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

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Research Article

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.

Keywords

Banach Space Variational Inequality Nonexpansive Mapping Strong Convergence Real Hilbert Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a real Banach space and let Open image in new window denote the normalized duality mapping from Open image in new window into Open image in new window given by

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

Let Open image in new window be a nonempty closed convex subset of Open image in new window . Recall that a self-mapping Open image in new window is said to be a contraction if there exists a constant Open image in new window such that

We use Open image in new window to denote the collection of all contractions on Open image in new window . That is, Open image in new window .

We use Open image in new window to denote the fixed point set of Open image in new window , that is, Open image in new window .

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

where Open image in new window .

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

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 Open image in new window be a real Hilbert space and let Open image in new window be a nonempty closed convex subset of Open image in new window . Let Open image in new window be Open image in new window strictly pseudocontractive self-mappings of Open image in new window such that Open image in new window . Let Open image in new window and let Open image in new window be a sequence in Open image in new window such that Open image in new window . Then the sequence Open image in new window defined by

where Open image in new window , converges weakly to a common fixed point of the mappings Open image in new window .

Theorem 1.

Let Open image in new window be a real Banach space and let Open image in new window be a nonempty closed convex subset of Open image in new window . Let Open image in new window be Open image in new window strictly pseudocontractive self-mappings of Open image in new window such that Open image in new window , and let Open image in new window be a real sequence satisfying the conditions Open image in new window , Open image in new window and Open image in new window . Let Open image in new window and let Open image in new window be defined by

where Open image in new window . Then Open image in new window converges strongly to a common fixed point of the mappings Open image in new window if and only if Open image in new window .

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 Open image in new window .

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

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 Open image in new window .

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

where Open image in new window .

Motivated by the works in [1, 2, 3, 4, 5, 6], 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 Open image in new window . 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 Open image in new window be a Banach space. Recall the norm of Open image in new window is said to be Gateaux differentiable (and Open image in new window is said to be smooth) if

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

Recall that if Open image in new window and Open image in new window are nonempty subsets of a Banach space Open image in new window such that Open image in new window is nonempty closed convex and Open image in new window , then a map Open image in new window is called a retraction from Open image in new window onto Open image in new window provided Open image in new window for all Open image in new window . A retraction Open image in new window is sunny provided Open image in new window for all Open image in new window and Open image in new window whenever Open image in new window . 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]).

converges strongly as Open image in new window to a fixed point of Open image in new window .

In particular, if Open image in new window is a constant, then (2.2) is reduced to the sunny nonexpansive retraction of Reich from Open image in new window onto Open image in new window ,

Lemma 2.2 (see [8]).

Let Open image in new window be a real uniformly smooth Banach space, then there exists a nondecreasing continuous function Open image in new window satisfying

(i) Open image in new window for all Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window , for all Open image in new window .

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

Lemma 2.3 (see [9]).

Let Open image in new window be a sequences of nonegative real numbers satisfying the property Open image in new window where Open image in new window and Open image in new window are such that

(i) Open image in new window ;

(ii)either Open image in new window or Open image in new window .

Then Open image in new window converges to Open image in new window .

3. Main Results

Theorem 3.1.

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

(i) Open image in new window ;

(ii) Open image in new window and Open image in new window ;

(iii) Open image in new window .

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

Proof.

First, we observe that Open image in new window is bounded. Indeed, if we take a fixed point Open image in new window of Open image in new window , noting that
It follows that
which implies that
Now, an induction yields

Hence Open image in new window is bounded, so are Open image in new window and Open image in new window for all Open image in new window .

Observe that
Set Open image in new window for all Open image in new window , it is well known that Open image in new window are all nonexpansive mappings and Open image in new window as a consequence of [10, Theorem 6]. Then we have

It also follows from (3.6) that Open image in new window .

Next, we claim that

where Open image in new window with Open image in new window being the fixed point of Open image in new window (see Lemma 2.1).

Indeed, Open image in new window solves the fixed point equation
Then we have
Thus we obtain
Noting that
Thus (3.11) gives
It follows that
Letting Open image in new window in (3.15) and noting (3.14) yields

where Open image in new window is a constant.

For (3.9), since Open image in new window strongly converges to Open image in new window , then Open image in new window is bounded. Hence we obtain immediately that the set Open image in new window is bounded. At the same time, we note that the duality map Open image in new window is norm-to-norm uniformly continuous on bounded sets of Open image in new window . By letting Open image in new window 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 Open image in new window strongly.

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

We observe that Open image in new window , then Open image in new window and Open image in new window is bounded. At the same time, from Open image in new window , we have that Open image in new window . This implies that Open image in new window .

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

Corollary 3.3.

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

(i) Open image in new window ;

(ii) Open image in new window and Open image in new window ;

(iii) Open image in new window .

For fixed Open image in new window and given Open image in new window arbitrarily, let the sequence Open image in new window be defined by
Then Open image in new window converges strongly to a common fixed point Open image in new window of the mappings Open image in new window , where Open image in new window is the unique solution of the following inequality:

where Open image in new window is a sunny nonexpansive retraction from Open image in new window onto Open image in new window .

Corollary 3.4.

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

(i) Open image in new window ;

(ii) Open image in new window and Open image in new window ;

(iii) Open image in new window .

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

Corollary 3.5.

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

(i) Open image in new window ;

(ii) Open image in new window and Open image in new window ;

(iii) Open image in new window .

For fixed Open image in new window and given Open image in new window arbitrarily, let the sequence Open image in new window be defined by
Then Open image in new window converges strongly to a common fixed point Open image in new window of the mappings Open image in new window , where Open image in new window is the unique solution of the following inequality:

where Open image in new window is a sunny nonexpansive retraction from Open image in new window onto Open image in new window .

Remark 3.6.

Corollary 3.5 improves Theorem CS without compactness assumption of mappings.

Notes

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.

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Copyright information

© 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.

Authors and Affiliations

  1. 1.Department of MathematicsTianjin Polytechnic UniversityTianjinChina
  2. 2.Department of Information ManagementCheng Shiu UniversityKaohsiungTaiwan

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