A new iterative method for equilibrium problems and fixed point problems for infinite family of nonself strictly pseudocontractive mappings
© Kim and Buong; licensee Springer. 2013
Received: 13 April 2013
Accepted: 16 September 2013
Published: 8 November 2013
The purpose of this paper is to present an iterative method for finding a common element of the set of solutions for an equilibrium problem and the set of common fixed points for a countably infinite family of nonself -strictly pseudocontractive mappings from a closed convex subset C into a Hilbert space H.
1 Introduction and preliminaries
The set of solutions of (1.1) is denoted by . The equilibrium problem (1.1) includes as special cases numerous problems in physics, optimization, economics, transportation, and engineering.
Assume that the bifunction G satisfies the following standard properties.
For each , is lower semicontinuous and convex;
It means that the class of λ-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.
Therefore, any λ-inverse strongly monotone mapping is monotone and Lipschitz continuous with the Lipschitz constant . It is well known  that if T is a nonexpansive mapping, then is -inverse strongly monotone. It is easy to verify that if T is λ-strictly pseudocontractive, then it is -inverse strongly monotone. Hence, is Lipschitz continuous with the Lipschitz constant .
If , , then problem (1.2) is reduced to (1.1) and shown in  and  to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, the Nash equilibria in noncooperative games, vector equilibrium problems, as well as certain fixed point problems (see also ). For finding approximative solutions of (1.1), there exist several approaches: the regularization approach [6–10], the gap-function approach [8, 11–13], and iterative procedure approach [14–30].
In the case that and , , (1.2) is a problem of finding a fixed point for a λ-strictly pseudocontractive mapping in C (see ).
In the case that and , , (1.2) is a problem of finding a common fixed point for a finite family of -strictly pseudocontractive mappings in C studied in , where the following algorithm is constructed:
where , is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings . The sequence converges weakly to a common fixed point of the maps .
If , , then (1.2) is a problem of finding a fixed point for a λ-strictly pseudocontractive mapping in C, which is an equilibrium point for G (see ).
Theorem 1.1 
Then and converge strongly to , where .
where α is the regularization parameter.
Theorem 1.2 
, , where d is a positive constant.
and and are the sequences of positive numbers.
Theorem 1.3 
Then the sequence converges strongly to the element , as .
- C2.For all ,
, or ,
, or ,
, or .
for all .
Then the sequence given by (1.5) converges strongly to the unique fixed point of , where is the metric projection from H onto ℱ.
For solving (1.2), Ceng and Yao  proposed the following algorithm:
Theorem 1.4 
Let f be a contraction of C into itself and given . Then the sequences and generated by (1.6), where is a sequence in for some , converge strongly to , where .
2 Main results
In this section, for solving problem (1.2), we present a new iterative method.
where , is a sequence of positive numbers satisfying , and α is a small regularization parameter tending to zero.
We need the following important lemmas for the proof of our main results.
Lemma 2.1 
Lemma 2.2 
- (ii)is firmly nonexpansive, i.e., for any ,
is closed and convex.
It is easy to see that is a nonexpansive mapping.
Lemma 2.3 
Assume that T is a λ-strictly pseudocontractive mapping of a closed convex subset C of a Hilbert space H. Then is demiclosed at zero; that is, whenever is a sequence in C weakly converging to some , and the sequence strongly converges to zero, it follows that .
Now, we are in a position to introduce and prove the main results.
, and , ,
and , the mapping A is well defined, and converges absolutely for each . It is easy to see that A is Lipschitz continuous with the Lipschitz constant and monotone.
It is not difficult to verify that for each , is a bifunction. Therefore, also is a bifunction, i.e., satisfies Assumption A. By using Lemma 2.2 with and , we can conclude that problem (2.5) (consequently (2.3)) has a unique solution for each .
By virtue of Lemma 2.4, we have . Since the closed convex set has only one element with the minimal norm, using (2.6) and the weak convergence of , we can conclude that all the sequence converges strongly to as .
for each . This completes the proof. □
Remark Obviously, if , where is the solution of (2.4) with , as , then .
This completes the proof. □
satisfy all the necessary conditions in Theorem 2.6.
involving a monotone hemicontinuous mapping . If is a λ-inverse strongly monotone mapping, and is a countably infinite family of nonexpansive self mappings in C, then problem (3.1) is recently studied in  and .
Then, . So, as are nonexpansive. Consequently, both mappings and are -strictly pseudocontractive. Thus, the mapping is also Lipschitz continuous with the Lipschitz constant . Then, algorithm (2.1) has the form:
where is defined by (3.3).
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the Republic of Korea (2013R1A1A2054617).
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