Convergence Theorem for Equilibrium and Variational Inequality Problems and a Family of Infinitely Nonexpansive Mappings in Hilbert Space

Abstract

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).

1. Introduction

Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of and the metric projection of onto . Let be a bifunction. We consider the equilibrium problem EP which is to find such that Let be the set of solutions. Some methods have been proposed to solve the equilibrium problem.

A mapping is said to be -inverse strongly monotone if there exists a real number such that , for all .

The classical variational inequality problem is to find an element such that The solution set of inequality (2) is denoted by . For given elements and , we have the following inequality: if and only if . It is known that the projection operator is nonexpansive. One can see that the variational inequality problem (2) is equivalent to a fixed point problem. Since an element is the solution of variational inequality (2) if and only if is a fixed point of the mapping . Recently, many researchers studied various iterative algorithms for finding an element of . Takahashi and Toyoda [1] introduced the following iterative scheme: They proved that the sequence converges weakly to a point . Y. Yao and J. C. Yao [2] introduced the following iterative scheme: Chang et al. [3] introduced the following iterative scheme: and obtained some strong convergence theorems.

In this paper, we will introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the variational inequality problem, and the equilibrium problem. Further, we obtain some strong convergence theorems and extend the results in [2–10].

2. Preliminaries

Let and be the weak convergence and strong convergence of the sequence in . Let be a nonempty closed convex subset of a Hilbert space . Let be a family of infinitely nonexpansive mappings and let be a sequence of positive numbers in . For , we define a mapping as follows: is the -mapping of into itself which is generated by , , , and , , , .

In order to prove our main results, the following Lemmas are needed.

Lemma 1 (see [11]). Let be a nonempty closed convex subset of a Banach space , let be a family of infinitely nonexpansive mappings, such that , and let be a sequence of positive numbers in for some . For any , let be the -mapping of into itself generated by , , , and , , , . Then is asymptotically regular and nonexpansive. Further, if is strictly convex, then .

Lemma 2 (see [4]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a family of infinitely nonexpansive mappings, such that , and let be a sequence of positive numbers in for some . Then for every and exists.
Using Lemma 2, we can define a mapping as follows: Such a is called the W-mapping generated by the sequence and . Throughout this paper, we always assume that is a sequence of positive numbers in for an element .

Lemma 3 (see [4]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a family of infinitely nonexpansive mappings such that and let be a sequence of positive numbers in for some . Then, is a nonexpansive mapping and .

Lemma 4 (see [4]). Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mappings, such that , and let be a sequence of positive numbers in for some . If is any bounded subset of , then .

Lemma 5 (see [10]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 6 (see [10]). Assume that is a sequence of nonnegative real numbers, such that where is some nonnegative integer, , and are sequences satisfying (1);(2) or ; then, .

Lemma 7 (see [10]). Let be a nonempty closed convex subset of a Hilbert space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive numbers with . Then, a mapping on defined by for all is well defined and nonexpansive and holds.

For solving the equilibrium problem for bifunction , assume that satisfies the following conditions: for all ; is monotone; that is, for all ;, for any ;for each is convex and lower semicontinuous.

If an equilibrium bifunction satisfies conditions , then we have the following two important results.

Lemma 8 (see [4]). Let be a nonempty closed convex subset of a Hilbert space and let be an equilibrium bifunction that satisfies conditions . Let and ; then, there exists such that , for all .

Lemma 9 (see [4]). Let be an equilibrium bifunction that satisfies conditions . For given and , define a mapping as follows: Then, the following conclusions hold: (1) is single-valued;(2) is firmly nonexpansive; that is, for any , ;(3)(4) is a closed and convex set.

3. The Main Results

Theorem 10. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -inverse strongly monotone mapping for each , where is some positive integer. Let be a -inverse strongly monotone mapping. Let be an equilibrium bifunction that satisfies conditions . Let be a family of infinite -strict pseudocontractive mappings with and let be a sequence of positive numbers in for some . is a family of infinitely nonexpansive mappings such that , where . Let be a strongly positive linear bounded operator with coefficient and let be a contraction with contraction constant and . Let be sequence generated by and where , and . If the following conditions are satisfied: and ;;;, ;, for all ,
then converges strongly to , where .

Proof. We define a bifunction by , for all , so the equilibrium problem is equivalent to the following equilibrium problem: find an element such that , for all and (11) can be written as

Step 1. First, we prove the sequences , , are bounded.
Let ; as , we have . Next we show that the mapping is nonexpansive for each . Consider Since is a strongly positive linear bounded operator, then , , so This implies that is bounded sequence in . Therefore , , , are all bounded.

Step 2. Next, we prove that and .
In fact, let us define a sequence by , for all ; then, we have Because , we have where , so Observing , , we have Putting in (19), in (20), adding up these two inequalities, and using condition to simplify, we have By condition , without loss of generality, we can assume that there exists a real number such that , so where .
Since are nonexpansive, so , where . Consider Using and the conditions , . By Lemma 5, we conclude that , . Consider So .

Step 3. Consider So we have .

Step 4. For any given , Simplifying it, we have So .

Step 5. Consider So .

Step 6. Consider So .

Step 7. Consider So .

Step 8. Next, we prove that , where .
To show it, we can choose a subsequence of such that Since is bounded, so there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that ; then, , for all ,, . So .
For any with and , let . Since and , from conditions and , we have This implies that . From condition , we have . So .
Define a mapping by , for all , where . From Lemma 7 we see that is nonexpansive such that Since every nonexpansive mapping is strictly pseudocontractive, so .
Now we prove that , and if not, we have . From Opial’s condition, we have Therefore, , so .

Step 9. Finally, we prove that . Since , so This implies that From Lemma 6, converges strongly to .
Taking for all , in Theorem 10, then Theorem 10 is reduced to the following.

Corollary 11. Let be a nonempty closed convex subset of a real Hilbert space . Let be a -inverse strongly monotone mapping and a -inverse strongly monotone mapping. Let be a family of infinite -strict pseudocontractive mappings with and let be a sequence of positive numbers in for some . is a family of infinitely nonexpansive mappings such that , where . Let be a strongly positive linear bounded operator with coefficient and let be a contraction with contraction constant and . Let and be generated in sequence by and where . If the following conditions are satisfied: , ;,
then converges strongly to , where .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors acknowledge the support of the Foundation of Langfang Teachers College (LSZY201307) and the Foundation of Education Department of Hebei Province (2011169).