Abstract

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi--asymptotically nonexpansive mappings, and null spaces of finite family of -inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors.

1. Introduction

Throughout this paper, we assume that is a real Banach space with a dual , is a nonempty closed convex subset of , and is the duality pairing between members of and . The mapping defined by is called the normalized duality mapping.

Let be a bifunction, let be a nonlinear mapping, and let be a proper extended real-valued function. The “so-called” generalized mixed equilibrium problem for , , is to find such that The set of solutions of (1.2) is denoted by , that is,

Special Examples
(1)If , then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by (2)If , then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by

These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–3]).

Let be a smooth, strictly convex, and reflexive Banach space, and let be a nonempty closed convex subset of . Throughout this paper, the Lyapunov function is defined by Following Alber [4], the generalized projection is defined by

Let be a nonempty closed convex subset of , let be a mapping, and let be the set of fixed points of . A point is said to be an asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by . A point is said to be a strong asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all strongly asymptotic fixed points of by .

A mapping is said to be nonexpansive if

A mapping is said to be relatively nonexpansive [5] if , and

A mapping is said to be weak relatively nonexpansive [6] if , and

A mapping is said to be closed if for any sequence with and , then .

A mapping is said to be quasi-ϕ-nonexpansive if and

A mapping is said to be quasi-ϕ-asymptotically nonexpansive, if and there exists a real sequence with such that

From the definition, it is easy to know that each relatively nonexpansive mapping is closed. The class of quasi--asymptotically nonexpansive mappings contains properly the class of quasi--nonexpansive mappings as a subclass. The class of quasi--nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, and the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

A mapping is said to be -inverse strongly monotone if there exists such that If is an -inverse strongly monotone mapping, then it is -Lipschitzian.

Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [5] obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Su et al. [6, 7], Zegeye and Shahzad [8], Wattanawitoon and Kumam [9], and Zhang [10] extend the notion from relatively nonexpansive mappings or quasi--nonexpansive mappings to quasi--asymptotically nonexpansive mappings and also prove some convergence theorems to approximate a common fixed point of quasi--nonexpansive mappings or quasi--asymptotically nonexpansive mappings.

Motivated and inspired by these facts, the purpose of this paper is to introduce a hybrid iterative scheme for finding a common element of null spaces of finite family of inverse strongly monotone mappings, set of common fixed points of an infinite family of quasi--asymptotically nonexpansive mappings, and the set of solutions of generalized mixed equilibrium problem.

2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

A Banach space is said to be strictly convex if for all with . It is said to be uniformly convex if for each , there exists such that for all with . The convexity modulus of is the function defined by for all . It is well known that is a strictly increasing and continuous function with , and is nondecreasing for all . Let , then is said to be -uniformly convex if there exists a constant such that , for all . The space is said to be smooth if the limit exists for all . And is said to be uniformly smooth if the limit exists uniformly in .

In the sequel, we will make use of the following lemmas.

Lemma 2.1 (see [11]). Let be a 2-uniformly convex real Banach space, then for all , the inequality holds, where , and is called the 2-uniformly convex constant of .

Lemma 2.2 (see [12]). Let be a smooth, strict convex, and reflexive Banach space, and let be a nonempty closed convex subset of , then the following conclusions hold:(i), for all , ,(ii)let and , then

Lemma 2.3 (see [12]). Let be a uniformly convex and smooth Banach space, and let , be sequences of . If and either or is bounded, then as .

Lemma 2.4 (see [10]). Let be a uniformly convex Banach space, let be a positive number, and let be a closed ball of . For any given points and for any given positive numbers with , there exists a continuous, strictly increasing, and convex function with such that for any , ,

For solving the generalized mixed equilibrium problem, let us assume that the bifunction satisfies the following conditions:(A1) for all ,(A2) is monotone, that is, , for all ,(A3), for all ,(A4) the function is convex and lower semicontinuous.

Lemma 2.5 (see [13]). Let be a smooth, strict convex, and reflexive Banach space, and let be a nonempty closed convex subset of . Let be a bifunction satisfying conditions (A1)–(A4). Let and , then there exists such that

By the same way as given in the proofs of [14, Lemmas  2.8 and 2.9], we can prove that the bifunction satisfies conditions (A1)–(A4) and the following conclusion holds.

Lemma 2.6. Let be a smooth, strictly convex, and reflexive Banach space, and let be a nonempty closed convex subset of . Let be a bifunction satisfying conditions (A1)–(A4), let be a -inverse strongly monotone mapping, and let be a lower semicontinuous and convex function. For given and , define a mapping by then the following hold:(i) is single valued,(ii) is a firmly nonexpansive-type mapping, that is, for all ,(iii), (iv) is closed and convex,(v), for all .

In the sequel, we make use of the function defined by for all and . Observe that for all and . The following lemma is well known.

Lemma 2.7 (see [4]). Let be a smooth, strict convex, and reflexive Banach space with as its dual, then for all and .

3. Main Results

In this section, we will propose the following new iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, the set of common fixed points of a family of quasi--asymptotically nonexpansive mappings, and null spaces of finite family of -inverse strongly monotone mappings in the setting of 2-uniformly convex and uniformly smooth real Banach spaces: where , is the mapping defined by (2.7), , for some and , where is the 2-uniformly convex constant of , for each , and for each .

Definition 3.1. A countable family of mappings is said to be uniformly quasi- -asymptotically nonexpansive mappings if there exists a sequence with such that for each

Theorem 3.2. Let be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space with a dual . Let be a countable family of closed and uniformly quasi--asymptotically nonexpansive mappings with a sequence such that . Suppose further that for each , is uniformly -Lipschitzian. Let , be a finite family of -inverse strongly monotone mappings, and let . Let be a finite family of equilibrium functions satisfying conditions (A1)–(A4), and be a finite family of lower semicontinuous convex function, and let be a finite family of -inverse strongly monotone mappings. If is a nonempty and bounded subset in and , then the sequence defined by (3.1) converges strongly to some point .

Proof. We divide the proof of Theorem 3.2 into five steps. (I) Sequences , , and all are bounded.
In fact, since , for any , from Lemma 2.2, we have This implies that the sequence is bounded, and so is bounded.
On the other hand, by Lemmas 2.1 and 2.7, we have that Thus, using the fact that , we have that Moreover, by the assumption that is a countable family of uniformly quasi--asymptotically nonexpansive mappings with a sequence such that (), hence for any given , from (3.6) we have that Hence, for each , is also bounded, denoted by By the way, from the definition of , it is easy to see that (II) For each , is a closed and convex subset of and .
It is obvious that is closed and convex. Suppose that is closed and convex for some . Since the inequality is equivalent to therefore, we have This implies that is closed and convex. Thus, for each , is a closed and convex subset of .
Next, we prove that for all . Indeed, it is obvious that . Suppose that for some . Since is uniformly smooth, is uniformly convex. For any given and for any positive integer , from Lemma 2.4, we have Having this together with (3.6), we have Hence, and for all .(III) is a Cauchy sequence.
Since and , we have that which implies that the sequence is nondecreasing and bounded, and so exists. Hence, for any positive integer , using Lemma 2.2 we have for all . Since exists, we obtain that Thus, by Lemma 2.3, we have that as . This implies that the sequence is a Cauchy sequence in . Since is a nonempty closed subset of Banach space , it is complete. Hence, there exists an in such that (IV) We show that .
Since by the structure of , we have that Again by (3.16) and Lemma 2.3, we get that . But Thus, This implies that as . Since is norm-to-norm uniformly continuous on bounded subsets of , we have that From (3.13), (3.20), and (3.21), we have that In view of condition , we see that It follows from the property of that Since and is uniformly continuous, it yields . Hence, from (3.24), we have Since is uniformly smooth and is uniformly continuous, it follows that Moreover, using inequalities (3.12) and (3.5), we obtain that This implies that that is, It follows from (3.1) and (3.29) that we have Furthermore, by the assumption that for each , is uniformly -Lipschitz continuous, hence, we have This together with (3.26) and (3.30) yields Hence, from (3.26), we have that is, In view of (3.26) and the closeness of , it yields that for all . This implies that . (IV) Now, we prove that .
It follows from (3.29) that Since , we have that for every subsequence of , and Let be an increasing sequence of natural numbers such that , for all , then and Since is -inverse strongly monotone, it is Lipschitz continuous, and thus Hence, Continuing this process, we obtain that , for all . Hence, (V) Next, we prove that .
Putting for and for all . For any , we have It follows from (3.22) that . Since is 2-uniformly convex and uniformly smooth Banach space and is bounded, we have that Since and , now we prove that for each , as . In fact, if , then we have that is, . By induction, the conclusion can be obtained. Since is norm-to-norm uniformly continuous on bounded subsets of , we get and since for some , we have that Next, since , for all , this implies that This implies that for some . Since is a convex and lower semicontinuous, we obtain from (3.45) and (3.47) that For any and , then . Since satisfies conditions (A1) and (A4), from (3.48), we have Delete , and then let , by condition (A3), we have that is, for each , we have Therefore, we have that This completes the proof.