Open Access

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

Journal of Inequalities and Applications20082008:280908

https://doi.org/10.1155/2008/280908

Received: 10 January 2007

Accepted: 2 December 2007

Published: 2 January 2008

Abstract

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.

1. Introduction

Let be a real Banach space and let denote the normalized duality mapping from into given by
(1.1)

where denotes the dual space of and denotes the generalized duality pairing. If is strictly convex, then is single valued. In the sequel, we will denote the single-value duality mapping by .

Let be a nonempty closed convex subset of . Recall that a self-mapping is said to be a contraction if there exists a constant such that
(1.2)

We use to denote the collection of all contractions on . That is, .

A mapping with domain and in is called pseudocontractive if, for all , there exists such that
(1.3)

We use to denote the fixed point set of , that is, .

Recently, Xu and Ori [1] have introduced an implicit iteration process below for a finite family of nonexpansive mappings. Let be self-mappings of and suppose that , the set of common fixed points of An implicit iteration process for a finite family of nonexpansive mappings is defined as follows with a real sequence in , :
(1.4)
which can be written in the following compact form:
(1.5)

where .

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 in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mapping and/or the parameters are sufficient to guarantee the strong convergence of the sequence .

Very recently, Osilike [2] first extended Xu and Ori [1] from 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 1.

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-mappings of such that . Let and let be a sequence in such that . Then the sequence defined by
(1.6)

where , converges weakly to a common fixed point of the mappings .

Theorem 1.

Let be a real Banach space and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-mappings of such that , and let be a real sequence satisfying the conditions , and . Let and let be defined by
(1.7)

where . Then converges strongly to a common fixed point of the mappings if and only if .

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 .

In 2005, Chidume and Shahzad [3] also proved the strong convergence of the implicit iteration process (1.5) to a common fixed point for a finite family of nonexpansive mappings. They gave the following theorem.

Theorem 1 CS.

Let be a uniformly convex Banach space, let be a nonempty closed convex subset of . Let be nonexpansive self-mappings of with . Suppose that one of the mappings in is semicompact. Let for some . From arbitrary , define the sequence by (1.5). Then converges strongly to a common fixed point of the mappings .

Remark 1.2.

Chidume and Shahzad gave an affirmative response to the question raised by Xu and Ori [1], but they imposed compactness condition on some mapping of .

In this paper, we will consider a process for a finite family of pseudocontractive mappings which include the nonexpansive mappings as special cases. Let be a contraction. Let , , and be three real sequences in and an initial point . Let the sequence be defined by
(1.8)
which can be written in the following compact form:
(1.9)

where .

Motivated by the works in [16], our purpose in this paper is to study the implicit iteration process (1.9) in the general setting of a uniformly smooth Banach space and prove the strong convergence of the iterative process (1.9) to a common fixed point of a finite family of pseudocontractive mappings . The results presented in this paper generalize and extend the corresponding results of Chidume and Shahzad [3], Osilike [2], Xu and Ori [1], and others.

2. Preliminaries

Let be a Banach space. Recall the norm of is said to be Gateaux differentiable (and is said to be smooth) if
(2.1)

exists for each in its unit sphere . It is said to be uniformly Frechet differentiable (and is said to be uniformly smooth) if the limit in (2.1) is attained uniformly for It is well known that a Banach space is uniformly smooth if and only if the duality map is single valued and norm-to-norm uniformly continuous on bounded sets of .

Recall that if and are nonempty subsets of a Banach space such that is nonempty closed convex and , then a map is called a retraction from onto provided for all . A retraction is sunny provided for all and whenever . 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 be a uniformly smooth Banach space, a closed convex subset of , a nonexpansive with . For each and every , then defined by
(2.2)

converges strongly as to a fixed point of .

In particular, if is a constant, then (2.2) is reduced to the sunny nonexpansive retraction of Reich from onto ,
(2.3)

Lemma 2.2 (see [8]).

Let be a real uniformly smooth Banach space, then there exists a nondecreasing continuous function satisfying

(i) for all ;

(ii) ;

(iii) , for all .

The inequality (iii) is called Reich's inequality.

Lemma 2.3 (see [9]).

Let be a sequences of nonegative real numbers satisfying the property where and are such that

(i) ;

(ii)either or .

Then converges to .

3. Main Results

Theorem 3.1.

Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be pseudocontractive self-mappings of such that . Let , and be three real sequences in satisfying the following conditions:

(i) ;

(ii) and ;

(iii) .

For and given arbitrarily, let the sequence be defined by (1.9). Then converges strongly to a common fixed point of the mappings , where is the unique solution of the following variational inequality:
(3.1)

Proof.

First, we observe that is bounded. Indeed, if we take a fixed point of , noting that
(3.2)
It follows that
(3.3)
which implies that
(3.4)
Now, an induction yields
(3.5)

Hence is bounded, so are and for all .

Observe that
(3.6)
Set for all , it is well known that are all nonexpansive mappings and as a consequence of [10, Theorem 6]. Then we have
(3.7)

It also follows from (3.6) that .

Next, we claim that
(3.8)

where with being the fixed point of (see Lemma 2.1).

Indeed, solves the fixed point equation
(3.9)
Then we have
(3.10)
Thus we obtain
(3.11)
Noting that
(3.12)
Thus (3.11) gives
(3.13)
where
(3.14)
It follows that
(3.15)
Letting in (3.15) and noting (3.14) yields
(3.16)

where is a constant.

For (3.9), since strongly converges to , then is bounded. Hence we obtain immediately that the set is bounded. At the same time, we note that the duality map is norm-to-norm uniformly continuous on bounded sets of . By letting in (3.16), it is not hard to find that the two limits can be interchanged and (3.8) is thus proven.

Finally, we show that strongly.

Indeed, using Lemma 2.2 and noting that (3.4), we obtain
(3.17)
where and
(3.18)

We observe that , then and is bounded. At the same time, from , we have that . This implies that .

Now, we apply Lemma 2.3 and use (3.8) to see that . 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 and at the same time extends the mappings from nonexpansive mappings to pseudocontractive mappings.

Corollary 3.3.

Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be pseudocontractive self-mappings of such that . Let , and be three real sequences in satisfying the following conditions:

(i) ;

(ii) and ;

(iii) .

For fixed and given arbitrarily, let the sequence be defined by
(3.19)
Then converges strongly to a common fixed point of the mappings , where is the unique solution of the following inequality:
(3.20)

where is a sunny nonexpansive retraction from onto .

Corollary 3.4.

Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be nonexpansive self-mappings of such that . Let , and be three real sequences in satisfying the following conditions:

(i) ;

(ii) and ;

(iii) .

For and given arbitrarily, let the sequence be defined by (1.9). Then converges strongly to a common fixed point of the mappings , where is the unique solution of the following variational inequality:
(3.21)

Corollary 3.5.

Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be nonexpansive self-mappings of such that . Let , and be three real sequences in satisfying the following conditions:

(i) ;

(ii) and ;

(iii) .

For fixed and given arbitrarily, let the sequence be defined by
(3.22)
Then converges strongly to a common fixed point of the mappings , where is the unique solution of the following inequality:
(3.23)

where is a sunny nonexpansive retraction from onto .

Remark 3.6.

Corollary 3.5 improves Theorem CS without compactness assumption of mappings.

Declarations

Acknowledgments

The authors are extremely grateful to the referee for his/her careful reading. The first author was partially supposed by National Natural Science Foundation of China, Grant no. 10771050. The second author was partially supposed by the Grant no. NSC 96-2221-E-230-003.

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Information Management, Cheng Shiu University

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Copyright

© Y. Yao and Y.-C. Liou. 2008

This article is published under license to BioMed Central Ltd. 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.