On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings
© G. Cai and C. S. Hu. 2009
Received: 17 April 2009
Accepted: 9 July 2009
Published: 4 August 2009
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.
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  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  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]).
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  and Ishikawa  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]).
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 , Inchan and Plubtieng , Zegeye and Shahazad , and many others.
We will use the following notations:
Lemma 2.1 ().
Lemma 2.2 ().
Lemma 2.3 ().
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see ):
The following lemma appears implicity in .
Lemma 2.4 ().
The following lemma was also given in .
Lemma 2.5 ().
Definition 2.6 (see ).
Lemma 2.7 ([10, Lemma 4.1]).
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:
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.
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
We divide the proof of Theorem 3.2 into four steps.
It follows from (2.1) and (3.53) that
Corollary 3.3 extends the Theorem of Tada and Takahashi  in the following senses:
from computation point of view, the algorithm in Corollary 3.3 is also simpler and, more convenient to compute than the one given in .
Corollary 3.5 extends Theorem 3.1 of Inchan and Plubtieng  from two asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
Corollary 3.7 extends Theorem 3.1 of Zegeye and Shahzad  from a finite family of asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
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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