# 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 .

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.

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]).

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]).

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 .

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 .

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 .

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].

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 .

for all Open image in new window .

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]).

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]).

The following lemma was also given in [2].

Lemma 2.5 ([2]).

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]).

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:

- (ii)if Open image in new window is bounded and Open image in new window , for every sequence Open image in new window in C,(2.9)

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

## 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.

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

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

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

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

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

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

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

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.

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.

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

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

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

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

- (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 .

- (iv)
Finally we show that Open image in new window , Open image in new window , where Open image in new window .

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.

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:

- (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.

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.

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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