn ×b

_{fxn−1 −pαn} αnβn

.

3.18

We observe that lim_n→∞αn 21 − δβn/αn βn 21 − δ, then ∞n0λn ∞ and

max{βn/1−γnxn−1 −p,1}fxn−1−pαn βn/αn 21 −δβn is bounded. At the same time, from limn→∞αn/αnβn 0, we have thatbfxn−1−pαn/αnβn→0.

This implies that lim sup_n→∞σn≤0.

Now, we apply Lemma 2.3and use 3.8to see thatxn −p → 0. This completes the proof.

Remark 3.2. Theorem 3.1 proves the strong convergence in the framework of real uniformly smooth Banach spaces. Our theorem extends Theorem O1 to the more general real Banach spaces. Our result improves Theorem O2 without condition lim infn→∞dxn, F 0 and at the same time extends the mappings from nonexpansive mappings to pseudocontractive mappings.

Corollary 3.3. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset

ofE. Let{T_i}N_i1beNpseudocontractive self-mappings ofCsuch that_iN₁ FixT_i/∅. Let{α_n},{β_n}, and{γn}be three real sequences in0,1satisfying the following conditions:

iαnβnγn1;

iilimn→∞βn 0andlimn→∞αn/βn 0;

iii∞_n0αn/αnβn ∞.

For fixedu∈Cand givenx0∈Carbitrarily, let the sequence{xn}be defined by

xnαnuβnxn−1γnTnxn, n≥1. 3.19 Then{xn}converges strongly to a common fixed pointpof the mappings{Ti}N_i1, wherep Quis the unique solution of the following inequality:

u−Qu, jz−Qu ≤0 ∀z∈ N

i1

FixTi , 3.20

Corollary 3.4. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. Let{Ti}N_i1beNnonexpansive self-mappings ofCsuch thatN_i1FixTi_/∅. Let{αn}, {βn}, and {γn}be three real sequences in0,1satisfying the following conditions:

iαnβnγn1;

iilimn→∞βn 0andlimn→∞αn/βn 0;

iii∞_n0αn/αnβn ∞.

Forf ∈ΠC and givenx0 ∈Carbitrarily, let the sequence{xn}be defined by1.9. Then{xn} converges strongly to a common fixed pointpof the mappings{Ti}N_i1, wherep Qfis the unique solution of the following variational inequality:

f−IQf, jz−Qf ≤0 ∀z∈ N

i1

FixTi . 3.21

Corollary 3.5. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset

ofE. Let{Ti}N_i1beNnonexpansive self-mappings ofCsuch thatN_i1FixTi_/∅. Let{αn}, {βn}, and {γn}be three real sequences in0,1satisfying the following conditions:

iαnβnγn1;

iilimn→∞βn 0andlimn→∞αn/βn 0; iii∞

n0αn/αnβn ∞.

For fixedu∈Cand givenx0∈Carbitrarily, let the sequence{xn}be defined by

xnαnuβnxn−1γnTnxn, n≥1. 3.22

Then{xn}converges strongly to a common fixed pointpof the mappings{Ti}N_i1, wherep Quis the unique solution of the following inequality:

u−Qup, jz−Qu ≤0 ∀z∈ N

i1

FixTi , 3.23

whereQis a sunny nonexpansive retraction fromContoN_i1FixTi.

Remark 3.6. Corollary 3.5 improves Theorem CS without compactness assumption of map-pings.

Acknowledgments

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