Journal of Applied Mathematics

1. Introduction

In 1994, Blum and Oettli [1] introduced equilibrium problems, which have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization.

Let be a real Banach space, the dual space of , and a nonempty closed convex subset of . Let be a bifunction and a real-valued function. The generalized mixed equilibrium problem (GMEP) of finding is such that Recently, Zhang [2] considered this problem. Here some special cases of problem (1.1) are stated as follows.

If , then problem (1.1) reduces to the following mixed equilibrium problem of finding such that which was considered by Ceng and Yao [3]. The set of solutions of this problem is denoted by MEP.

If , then problem (1.1) reduces to the following generalized equilibrium problem of finding such that which was studied by S. Takahashi and W. Takahashi [4].

If and , then problem (1.1) reduces to the following equilibrium problem of finding such that The set of solutions of problem (1.4) is denoted by EP.

If , then problem (1.1) reduces to the following classical variational inequality problem of finding such that The set of solutions of problem (1.5) is denoted by .

The problem (1.1) is very general in the sense that it includes, as special cases, numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 3–7].

The normalized duality mapping from to is defined by where denotes the generalized duality pairing. It is well known that if is smooth then is single valued and if is uniformly smooth then is uniformly continuous on bounded subsets of . Moreover, if is a reflexive and strictly convex Banach space with a strictly convex dual, then is single valued, one to one, surjective, and it is the duality mapping from into and thus and (see [8]).

On the other hand, let be a set-valued mapping. The problem of finding satisfying contains numerous problems in economics, optimization, and physics. Such is called a zero point of .

A set-valued mapping with graph , domain , and range is said to be monotone if for all . is said to be maximal monotone if the graph of is not properly contained in the graph of any other monotone operator. It is known that is a maximal monotone if and only if for all when is a reflexive, strictly convex, and smooth Banach space (see [9]).

Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of , and let be a monotone operator satisfying . Then the resolvent of defined by is a single-valued mapping from to for all . For , the Yosida approximation of is defined by for all .

A mapping is said to be monotone if, for each , is said to be -inverse strongly monotone if there exists a positive real number such that If is -inverse strongly monotone, then it is Lipschitz continuous with constant , that is,

Let be a smooth Banach space. The function defined by is studied by Alber [10], Kamimura and Takahashi [11], and Reich [12]. It follows from the definition of the function that Observe that, in a Hilbert space , .

Lemma 1.1 (see [10]). Let be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space , and let . Then there exists a unique element such that .

Let be a reflexive, strictly convex, and smooth Banach space and a nonempty closed and convex subset of . The generalized projection mapping, introduced by Alber [10], is a mapping that assigns to an arbitrary point , the minimum point of the functional , that is, due to Lemma 1.1, where is the solution to the minimization problem .

Let be a mapping from into itself. denotes the set of fixed points of . A point in is said to be an asymptotic fixed point of if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called nonexpansive if for all and relatively nonexpansive (see [13, 14]) if and for all and . is said to be -nonexpansive if for all . is said to be relatively quasi-nonexpansive if and for all and . Note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the strong restriction: .

When is a maximal monotone operator, a well-known method for solving the equation in a Hilbert space is the proximal point algorithm (see [15]): and where and for all is the resolvent operator for , then Rockafellar proved that the sequence converges weakly to an element of .

The modifications of the proximal point algorithm for different operators have been investigated by many authors. Kohsaka and Takahashi [16] considered the following Algorithm (1.13) in a smooth and uniformly convex Banach space: and Kamimura et al. [17] considered Algorithm (1.14) in a uniformly smooth and uniformly convex Banach space: They showed that Algorithm (1.13) converges strongly and Algorithm (1.14) converges weakly provided that the sequences , of real numbers are chosen appropriately.

Recently, Saewan and Kumam [18] proposed the following iterative scheme: for an initial with and where is the same as in Lemma 2.8 and they obtained a strong convergence theorem.

In 2010, Zeng et al. [19] introduced the following hybrid iterative process: let be chosen arbitrarily, Then they proved some strong and weak convergence theorems.

Very recently, for mixed equilibrium problems, variational inequality problems, fixed point problems, and zeros of maximal monotone operators, many authors have studied them and obtained many new results, see, for instance, [20–23].

On the other hand, Nakajo et al. [24] introduced the following condition. Let be a nonempty closed convex subset of a Hilbert space , let be a family of mappings of into itself with , and denotes the set of all weak subsequential limits of a bounded sequence in . is said to satisfy the NST-condition if, for every bounded sequence in ,

Motivated and inspired by the above work, the purpose of this paper is to introduce a new hybrid projection iterative scheme which converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings and zero of maximal monotone operators in Banach spaces.

2. Preliminaries

Let be a normed linear space with dim . The modulus of smoothness of is the function defined by The space is said to be smooth if , and is called uniformly smooth if and only if .

The modulus of convexity of is the function defined by is called uniformly convex if and only if for every . Let , then is said to be -uniformly convex if there exists a constant such that for every . Observe that every -uniformly convex is uniformly convex. It is well known (see, e.g., [7]) that

In what follows, we will make use of the following lemmas.

Lemma 2.1 (see [7]). Let be a 2-uniformly convex and smooth Banach space. Then, for all , one has where is the normalized duality mapping of and is the 2-uniformly convex constant of .

Lemma 2.2 (see [10, 11]). Let be a real smooth, strictly convex, and reflexive Banach space and a nonempty closed convex subset. Then the following conclusions hold: (1); (2)suppose and , then

Lemma 2.3 (see [11]). Let be a real smooth and uniformly convex Banach space, and let and be two sequences of . If either or is bounded and as , then as .

Lemma 2.4 (see [25]). Let be a real smooth Banach space, and let be a maximal monotone mapping, then is a closed and convex subset of .

We denote by the normal cone for at a point , that is, . In the following, we will use the following Lemma.

Lemma 2.5 (see [15]). Let be a nonempty closed convex subset of a Banach space , and let be a monotone and hemicontinuous operator of into . Let be an operator defined as follows: Then is maximal monotone and .

We make use of the function defined by studied by Alber [10]; that is, for all and . We know the following lemma.

Lemma 2.6 (see [10]). Let be a reflexive strictly convex and smooth Banach space with as its dual. Then for all and .

Lemma 2.7 (see [26]). Let be a uniformly convex Banach space, and let be a closed ball of . Then there exists a continuous strictly increasing convex function with such that for all and with .

For solving the equilibrium problem, let us assume that satisfies 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 2.8 (see [2]). Let be a closed subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, let be a lower semicontinuous and convex function, and let be a bifunction from to satisfying (A1)–(A4). For and , then there exists such that Define a mapping as follows: for all . Then, the following conclusions hold:(1) is single valued;(2) is firmly nonexpansive, that is, for all , ;(3);(4) is closed and convex;(5).

Lemma 2.9 (see [27]). Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed convex subset of , and let be a monotone operator satisfying . Let , and let and be the resolvent and the Yosida approximation of , respectively. Then the following hold:(i), for all ;(ii), for all ;(iii).

Lemma 2.10 (see [28]). Let be a real uniformly smooth and strictly convex Banach space and a nonempty closed convex subset of . Let be a relatively quasi-nonexpansive mapping. Then is a closed convex subset of .

3. Strong Convergence Theorems

In this section, let be two maximal monotone operators satisfying . We denote the resolvent operators of and by and for each , respectively. For each , the Yosida approximations of and are defined by and , respectively. It is known that

Theorem 3.1. Let be a real uniformly smooth and 2-uniformly convex Banach space and a nonempty, closed, and convex subset of . Let be a -inverse strongly monotone mapping of into satisfying for all and . Let be a monotone continuous mapping, and let be two countable families of relatively quasi-nonexpansive mappings from into itself satisfying NST-conditions such that . Suppose that , where is the constant in (2.4). Let for some and satisfy . Let be the sequence generated by where is the normalized duality mapping, , and are four sequences in , and is defined by (*). The following conditions hold:(i); (ii); (iii).

Then converges strongly to .

Proof. We have the following steps.Step 1. First we prove that and are both closed and convex and .
In fact, it follows from Lemmas 2.4, 2.5, 2.8, and 2.10 that is closed and convex. It is obvious that is closed and convex for each . Let . For any , is equivalent to which implies that is closed and convex for each . Next, we prove that . For any given , by Lemma 2.9 we have Since are relatively quasi-nonexpansive, from the definition of , the convexity of , and (3.5), we have Moreover, it follows from Lemmas 2.2 and 2.6 that Since , is -inverse strongly monotone, from the above inequality, Lemma 2.1, and the fact that for all and , we obtain From (3.7)-(3.9), we have So from (3.6) and (3.10), we have By Lemma 2.8(5) and (3.11), we have Therefore, ; that is for each .
Next we prove that for each by induction. For , thus . Suppose that for some . Since , by Lemma 2.2, for any , we have Since , for any , we have which implies that , that is, for each .Step 2. Next we prove that .
Similar to the proof of Step  3 in [19, Theorem  3.1], we have that is nondecreasing and bounded and . So is bounded, then, by (3.5)–(3.12), we obtain that , , and are all bounded. Furthermore, and are both bounded. By Lemma 2.3, we have Since and by condition (i), we have which together with Lemma 2.3 implies that . So we obtain Let . It follows from the boundedness of that and are bounded. Let . By Lemma 2.7, (3.5), and (3.9), for any , we obtain It follows from (3.10), (3.12), and (3.19) that From (3.17) and (3.20), we have which together with condition (iii) implies that Since is uniformly continuous on bounded sets, then By (3.18) and (3.19), we have It follows from (3.20) and (3.24) thatss So by (3.17) and (3.25), we obtain which together with conditions (i) and (ii) implies that Since is uniformly continuous on bounded sets, then It follows from (3.2) that Therefore, from (3.27) and condition (i), we have Thus, From (3.7) and (3.9), we have which together with (3.31) and condition (i) implies that Hence, From Lemmas 2.1, 2.2, and 2.6, (3.34), and the fact that for all and , we have This implies that Combining (3.28) and (3.36), we have It follows from (3.6), (3.7), and (3.12) that By Lemma 2.9, (3.9), and (3.38), we have So, by (3.17) and conditions (i) and (ii), we have which implies that Combining (3.37) and (3.41), we obtain It follows from (3.6), (3.10), and (3.12) that which implies that Combining the above inequality, (3.10), and Lemma 2.9, we have By conditions (i) and (iii), (3.17), and (3.45), we have which implies that It follows from (3.23) and (3.47) that Step 3. Now we show that , where
Indeed, since is bounded and is reflexive, we know that . For any arbitrary , there exists a subsequence of such that . From (3.36), we have . Since satisfy NST-conditions, from (3.42) and (3.48) we have .
Let be an operator as follows: By Lemma 2.5, is maximal monotone and . Let . Since , we have . Moreover, implies that
On the other hand, it follows from and Lemma 2.2 that and hence So, from (3.51), (3.53), and being -Lipschitz continuous, we obtain Since is uniformly continuous on bounded sets, by (3.36) and replacing by in (3.54), as we have . Thus, , and hence .
Next we show that . Let , . It follows from (3.2) that hence, from (3.22) and condition (i), we have which implies that From (3.17), (3.31), and (3.57), we obtain Thus, . Since is uniformly continuous on bounded sets, from (3.58) we have . Therefore, it follows from that . Since , we have Combining the above inequality, (A2) and (A4), we get Replacing by and taking the limit as in the above inequality and by (A4), we have . For any and , define . So . From (A1) and (A4), we have that is, . Thus, from (A3), let , we have , which implies that .
From (3.31), (3.36), (3.41), and (3.47), similar to the proof of Step  5 in Theorem  3.1 of [19], we have . Therefore, .Step 4. Finally we prove that converges strongly to .
It follows from Step  6 in Theorem  3.1 of [19] that we have the conclusion. This completes the proof.