#### Abstract

The purpose of this paper is to introduce a new iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi--nonexpansive mappings in Banach spaces. Moreover, the strong convergence theorems of this method are established under the suitable conditions of the parameter imposed on the algorithm. Finally, we apply our results to finding a zero point of inverse-strongly monotone operators and complementarity problems. Our results presented in this paper improve and extend the recently results by many others.

#### 1. Introduction

Equilibrium problem theory is the most important area of mathematical sciences and widely popular among mathematicians and researchers in other fields due to its applications in a wide class of problems which arise in economics, finance, optimization, network and transportation, image reconstruction, ecology, and many others. It has been improved and extended in many directions. Furthermore, equilibrium problems are related to the problem of finding fixed point of nonexpansive mappings. In this way, they have been extensively studied by many authors; see [1–9]. They introduced new iterative schemes for finding a common element of the set of the solutions of equilibrium problems and the set of fixed points. In this paper, we are interested a new hybrid iterative method for finding a common elements of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi--nonexpansive mappings in the framework of Banach spaces.

Let be a Banach space with norm and a nonempty closed convex subset of and let denote the dual of .

A mapping is said to be(1)*nonexpansive* [1] if for all ,(2)*relatively nonexpansive* [10–12] if and for all and , where the functional defined by (2.6). The asymptotic behavior of a relatively nonexpansive mapping was studied in [13, 14],(3)*-nonexpansive*, if for ,(4)*quasi- **-nonexpansive* if and for and .

In the sequel, we denote as the set of fixed points of . If is a bounded closed convex set and is a nonexpansive mapping of into itself, then is nonempty (see [15]).

A point in is said to be an asymptotic fixed point of [16] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by .

Let be an operator from into , and is said to be *-inverse-strongly monotone* if there exists a positive real number such that

If an operator is an -inverse-strongly monotone, then we can said that is *Lipschitz continuous*; that is, for all .

Let be a bifunction, a real-valued function, and be a nonlinear mapping. The *generalized mixed equilibrium problem* is to find such that

We denote as the set of solutions to (1.2) that is,

If , the problem (1.2) reduced into the *mixed equilibrium problem for *, denoted by , is to find such that

If , the problem (1.2) reduced into the *mixed variational inequality* of Browder type, denoted by , is to find such that

If and the problem (1.2), reduced into the *equilibrium problem for *, denoted by , is to find such that

In addition, fixed point problem, optimization problem, and many problems can be written in the form of . There are the development of researches in this area as seen in many papers which appeared in the literature on the existence of the solutions of ; see, for example [17–21] and reference therein. Furthermore, there are many solution methods proposed continuously to solve the as shown in [2, 3, 18, 20, 22–26] and many others.

Next, we let be a *monotone* operator of into . The so-called *variational inequality problem* is to find a point such that
The set of solutions of the variational inequality problem is denoted by .

As we know that the classical variational inequality was first introduced and studied by Stampacchia [27] in 1964. Its solution can be computed by using iterative projection method. There are many results with corresponding to variational inequality; for example, Yao et al. [28] proposed the strong convergence theorem for a system of nonlinear variational inequalities in Banach spaces, and then, they studied the two-step projection methods, and they established the convergence theorem for a system of variational inequality problems in the framework of Banach spaces. Moreover, the important generalized variational inequalities called variational inclusion also have been extensively studied and extended in many different directions. Yao et al. [29] considered the algorithm and proved the strong convergence of common solutions for variational inclusions, mixed equilibrium problems, and fixed point problems.

The one classical way to approximate a fixed point of a nonlinear self mapping on was firstly introduced by Halpern [30], and then, Aoyama et al. [31] extended the mapping in the Halpern-type iterative sequence to be a countable family of nonexpansive mappings. They introduced the following iterative sequence: let and for all , where is a nonempty closed convex subset of a Banach space, is a sequence in , and is a sequence of nonexpansive mappings with some conditions. They proved that converges strongly to a common fixed point of .

Recently, Nakajo et al. [32] introduced the more general condition so-called the NST*-condition, and is said to satisfy the NST*-condition if for every bounded sequence in , They also prove strong convergence theorems by the hybrid method for families of mappings in a uniformly convex Banach space whose norm is Gâteaux differentiable.

In Hilbert space , Iiduka et al. [33] introduced an iterative scheme and proved that the sequence generated by the following algorithm: , and where is the metric projection of onto and is a sequence of positive real numbers, converges weakly to some element of .

Later, Iiduka and Takahashi [34] are interested in the similar problem in the framework of Banach spaces, they introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator , and for every , where is the generalized metric projection from onto , is the duality mapping from into , and is a sequence of positive real numbers. They proved that the sequence generated by (1.11) converges weakly to some element of .

In 1974, Rockafellar interested in the following problem of finding:
where is an operator from into . Such is called a *zero point* of . He introduced a well-known method, *proximal point algorithm*, for solving (1.12) in a Hilbert space as shown in the following: and
where , is a maximal monotone and . He proved that the sequence converges weakly to an element of .

In 2004, Kamimura et al. [35] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space ; namely, They proved that the algorithm generated by (1.14) converges weakly to some element of .

In 2008, Li and Song [36] established a strong convergence theorem in a Banach space. They introduced the following algorithm: and Under the suitable conditions of the coefficient sequences , , and , they proved that the sequence generated by the above scheme converges strongly to , where is the generalized projection from onto .

In 2010, Petrot et al. [37] introduced a hybrid projection iterative scheme for approximating a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of two quasi--nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space by the following manner: They proved that converges strongly to , where .

Recently, Klin-eam et al. [38], obtained the strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Saewan and Kumam [39] introduced a new hybrid projection method for finding a common solution of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an -inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem, and zeros of a maximal monotone operator in a real uniformly smooth and 2-uniformly convex Banach space. Wattanawitoon and Kumam [40] proved the strong convergence theorem by using modified hybrid projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solution of variational inequality operators of an inverse strongly monotone, the zero point of a maximal monotone operator, and the set of fixed point of two relatively quasi-nonexpansive mappings in Banach space.

Motivated and inspired by the ongoing research and the above-mentioned results, we are also interested in generalized mixed equilibrium problem, variational inequality problems, and the zero point of maximal monotone operators. In this paper, we extend the fixed point problems of two relatively quasi-nonexpansive mappings in [40] to the countable families of two quasi--nonexpansive mappings and improve the iterative scheme to be more general as shown in the following: , By the new iterative scheme, we will prove the strong convergence theorems of the sequence which could be converged to the point . Furthermore, we propose the new better appropriate conditions of the coefficient sequences , and . Finally, we will apply our result to find a zero point of inverse-strongly monotone operators and complementarity problem in the last section. The results presented in this paper extend and improve the corresponding ones announced by Kamimura et al. [35], Petrot et al. [37], Wattanawitoon and Kumam [40], and some authors in the literature.

#### 2. Preliminaries

In this section, we propose the following preliminaries and lemmas which will be used in our proof.

Throughout this paper, we let be a Banach space with norm , and a nonempty closed convex subset of , and let denote the dual of . We write to indicate that the sequence converges weakly to and implies that the sequence converges strongly to .

Let be the unit sphere of . A Banach space is said to be *strictly convex* if for any ,

It is also said to be *uniformly convex* if for each , there exists such that for any
We know that a uniformly convex Banach space is reflexive and strictly convex; see [41, 42] for more details.

The *modulus of convexity* of is the function defined by

Furthermore, it is said to be *smooth*, provided that
exists for all . It is also said to be *uniformly smooth* if the limit is attained uniformly for .

Let be a fixed real number with . Observe that every -uniformly convex is uniformly convex. One should note that no a Banach space is -uniformly convex for . It is well known that a Hilbert space is *2*-uniformly convex and uniformly smooth. For each , the *generalized duality mapping * is defined by
for all .

In particular, is called *the normalized duality mapping*. If is a Hilbert space, then , where is the identity mapping. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

We know the following (see [43]):(1)if is smooth, then is single-valued,(2)if is strictly convex, then is one-to-one and holds for all with ,(3)if is reflexive, then is surjective,(4)if is uniformly convex, then it is reflexive,(5)if is uniformly convex, then is uniformly norm-to-norm continuous on each bounded subset of .

The duality from a smooth Banach space into is said to be *weakly sequentially continuous* [44] if implies , where implies the weak* convergence.

Let be a smooth, strictly convex and reflexive Banach space, and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by

*Remark 2.1. *We know the following: for each ,(i),(ii),(iii) in a real Hilbert space.

The *generalized projection*, introduced by Alber [45], is a map that assigns to an arbitrary point the minimum point of the function ; that is, , where is the solution to the minimization problem
existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping .

If is a reflexive, strictly convex and smooth Banach space, then for , , if and only if . It is sufficient to show that if , then . From Remark 2.1 (i), we have . This implies that . From the definition of , one has . Therefore, we have ; see [43, 46] for more details.

Lemma 2.2 (see [47, 48]). *If be a 2-uniformly convex Banach space, then for all , one has
**
where is the normalized duality mapping of and .*

The best constant in the Lemma is called the *2*-uniformly convex constant of ; see [41].

Lemma 2.3 (see [47, 49]). *If is a p-uniformly convex Banach space and a given real number with , then for all , and **
where is the generalized duality mapping of and is the p-uniformly convexity constant of .*

Lemma 2.4 (Xu [48]). *Let be a uniformly convex Banach space, then for each , there exists a strictly increasing, continuous and convex function such that and
**
for all and .*

Lemma 2.5 (Kamimura and Takahashi [50]). *Let be a uniformly convex and smooth real Banach space and two sequences of . If and either or is bounded, then .*

Lemma 2.6 (Alber [45]). *Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if
*

Lemma 2.7 (Alber [45]). *Let be a reflexive, strictly convex and smooth Banach space and a nonempty closed convex subset of and let . Then,
*

Let be a strictly convex, smooth, and reflexive Banach space and the duality mapping from into . Then, is also single-valued, one-to-one, and surjective, and it is the duality mapping from into . Define a function as follows (see [51]): for all and . Then, it is obvious that and .

Lemma 2.8 (Kohsaka and Takahashi [51, Lemma 3.2]). *Let be a strictly convex, smooth, and reflexive Banach space and as in (2.13). Then,
**
for all and . *

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

Lemma 2.9 (Blum and Oettli [17]). *Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space and a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that
*

Lemma 2.10 (Takahashi and Zembayashi [52]). *Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space and a bifunction from to satisfying (A1)–(A4). For all and , define a mapping as follows:
**
for all . Then, the following hold:*(1)* is single-valued,*(2)* is a firmly nonexpansive-type mapping, that is, for all ,
*(3)*,
*(4)* is closed and convex.*

Lemma 2.11 (Takahashi and Zembayashi [52]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , and a bifunction from to satisfying (A1)–(A4) and let . Then, for and ,
*

Lemma 2.12 (Zhang [53]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, convex and lower semi-continuous, and a bifunction from to satisfying (A1)–(A4). For and , then there exists such that
**
Define a mapping as follows:
**
for all . Then, the following hold:*(i)* is single-valued,*(ii)* is firmly nonexpansive, that is, for all , ,*(iii)*,
*(iv)* is closed and convex,*(v)* for all , .*

It follows from Lemma 2.10 that the mapping defined by (2.21) is a relatively nonexpansive mapping. Thus, it is quasi--nonexpansive.

Let be a reflexive, strictly convex and smooth Banach space. Let be a closed convex subset of . Because is strictly convex and coercive in the first variable, we know that the minimization problem has a unique solution. The operator is said to be the generalized projection of on .

Let be a set-valued mapping from to with graph , domain , and range . We denote a set-valued operator from to by . is said to be *monotone* if for all . A monotone operator is said to be *maximal monotone* if it graph is not properly contained in the graph of any other monotone operator. We know that if is maximal monotone, then the solution set is closed and convex.

Let be a reflexive, strictly convex and smooth Banach space, it is known that is maximal monotone if and only if for all .

Define the *resolvent* of by . In other words, for all . is a single-valued mapping from to . Also, for all , where is the set of all fixed points of . Define, for , the *Yosida approximation* of by . We know that for all and .

Lemma 2.13 (Kohsaka and Takahashi [51, Lemma 3.1]). *Let be a smooth, strictly convex, and reflexive Banach space, let be a maximal monotone operator with , , and . Then,
**
for all and .*

Let be an inverse-strongly monotone mapping of into which is said to be *hemicontinuous* if for all , the mapping of into , defined by , is continuous with respect to the weak* topology of . We define by the *normal cone* for at a point ; that is,

Theorem 2.14. *(Rockafellar [54]). Let be a nonempty, closed convex subset of a Banach space and a monotone, hemicontinuous operator of into . Let be an operator defined as follows:
**
Then , is maximal monotone and .*

Lemma 2.15 (Tan and Xu [55]). *Let and be two sequences of nonnegative real numbers satisfying
**
If , then exists.*

#### 3. The Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of mixed equilibrium problems, the set of solutions of the variational inequality problem, the zero point of a maximal monotone operator, and the set of two families of quasi--nonexpansive mappings in a Banach space by using the shrinking hybrid projection method.

Theorem 3.1. *Let be a 2-uniformly convex and uniformly smooth Banach space and a nonempty closed convex subset of . Let be a bifunction from to satisfying (A1)–(A4), let be a proper lower semicontinuous and convex function, and let be a maximal monotone operator satisfying . Let for , let be an -inverse-strongly monotone operator of into , and let be a continuous and monotone mapping. Let and be two families of quasi--nonexpansive mappings of into itself satisfies the NST-condition, with and for all and . Let be a sequence generated by , and
**
for all . If the coefficient sequence , , and , satisfy , , , , and for some with , is the 2-uniformly convexity constant of . Then the sequence converges strongly to .*

*Proof. *We first show that is bounded. Let , and let
Put and .

With its relatively nonexpansiveness of and by Lemma 2.8, the convexity of the function in the second variable, we have
Since and is -inverse-strongly monotone, we consider
Therefore, by Lemma 2.2, we obtain
We can rewrite (3.3), which yield that
Apply the Lemma 2.8, Lemma 2.13 and (3.6), we consider
hence, we obtain
By (3.1), again,
This shows that . Consequently, , for all .

Next, we show that exists. Since , it follows from Lemma 2.7 that
for each . Then, is bounded. It implies that is bounded and , , , and are also bounded.

From and , we have
Therefore, is nondecreasing. It follows that the limit of exists, and from Lemma 2.7, we have
for all . Thus, we have
Since , it follows from the definition of that
By Lemma 2.5, (3.13), and (3.14), we note that
Since is uniformly norm-to-norm continuous on the bounded set, we obtain
Since for any positive integer , it follows from Lemma 2.7 that
Taking , we have as . It follows from Lemma 2.5, that as . Hence, is a Cauchy sequence. Since is a Banach space and is closed and convex, we can assume that as .

Next, we show that .

Since is a uniformly smooth Banach space, we know that is a uniformly convex Banach space. Let . From Lemma 2.4, we have
This implies that
On the other hand, we have
Noticing (3.15) and (3.16), we obtain
Since and (3.21), it follows from (3.19) that
It follows from the property of that
Since is uniformly norm-to-norm continuous on bounded sets, we see that
Similarly, using the condition , one can obtain
By (3.6), (3.8), and (3.18), we have
This implies that
By assumption, and (3.21), we get that
From Lemma 2.7, Lemma 2.8, and (3.5), we have
By Lemma 2.8 and Lemma 2.13, we have
From Lemma 2.5 and (3.28), we obtain
Since is uniformly norm-to-norm continuous on bounded sets, we note that
Since as , as . Combining (3.15), (3.25), and (3.28), we also obtain
By (3.15) and (3.31), we obtain that
By (3.15), (3.24), (3.33), and (3.34), and , satisfies the NST*-condition and , then we have .

Since is bounded, there exists a subsequence of such that . It follows from (3.31) that we have as . Next, we show that .

By (3.6), (3.8), and (3.9), we obtain
This implies that
By (3.21), we have
Since is uniformly norm-to-norm continuous on bounded sets, we note that
Indeed, since , it follows from (3.38) that
If , then it holds from the monotonicity of that
for all . Letting , we get . Then, the maximality of implies .

Next, we show that . Let be an operator as follows:
By Theorem 2.14, is maximal monotone and .

Let . Since , we get . From , we have
On the other hand, since , then by Lemma 2.6, we have
thus
It follows from (3.42) and (3.44) that
where . From (3.31) and (3.32), we obtain . By the maximality of , we have and hence .

Next, we show that . From and Lemma 2.12, we obtain
On the other hand, we have
Noticing (3.15) and (3.16), we obtain
It follows that
By Lemma 2.5, we have
Since is uniformly norm-to-norm continuous on bounded sets, we get
From the assumption , we get
Noticing that , we have
Hence,
From the (A2), we note that
Taking the limit as in the above inequality, and from (A4) and , we have . For and , define . Noticing that , we obtain , which yields that . It follows from (A1) that
That is, .

Let , from (A3), we obtain