Fixed Point Theory and Applications

, 2009:798319 | Cite as

On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings

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Abstract

We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others.

Keywords

Hilbert Space Variational Inequality Equilibrium Problem Nonexpansive Mapping Lipschitz Constant 

1. Introduction

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window . A mapping Open image in new window is said to be nonexpansive if for all Open image in new window we have Open image in new window . It is said to be asymptotically nonexpansive [1] if there exists a sequence Open image in new window with Open image in new window and Open image in new window such that Open image in new window for all integers Open image in new window and for all Open image in new window . The set of fixed points of Open image in new window is denoted by Open image in new window .

Let Open image in new window be a bifunction, where Open image in new window is the set of real number. The equilibrium problem for the function Open image in new window is to find a point Open image in new window such that

The set of solutions of (1.1) is denoted by Open image in new window . In 2005, Combettes and Hirstoaga [2] introduced an iterative scheme of finding the best approximation to the initial data when Open image in new window is nonempty, and they also proved a strong convergence theorem.

For a bifunction Open image in new window and a nonlinear mapping Open image in new window , we consider the following equilibrium problem:
The set of such that Open image in new window is denoted by Open image in new window , that is,

In the case of Open image in new window , Open image in new window . In the case of Open image in new window , Open image in new window is denoted by Open image in new window . The problem (1.2) is very general in the sense that it includes, as special cases, some optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [3, 4]).

Recall that a mapping Open image in new window is called monotone if
A mapping Open image in new window of Open image in new window into Open image in new window is called Open image in new window -inverse strongly monotone, see [5, 6, 7], if there exists a positive real number Open image in new window such that

for all Open image in new window . It is obvious that any Open image in new window inverse strongly monotone mapping Open image in new window is monotone and Lipschitz continuous.

Construction of fixed points of nonexpansive mappings and asymptotically nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [1, 8, 9, 10]). Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann [11] and Ishikawa [12] iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities; see, for example, [11, 12, 13]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [11, 12]).

Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made. In 2003, Nakajo and Takahashi [14] proposed the following modification of the Mann iteration method for a nonexpansive mapping Open image in new window in a Hilbert space Open image in new window :

where Open image in new window denotes the metric projection from Open image in new window onto a closed convex subset Open image in new window of Open image in new window . They proved that if the sequence Open image in new window bounded above from one, then Open image in new window defined by (1.6) converges strongly to Open image in new window .

Recently, Kim and Xu [15] adapted the iteration (1.6) to an asymptotically nonexpansive mapping in a Hilbert space Open image in new window :

where Open image in new window , as Open image in new window . They proved that if Open image in new window for all Open image in new window and for some Open image in new window , then the sequence Open image in new window generated by (1.7) converges strongly to Open image in new window .

Very recently, Inchan and Plubtieng [16] introduced the modified Ishikawa iteration process by the shrinking hybrid method [17] for two asymptotically nonexpansive mappings Open image in new window and Open image in new window , with Open image in new window a closed convex bounded subset of a Hilbert space Open image in new window . For Open image in new window and Open image in new window , define Open image in new window as follows:

where Open image in new window , as Open image in new window and Open image in new window and Open image in new window for all Open image in new window . They proved that the sequence Open image in new window generated by (1.8) converges strongly to a common fixed point of two asymptotically nonexpansive mappings Open image in new window and Open image in new window .

Zegeye and Shahzad [18] established the following hybrid iteration process for a finite family of asymptotically nonexpansive mappings in a Hilbert space Open image in new window :

where Open image in new window , as Open image in new window . Under suitable conditions strong convergence theorem is proved which extends and improves the corresponding results of Nakajo and Takahashi [14] and Kim and Xu [15].

On the other hand, for finding a common element of Open image in new window , Tada and Takahashi [19] introduced the following iterative scheme by the hybrid method in a Hilbert space: Open image in new window and let

for every Open image in new window , where Open image in new window for some Open image in new window and Open image in new window satisfies Open image in new window . Further, they proved that Open image in new window and Open image in new window converge strongly to Open image in new window , where Open image in new window .

Inspired and motivated by the above facts, it is the purpose of this paper to introduce the Mann iteration process for finding a common element of the set of common fixed points of an infinite family of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem. Then we prove some strong convergence theorems which extend and improve the corresponding results of Tada and Takahashi [19], Inchan and Plubtieng [16], Zegeye and Shahazad [18], and many others.

2. Preliminaries

We will use the following notations:

(1)" Open image in new window " for weak convergence and " Open image in new window " for strong convergence;

(2) Open image in new window denotes the weak Open image in new window -limit set of Open image in new window .

Let Open image in new window be a real Hilbert space. It is well known that

for all Open image in new window .

It is well known that Open image in new window satisfies Opial's condition [20], that is, for any sequence Open image in new window with Open image in new window , the inequality

holds for every Open image in new window with Open image in new window . Hilbert space Open image in new window satisfies the Kadec-Klee property [21, 22], that is, for any sequence Open image in new window with Open image in new window and Open image in new window together imply Open image in new window .

We need some facts and tools in a real Hilbert space Open image in new window which are listed as follows.

Lemma 2.1 ([23]).

Let Open image in new window be an asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset Open image in new window of a Hilbert space Open image in new window . If Open image in new window is a sequence in Open image in new window such that Open image in new window and Open image in new window , then Open image in new window .

Lemma 2.2 ([24]).

Let Open image in new window be a nonempty closed convex subset of Open image in new window and also give a real number Open image in new window . The set Open image in new window is convex and closed.

Lemma 2.3 ([22]).

Let Open image in new window be a nonempty closed convex subset of Open image in new window and let Open image in new window be the (metric or nearest) projection from Open image in new window onto Open image in new window Open image in new window i.e., Open image in new window is the only point in Open image in new window such that Open image in new window . Given Open image in new window and Open image in new window . Then Open image in new window if and only if it holds the relation:

For solving the equilibrium problem, let us assume that the bifunction Open image in new window satisfies the following conditions (see [3]):

(A1) Open image in new window for all Open image in new window ;

(A2) Open image in new window is monotone, that is, Open image in new window for any Open image in new window ;

(A3) Open image in new window is upper-hemicontinuous, that is, for each Open image in new window

(A4) Open image in new window is convex and weakly lower semicontinuous for each Open image in new window .

The following lemma appears implicity in [3].

Lemma 2.4 ([3]).

Let Open image in new window be a nonempty closed convex subset of Open image in new window and let Open image in new window be a bifunction of Open image in new window into Open image in new window satisfying (A1)–(A4). Let Open image in new window and Open image in new window . Then, there exists Open image in new window such that

The following lemma was also given in [2].

Lemma 2.5 ([2]).

Assume that Open image in new window satisfies (A1)–(A4). For Open image in new window and Open image in new window , define a mapping Open image in new window as follows:

for all Open image in new window . Then, the following holds

(1) Open image in new window is single-valued;

(2) Open image in new window is firmly nonexpansive, that is, for any Open image in new window , Open image in new window .

This implies that Open image in new window , that is, Open image in new window is a nonexpansive mapping:

(3) Open image in new window ;

(4) Open image in new window is a closed and convex set.

Definition 2.6 (see [25]).

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 a family of asymptotically nonexpansive mappings of Open image in new window into itself, and let Open image in new window be a sequence of real numbers such that Open image in new window for every Open image in new window with Open image in new window . For any Open image in new window define a mapping Open image in new window as follows:

Such a mapping Open image in new window is called the modified Open image in new window -mapping generated by Open image in new window and Open image in new window .

Lemma 2.7 ([10, Lemma 4.1]).

(iii) if Open image in new window , Open image in new window and Open image in new window is closed convex.

Lemma 2.8 ([10, Lemma 4.4]).

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 a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window . Let Open image in new window for every Open image in new window , where Open image in new window for every Open image in new window and Open image in new window with Open image in new window for every Open image in new window and Open image in new window for every Open image in new window and let Open image in new window for every Open image in new window . Then, the following holds:

3. Main Results

In this section, we will introduce two iterative schemes by using hybrid approximation method for finding a common element of the set of common fixed points for a family of infinitely asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert space. Then we show that the sequences converge strongly to a common element of the two sets.

Theorem 3.1.

Let Open image in new window be a nonempty bounded closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a bifunction satisfying the conditions (A1)–(A4), let Open image in new window be an Open image in new window -inverse strongly monotone mapping of Open image in new window into Open image in new window , let Open image in new window be a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window , where Open image in new window and let Open image in new window be a sequence of real numbers with Open image in new window for all Open image in new window and Open image in new window for every Open image in new window and Open image in new window for some Open image in new window . Let Open image in new window be the modified Open image in new window -mapping generated by Open image in new window and Open image in new window . Assume that Open image in new window Open image in new window Open image in new window for every Open image in new window and 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 sequences generated by the following algorithm:

where Open image in new window and Open image in new window and Open image in new window and Open image in new window . Then Open image in new window and Open image in new window converge strongly to Open image in new window .

Proof.

We show first that the sequences Open image in new window and Open image in new window are well defined.

We observe that Open image in new window is closed and convex by Lemma 2.2. Next we show that Open image in new window for all Open image in new window . we prove first that Open image in new window is nonexpansive. Let Open image in new window . Since Open image in new window is Open image in new window -inverse strongly monotone and Open image in new window , we have

Thus Open image in new window is nonexpansive.

Since

we obtain

By Lemma 2.5, we have Open image in new window , Open image in new window .

Let Open image in new window , it follows the definition of Open image in new window that

Again by Lemma 2.5, we have Open image in new window , Open image in new window .

Since Open image in new window and Open image in new window are nonexpansive, one has

Then using the convexity of Open image in new window and Lemma 2.7 we obtain that

So Open image in new window for all Open image in new window and hence Open image in new window for all Open image in new window . This implies that Open image in new window is well defined. From Lemma 2.4, we know that Open image in new window is also well defined.

Next, we prove that Open image in new window , Open image in new window , Open image in new window , Open image in new window , as Open image in new window .

It follows from Open image in new window that

This implies that
and hence

Since Open image in new window is bounded, then Open image in new window and Open image in new window are bounded.

From Open image in new window and Open image in new window , we have

This implies that

Hence, Open image in new window is nodecreasing, and so Open image in new window exists.

Next, we can show that Open image in new window . Indeed, From (2.1) and (3.13), we obtain

On the other hand, it follows from Open image in new window that
It follows that

Next, we claim that Open image in new window . Let Open image in new window , it follows from (3.8) that

This implies that
It follows from (3.19) that
From Lemma 2.5, one has
This implies that
By (3.8), we have
Substituting (3.24) into (3.25), we obtain
which implies that
Noticing that Open image in new window and (3.19), it follows from (3.27) that
From (3.17) and (3.28), we have
Similarly, from (3.19) and (3.28), one has
Noticing that the condition Open image in new window , it follows that
which implies that

Next, we prove that there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window , where Open image in new window .

Since Open image in new window is bounded and Open image in new window is closed, there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window , where Open image in new window . From (3.28), we have Open image in new window . Noticing (3.29) and (3.32), it follows from Lemma 2.7 that Open image in new window . Next we prove that Open image in new window . Since Open image in new window , for any Open image in new window we have

From (A2), one has
Since Open image in new window , we have Open image in new window . Further, from monotonicity of Open image in new window , we have Open image in new window . So, from (A4) we have
as Open image in new window . From (A1) and (A4), we also have
and hence

This implies that Open image in new window . Therefore Open image in new window .

Finally we show that Open image in new window , Open image in new window , where Open image in new window .

Putting Open image in new window and consider the sequence Open image in new window . Then we have Open image in new window and by the weak lower semicontinuity of the norm and by the fact that Open image in new window for all Open image in new window which is implied by the fact that Open image in new window , we obtain

This implies that Open image in new window (hence Open image in new window by the uniqueness of the nearest point projection of Open image in new window onto Open image in new window ) and that

It follows that Open image in new window , and hence Open image in new window . Since Open image in new window is an arbitrary (weakly convergent) subsequence of Open image in new window , we conclude that Open image in new window . From (3.28), we know that Open image in new window also. This completes the proof of Theorem 3.1.

Theorem 3.2.

Let Open image in new window be a nonempty bounded closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a bifunction satisfying the conditions (A1)–(A4), let Open image in new window be an Open image in new window -inverse strongly monotone mapping of Open image in new window into Open image in new window , and let Open image in new window be a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window , where Open image in new window . Let Open image in new window for every Open image in new window , where Open image in new window for every Open image in new window and Open image in new window with Open image in new window for each Open image in new window and Open image in new window for every Open image in new window and assume that Open image in new window for every 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 sequences generated by

where Open image in new window and Open image in new window and Open image in new window . Then Open image in new window and Open image in new window converge strongly to Open image in new window .

Proof.

We divide the proof of Theorem 3.2 into four steps.

(i)We show first that the sequences Open image in new window and Open image in new window are well defined.

From the definition of Open image in new window and Open image in new window , it is obvious that Open image in new window is closed and Open image in new window is closed and convex for each Open image in new window . We prove that Open image in new window is convex. Since
is equivalent to

it follows that Open image in new window is convex. So, Open image in new window is a closed convex subset of Open image in new window for any Open image in new window .

Next, we show that Open image in new window . Indeed, let Open image in new window and let Open image in new window be a sequence of mappings defined as in Lemma 2.5. Similar to the proof of Theorem 3.1, we have

By virtue of the convexity of norm Open image in new window , (3.46), and Lemma 2.8, we have

Therefore, Open image in new window for all Open image in new window .

Next, we prove that Open image in new window , Open image in new window . For Open image in new window , we have Open image in new window . Assume that Open image in new window . Since Open image in new window is the projection of Open image in new window onto Open image in new window , by Lemma 2.3, we have

In particular, we have

This implies that Open image in new window is well defined. From Lemma 2.4, we know that Open image in new window is also well defined.

(ii)We prove that Open image in new window , Open image in new window , Open image in new window , Open image in new window , as Open image in new window .

Since Open image in new window is a nonempty closed convex subset of Open image in new window , there exists a unique Open image in new window such that Open image in new window .

From Open image in new window , we have

Since Open image in new window is bounded, we have Open image in new window , Open image in new window and Open image in new window are bounded. From the definition of Open image in new window , we have Open image in new window , which together with the fact that Open image in new window implies that

This shows that the sequence Open image in new window is nondecreasing. So, Open image in new window exists.

It follows from (2.1) and (3.53) that

Noticing that Open image in new window exists, this implies that
So, we have Open image in new window . It follows that
Similar to the proof of Theorem 3.1, we have
From (3.55) and (3.58), we have
Similarly, from (3.57) and (3.58), one has
Noticing the condition Open image in new window , it follows that
which implies that
  1. (iii)

    We prove that there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window , where Open image in new window .

     
Since Open image in new window is bounded and Open image in new window is closed, there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window , where Open image in new window . From (3.58), we have Open image in new window . Noticing (3.59) and (3.62), it follows from Lemma 2.8 that Open image in new window . By using the same method as in the proof of Theorem 3.1, we easily obtain that Open image in new window .
It follows from Open image in new window and the weak lower-semicontinuity of the norm that
Thus, we obtain that Open image in new window . Using the Kadec-Klee property of Open image in new window , we obtain that

Since Open image in new window is an arbitrary subsequence of Open image in new window , we conclude that Open image in new window converges strongly to Open image in new window . By (3.58), we have Open image in new window also. This completes the proof of Theorem 3.2.

Corollary 3.3.

Let Open image in new window be a nonempty bounded closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a bifunction satisfying the conditions (A1)–(A4), let Open image in new window be a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window , where Open image in new window and let Open image in new window be a sequence of real numbers with Open image in new window for all Open image in new window and Open image in new window for every Open image in new window and Open image in new window for some Open image in new window . Let Open image in new window be the modified Open image in new window -mapping generated by Open image in new window and Open image in new window . Assume that Open image in new window Open image in new window Open image in new window Open image in new window for every Open image in new window and 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 sequences generated by the following algorithm:

where Open image in new window and Open image in new window and Open image in new window and Open image in new window such that Open image in new window . Then Open image in new window and Open image in new window converge strongly to Open image in new window .

Proof.

Putting Open image in new window , the conclusion of Corollary 3.3 can be obtained as in the proof of Theorem 3.1.

Remark 3.4.

Corollary 3.3 extends the Theorem of Tada and Takahashi [19] in the following senses:

(1)from one nonexpansive mapping to a family of infinitely asymptotically nonexpansive mappings;
  1. (2)

    from computation point of view, the algorithm in Corollary 3.3 is also simpler and, more convenient to compute than the one given in [19].

     

Corollary 3.5.

Let Open image in new window be a nonempty bounded closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window and let Open image in new window be a sequence of real numbers with Open image in new window for all Open image in new window and Open image in new window for every Open image in new window and Open image in new window for some Open image in new window . Let Open image in new window be the modified Open image in new window -mapping generated by Open image in new window and Open image in new window . Assume that Open image in new window Open image in new window Open image in new window Open image in new window for every Open image in new window and Open image in new window such that Open image in new window . Let Open image in new window be a sequence generated by the following algorithm:

where Open image in new window and Open image in new window . Then Open image in new window converges strongly to Open image in new window .

Proof.

Putting Open image in new window , Open image in new window , Open image in new window and Open image in new window , for all Open image in new window in Theorem 3.1, we have Open image in new window , therefore Open image in new window . The conclusion of Corollary 3.5 can be obtained from Theorem 3.1 immediately.

Remark 3.6.

Corollary 3.5 extends Theorem 3.1 of Inchan and Plubtieng [16] from two asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.

Corollary 3.7.

Let Open image in new window be a nonempty bounded closed convex subset of a real Hilbert space Open image in new window , and let Open image in new window be a family of asymptotically nonexpansive mappings of Open image in new window into itself with Lipschitz constants Open image in new window , that is, Open image in new window ( Open image in new window ) such that Open image in new window . Let Open image in new window for every Open image in new window , where Open image in new window for every Open image in new window and Open image in new window with Open image in new window for each Open image in new window and Open image in new window for every Open image in new window and assume that Open image in new window for every Open image in new window such that Open image in new window . Let Open image in new window be a sequence generated by

where Open image in new window and Open image in new window . Then Open image in new window converges strongly to Open image in new window .

Proof.

Putting Open image in new window , Open image in new window , Open image in new window and Open image in new window , for all Open image in new window in Theorem 3.2, we have Open image in new window , therefore Open image in new window . The conclusion of Corollary 3.7 can be obtained from Theorem 3.2.

Remark 3.8.

Corollary 3.7 extends Theorem 3.1 of Zegeye and Shahzad [18] from a finite family of asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.

Notes

Acknowledgments

This research is supported by the National Science Foundation of China under Grant (10771175), and by the key project of chinese ministry of education(209078) and the Natural Science Foundational Committee of Hubei Province (D200722002).

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Copyright information

© G. Cai and C. S. Hu. 2009

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 MathematicsHubei Normal UniversityHuangshiChina

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