- Research Article
- Open Access

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

- Gang Cai
^{1}Email author and - Changsong Hu
^{1}Email author

**2009**:798319

https://doi.org/10.1155/2009/798319

© G. Cai and C. S. Hu. 2009

**Received:**17 April 2009**Accepted:**9 July 2009**Published:**4 August 2009

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

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

In the case of , . In the case of , is denoted by . 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 . It is obvious that any inverse strongly monotone mapping 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, 8–10]). 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, [11–13]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [11, 12]).

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

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

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

where , as . 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 , where for some and satisfies . Further, they proved that and converge strongly to , where .

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)" " for weak convergence and " " for strong convergence;

(2) denotes the weak -limit set of .

for all .

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

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

Lemma 2.1 ([23]).

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

Lemma 2.2 ([24]).

Let be a nonempty closed convex subset of and also give a real number . The set is convex and closed.

Lemma 2.3 ([22]).

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

(A1) for all ;

(A2) is monotone, that is, for any ;

(A3) is upper-hemicontinuous, that is, for each

(A4) is convex and weakly lower semicontinuous for each .

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 . Then, the following holds

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any , .

This implies that , that is, is a nonexpansive mapping:

(3) ;

(4) is a closed and convex set.

Definition 2.6 (see [25]).

Such a mapping is called the modified -mapping generated by and .

Lemma 2.7 ([10, Lemma 4.1]).

- (i)
for all , and ;

(iii) if , and is closed convex.

Lemma 2.8 ([10, Lemma 4.4]).

Let be a nonempty closed convex subset of . Let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, ( ) such that . Let for every , where for every and with for every and for every and let for every . Then, the following holds:

- (iii)
if , and 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 and and and . Then and converge strongly to .

Proof.

We show first that the sequences and are well defined.

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

Thus is nonexpansive.

Since

By Lemma 2.5, we have , .

Let , it follows the definition of that

Again by Lemma 2.5, we have , .

Since and are nonexpansive, one has

So for all and hence for all . This implies that is well defined. From Lemma 2.4, we know that is also well defined.

Next, we prove that , , , , as .

It follows from that

Since is bounded, then and are bounded.

From and , we have

Hence, is nodecreasing, and so exists.

Next, we can show that . Indeed, From (2.1) and (3.13), we obtain

Next, we claim that . Let , it follows from (3.8) that

Next, we prove that there exists a subsequence of which converges weakly to , where .

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

This implies that . Therefore .

Finally we show that , , where .

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

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

Theorem 3.2.

where and and . Then and converge strongly to .

Proof.

We divide the proof of Theorem 3.2 into four steps.

(i)We show first that the sequences and are well defined.

it follows that is convex. So, is a closed convex subset of for any .

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

Therefore, for all .

Next, we prove that , . For , we have . Assume that . Since is the projection of onto , by Lemma 2.3, we have

This implies that is well defined. From Lemma 2.4, we know that is also well defined.

(ii)We prove that , , , , as .

Since is a nonempty closed convex subset of , there exists a unique such that .

From , we have

This shows that the sequence is nondecreasing. So, exists.

It follows from (2.1) and (3.53) that

- (iii)
We prove that there exists a subsequence of which converges weakly to , where .

- (iv)
Finally we show that , , where .

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

Corollary 3.3.

where and and and such that . Then and converge strongly to .

Proof.

Putting , 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 and . Then converges strongly to .

Proof.

Putting , , and , for all in Theorem 3.1, we have , therefore . 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 and . Then converges strongly to .

Proof.

Putting , , and , for all in Theorem 3.2, we have , therefore . 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.

## Declarations

### 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).

## Authors’ Affiliations

## References

- Goebel K, Kirk WA:
**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35:**171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Moudafi A, Théra M:
**Proximal and dynamical approaches to equilibrium problems.**In*Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar - Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Liu F, Nashed MZ:
**Regularization of nonlinear ill-posed variational inequalities and convergence rates.***Set-Valued Analysis*1998,**6**(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong and weak convergence theorems by an improved splitting method.***Communications on Applied Nonlinear Analysis*2002,**9**(2):99–107.MathSciNetMATHGoogle Scholar - Podilchuk CI, Mammone RJ:
**Image recovery by convex projections using a least-squares constraint.***Journal of the Optical Society of America A*1990,**7:**517–521.View ArticleGoogle Scholar - Byrne C:
**A unified treatment of some iterative algorithms in signal processing and image reconstruction.***Inverse Problems*2004,**20**(1):103–120. 10.1088/0266-5611/20/1/006MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Shimoji K, Takahashi W:
**On strong convergence by the hybrid method for families of mappings in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(1–2):112–119. 10.1016/j.na.2008.10.034MathSciNetView ArticleMATHGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Ishikawa S:
**Fixed points by a new iteration method.***Proceedings of the American Mathematical Society*1974,**44:**147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu HK:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(5):1140–1152. 10.1016/j.na.2005.05.059MathSciNetView ArticleMATHGoogle Scholar - Inchan I, Plubtieng S:
**Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces.***Nonlinear Analysis: Hybrid Systems*2008,**2**(4):1125–1135. 10.1016/j.nahs.2008.09.006MathSciNetMATHGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar - Zegeye H, Shahzad N:
**Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4496–4503. 10.1016/j.na.2007.11.005MathSciNetView ArticleMATHGoogle Scholar - Tada A, Takahashi W:
**Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.***Journal of Optimization Theory and Applications*2007,**133**(3):359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Lin P-K, Tan K-K, Xu HK:
**Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*1995,**24**(6):929–946. 10.1016/0362-546X(94)00128-5MathSciNetView ArticleMATHGoogle Scholar - Martinez-Yanes C, Xu H-K:
**Strong convergence of the CQ method for fixed point iteration processes.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
**Weak and strong convergence theorems for families of nonexpansive mappings and their applications.***Annales Universitatis Mariae Curie-Sklodowska. Sectio A*1997,**51**(2):277–292.MathSciNetMATHGoogle Scholar

## Copyright

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.