Viscosity-Projection Method for a Family of General Equilibrium Problems and Asymptotically Strict Pseudocontractions in the Intermediate Sense

In this paper, a Meir-Keeler contraction is introduced to propose a viscosity-projection approximation method for finding a common element of the set of solutions of a family of general equilibrium problems and the set of fixed points of asymptotically strict pseudocontractions in the intermediate sense. Strong convergence of the viscosity iterative sequences is obtained under some suitable conditions. Results presented in this paper extend and unify the previously known results announced by many other authors.


Introduction
Let be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Let be a nonempty closed convex subset of . Let : → be a nonlinear mapping and : × → R be a bifunction, where R denotes the set of real numbers. We consider the following generalized equilibrium problem: Find ∈ such that ( , ) + ⟨ , − ⟩ ≥ 0, ∀ ∈ . (1) We use EP( , ) to denote the set of solution of problem (1). If ≡ 0, the zero mapping, then the problem (1) reduces to the normal equilibrium problem: Find ∈ such that ( , ) ≥ 0, ∀ ∈ .
We use VI( , ) to denote the set of solution of problem (3). The generalized equilibrium problem (1) is very general in the sense that it includes, as special cases, saddle point problems, variational inequalities, optimization problems, mini-max problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [1][2][3][4]).
Recall that a nonlinear mapping : → is said to be nonexpansive if is said to be uniformly -Lipschitz continuous if there exists a constant > 0 such that is said to be asymptotically nonexpansive if there exists a sequence ∈ [1, ∞) with → 1 as → ∞ such that − ≤ − , ≥ 1, ∀ , ∈ . (6) is said to be asymptotically nonexpansive in the intermediate sense [5] if it is continuous and the following inequality holds: Putting = max{0, sup , ∈ (‖ − ‖ − ‖ − ‖)}, we see that → 0 as → ∞. Then scheme (7) is reduced to the following: 2 The Scientific World Journal The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [5] as a generalization of the class of asymptotically nonexpansive mappings. It is known that, if is a nonempty bounded closed convex subset of a real Hilbert space , then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point (see, e.g., [6]).
Recall also that is said to be a -strict pseudocontraction [7,8] if there exists a coefficient ∈ [0, 1) such that is said to be an asymptotically -strict pseudocontraction [9,10] if there exists a sequence ∈ [1, ∞) with → 1 as → ∞ and a constant ∈ [0, 1) such that is said to be an asymptotically -strict pseudocontraction in the intermediate sense [11,12] we see that → 0 as → ∞. Then scheme (11) is reduced to the following: We use Fix( ) to denote the set of fixed point of , that is, Fix( ) = { ∈ : = }. The class of asymptotically strict pseudocontractions in the intermediate sense was introduced as a generalization of the asymptotically strict pseudocontractions and asymptotically nonexpansive in the intermediate sense. Clearly, a nonexpansive mapping is a 0strict pseudocontraction, and an asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction. (see, e.g., [7][8][9][10][11][12]).
Fixed point technique represent an important tool for finding the approximate solution of equilibrium problem and its variant forms, which have been studied extensively in recent years due to their applications in physics, economics, optimization, and pure and applied sciences. Some numerical methods have been proposed for finding a common element of the set of fixed point of various types of nonexpansive mappings and the set of solution of equilibrium problems with bifunctions satisfying certain conditions; see [8][9][10][11][12][13][14][15][16][17][18][19][20] and references therein.
Recently, Sahu et al. [11] considered a new iterative scheme for asymptotically strictly pseudocontractive mappings in the intermediate sense.
On the other hand, Moudafi [13] introduced the following viscosity approximation method for fixed point problem of nonexpansive mapping where is a contractive mapping. He proved that the viscosity iterative sequence { } convergence strongly to a fixed point of , which is the unique solution of the variational inequality: Furthermore, S. Takahashi and W. Takahashi [14] and Inchan [15] modified the viscosity approximation methods for finding a common element of the set of fixed point problems and equilibrium problems. In 2012, Kimura and Nakajo [16] introduced a Meir-Keeler contraction and proposed a modified viscosity approximations by the shrinking projection method in Hilbert spaces, the socalled viscosity-projection method. To be more precise, they proved the following theorem.
In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way: where Our purpose is not only to extend the viscosity-projection method with a Meir-Keeler contraction to the case of a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, but also to obtain a strong convergence theorem by using the proposed schemes under some appropriate conditions. Results presented in this paper extend and unify the corresponding ones of [10][11][12][13]16].

Preliminaries
Let be a nonempty closed convex subset of a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. We use notation ⇀ for weak convergence and → for strong convergence of a sequence. For every point ∈ , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto defined by ( ) = arg min ∈ ‖ − ‖. It is well known that is nonexpansive mapping, and = is equivalent to (see, e.g., [21]) the following: Recall that a mapping : → is said to be monotone if is said to be -strongly monotone if there exists a constant > 0 such that is said to be -inverse strongly monotone if there exists a constant > 0 such that It is easy to see that if is an -inverse strongly monotone mapping from into , then is 1/ -Lipschitz continuous.
Lemma 3 (see [8]). Let be a nonempty closed convex subset of a real Hilbert space . For any , , ∈ and given also a real number ∈ R, the set is closed and convex.
Lemma 4 (see [11] Lemma 6 (see [11]). Let be a nonempty closed convex subset of a real Hilbert space and : → be an asymptotically Recall also that a mapping of a complete metric space It is known that has a unique fixed point (see, e.g., [22] We know that Meir-Keeler contraction is a generalization of contraction, and the following result, which extends the Banach contraction principle, is proved in [23]. Lemma 7 (see [23]). A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.
Step 2. We show that is closed convex subset of for each ≥ 1. By the assumption of +1 , it is easy to see that is closed for each ≥ 1. We only show that is convex for each ≥ 1. It is obvious that 1 = is closed and convex. Suppose that is closed and convex for some ≥ 1. For any ∈ , we see that is equivalent to Taking 1 and 2 in +1 and putting = 1 + (1 − ) 2 , it follows that 1 , 2 ∈ , and so Combing (36) and (37), we obtain that That is, In view of the convexity of , we see that ∈ . This implies that ∈ +1 . Therefore, +1 is convex. Consequently, is closed and convex for each ≥ 1.
We also obtain the following results by using the viscosityhybrid projection methods, which extend and improve the hybrid method (CQ) proposed by Sahu et al. [11] and Hu and Cai [12]. Proof. We have that and are closed convex subsets of and Ω ⊂ for every ∈ N. We only prove that Ω ⊂ for every ∈ N and that a sequence { } is well-defined. We have 1 ∈ and Ω ⊂ 1 = . Assume that ∈ and Ω ⊂ for some ∈ N. Since Ω ⊂ ∩ , there exists a unique element +1 = ∩ ( ), and hence This implies that That is, Ω ⊂ +1 . Therefore, we prove that Ω ⊂ . On the other hand, ⋂ ∞ =1 is a Meir-Keeler contraction on , there exists a unique element , it follows from Lemma 9 that → = ⋂ ∞ =1 ( ). We also have = ( −1 ) by the definition of . Therefore, as in the proof of Theorem 10, we get → , and the desired conclusion follows immediately from Theorem 10. This completes the proof.

If
= = 1, we obtain the following corollary for a general equilibrium problem and asymptotically strict pseudocontraction in the intermediate sense as a special cases.