#### Abstract

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.

#### 1. 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 be a bifunction, where denotes the set of real numbers. We consider the following generalized equilibrium problem: Find such that We use to denote the set of solution of problem (1). If , the zero mapping, then the problem (1) reduces to the normal equilibrium problem: Find such that We use to denote the set of solution of problem (2). If , then the problem (1) reduces to the classical variational inequality problem: Find such that We use 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., [1–4]).

Recall that a nonlinear mapping is said to be nonexpansive if is said to be uniformly -Lipschitz continuous if there exists a constant such that is said to be asymptotically nonexpansive if there exists a sequence with as such that is said to be asymptotically nonexpansive in the intermediate sense [5] if it is continuous and the following inequality holds: Putting , we see that as . Then scheme (7) is reduced to the following: 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 such that is said to be an asymptotically -strict pseudocontraction [9, 10] if there exists a sequence with as and a constant such that is said to be an asymptotically -strict pseudocontraction in the intermediate sense [11, 12] if there exists a sequence with as and a constant such that Putting , we see that as . Then scheme (11) is reduced to the following: We use to denote the set of fixed point of , that is, . 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 0-strict pseudocontraction, and an asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction. (see, e.g., [7–12]).

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 [8–20] and references therein.

In 2009, Qin et al. [10] introduced the following explicit iterative algorithm for finding a common fixed point of a finite family of asymptotically -strictly pseudocontractions for each where , is a sequence in (0,1) and , . They also obtain weak and strong convergence theorems based on the cyclic scheme above.

Recently, Sahu et al. [11] considered a new iterative scheme for asymptotically strictly pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.

Theorem SXY. *Let be a nonempty closed and convex subset of a real Hilbert space and be a uniformly continuous asymptotically -strictly pseudocontractive mapping in the intermediate sense with a sequence such that is nonempty and bounded. Let be a sequence in such that for all . Let be a sequence generated by the following (CQ) algorithm:
**
where and . Then, converges strongly to , where is metric projection from onto .**In 2011, Hu and Cai [12] modified schemes (13) and (14) to the case of asymptotically strictly pseudocontractive mappings in the intermediate sense concerning the equilibrium problem and proposed the following modified hybrid method:
**
where as and . Moreover, they obtained convergence theorems under some suitable conditions.**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 so-called viscosity-projection method. To be more precise, they proved the following theorem.*

Theorem KN. *Let be a nonempty closed convex subset of , and let be a sequence of mappings of into itself with which satisfies the following condition: there exists with such that for every , and . Let be a Meir-Keeler contraction of into itself, and let be a sequence generated by
**
for each . For every sequence and and imply that . Then, converges strongly to , which satisfies .*

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 as and .

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 [10–13, 16].

#### 2. 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 . 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 such that is said to be -inverse strongly monotone if there exists a constant such that It is easy to see that if is an -inverse strongly monotone mapping from into , then is -Lipschitz continuous.

To study the generalized equilibrium problem (1), we may assume that the bifunction satisfies the following conditions: (A1)for all ; (A2) is monotone, that is, for all ; (A3) for each , ; (A4) for each is convex and lower semi-continuous.

Lemma 1 (see [1, 3]). * Let be a bifunction satisfying (A1)–(A4). Then, for any and , there exists such that
**
Further, if , then the following hold: *(1)* is single-valued; *(2)* is firmly nonexpansive, that is, for all ; *(3)*; *(4)* is closed and convex.*

Lemma 2 (see [8]). *In a Hilbert space , there hold the following identities: *(i)*, for all ; *(ii)*, for all , for all .*

Lemma 3 (see [8]). *Let be a nonempty closed convex subset of a real Hilbert space . For any and given also a real number , the set
**
is closed and convex.*

Lemma 4 (see [11]). * Let be a nonempty closed convex subset of a real Hilbert space and be a uniformly -Lipschitz continuous and asymptotically -strict pseudocontraction in the intermediate sense. Then is closed and convex.*

Lemma 5 (see [11]). * Let be a nonempty closed convex subset of a real Hilbert space and be a uniformly -Lipschitz continuous and asymptotically -strict pseudocontraction in the intermediate sense. Then is demiclosed at zero, that is, if the sequence such that and as , then .*

Lemma 6 (see [11]). *Let be a nonempty closed convex subset of a real Hilbert space and be an asymptotically -strict pseudocontraction in the intermediate sense with . Then
**Recall also that a mapping of a complete metric space into itself is called a contraction with coefficient if , for all . It is known that has a unique fixed point (see, e.g., [22]). On the other hand, Meir and Keeler [23] defined the following mapping, called the Meir-Keeler contraction. A mapping is called a Meir-Keeler contraction if, for every , there exists such that implies that
**
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.*

Lemma 8 (see [24]). * Let be a Meir-Keeler contraction on a convex subset of a Banach space . Then, for every , there exists such that implies that
**Let be a sequence of nonempty closed convex subsets of . We define a subset of as follows: if and only if there exists such that and for all . Similarly, a subset of is defined by if and only if there exists a subsequence of and a sequence such that and for all . If satisfies
**
it is said that converges to in the sense of Mosco [25], and we write . One of the simplest examples of Mosco convergence is a decreasing sequence with respect to inclusion. The Mosco limit of such a sequence is . For more details, see [26].*

Lemma 9 (see [27]). *Let be a sequence of nonempty closed convex subsets of . If exists and is nonempty, then, for each , converges strongly to .**For the rest of this paper, let be a bifunction satisfying (A1)–(A4) and be an -inverse strongly monotone mapping, for some . For each and , define a mapping as follows:
**
It follows from Lemma 1 that for each .**Let be a uniformly -Lipschitz continuous and asymptotically -strict pseudocontractive mapping in the intermediate sense with the sequences such that and such that , for some , that is,
**
Remark that , , and .*

#### 3. Main Results

Theorem 10. *Let be a nonempty closed convex subset of Hilbert space . Let be a bifunction satisfying (A1)–(A4), and let be an -inverse strongly monotone mapping, for each . Let be a uniformly -Lipschitz continuous and asymptotically -strict pseudocontractive mapping in the intermediate sense with the sequences and for each . If is a Meir-Keeler contraction of into itself and is nonempty and bounded. Assume that are sequences in such that , and such that , for each and . Then the sequence generated by (19) converges strongly to .*

*Proof. *We split the proof into six steps.*Step 1.* We prove that exists a unique fixed point. To do this, we first show that is nonexpansive for each . Indeed,
It follows that is nonexpansive. By Lemma 1, we know that is closed and convex. We also know from Lemma 4 that is closed, and convex. Hence, is a nonempty, closed and convex subset of . Consequently, is well-defined. Since is nonexpansive, the composed mapping of into itself is a Meir-Keeler contraction on ; see [24, Proposition 3]. By Lemma 7, there exists a unique fixed point of .*Step 2*. We show that is closed convex subset of for each . By the assumption of , it is easy to see that is closed for each . We only show that is convex for each . It is obvious that is closed and convex. Suppose that is closed and convex for some . For any , we see that
is equivalent to
Taking and in and putting , it follows that , and so
Combing (36) and (37), we obtain that
That is,
In view of the convexity of , we see that . This implies that . Therefore, is convex. Consequently, is closed and convex for each .*Step 3.* We show that for each . Put for every and for all . Note that is nonexpansive. Therefore, . It is obvious that . Suppose that for some . Taking , it follows from Lemma 1 that
From (19), we observe that
By virtue of convexity of , combining (40) and (41), we have
where as and . Setting in (40)–(43), it follows that . Therefore, for each .*Step 4.* Next, we prove that , where . Note that is a closed convex subset of and for all . Thus, is well-defined. Since the composed mapping is a Meir-Keeler contraction on , there exists a unique fixed point by Lemma 7. Let for each . We get , since for every . Thus, from Lemma 9, we get

We prove that in the following. If it were so, it would hold that . Let , such that . By the definition of Meir-Keeler contraction, there exists with such that implies that
From Lemma 8, there exists such that implies that
By (44), there exists such that for each . As in the proof of [24, Theorem 8], we consider the following two cases: (i)There exists such that . (ii) for every .

In case (i), it holds that
Thus we get
which means that
This is a contradiction. In case (ii), we have
Thus we get
It follows from (44) that
This is a contradiction again. Therefore, we obtain that . Moreover, since , we have for each . By , we have
which is equivalent to from the property of metric projection.*Step 5.* Now, we prove that . From as , we have
Since , we have
It follows from (55) and Lemma 2 that
Since as and (54), we obtain

For each , , it follows from (33) and (40) that
By (40), (42), and (58), we have
which implies that
Since for each and as . From (57), we have
On the other hand, it follows from the nonexpansive and Lemma 1 that
which implies that
From (40), (42), and (63), we obtain
which implies that
It follows from (57) and (65) that
Note that
Therefore,
*Step 6.* Finally, we prove that . To do this, we first show that . Note that
From (54) and (68), we get
It follows that
For any positive integer , note that , where . By (19) and the conditions and , we have
From (57) and (68), we get
By the fact that and , we observe that
Applying (71), (73), and Lemma 6, we obtain
By the uniformly -Lipschitzian of , we have
It follows from (73) and (75) that
Since
Combining (71) and (77), we obtain
Moreover, for each , we have
This implies that
Note that as . It follows from (81) and Lemma 5 that .

Next, we show that . From Lemma 1 and since , , we have
By (A2), we have
Let for all and . This implies that . Then, we have
From (66), we have as . Moreover, by (A4) and the monotonicity of , we obtain
Using (A1), (A4), and (85), we obtain
and hence
Let , from (A3) and (87), we have
This implies that , . Therefore, . Consequently, we obtain that . This completes the proof.

We also obtain the following results by using the viscosity-hybrid projection methods, which extend and improve the hybrid method (CQ) proposed by Sahu et al. [11] and Hu and Cai [12].

Theorem 11. *Let be a nonempty closed convex subset of Hilbert space . Let be a bifunction satisfying (A1)–(A4), and let be an -inverse strongly monotone mapping, for each . Let be a uniformly -Lipschitz continuous and asymptotically -strict pseudocontractive mapping in the intermediate sense with the sequences and for each . If is a Meir-Keeler contraction of into itself and is nonempty and bounded. Let be a sequence defined by
**
where as and . Assume that and are sequences in such that , and such that , for each . Then the sequence generated by (89) converges strongly to .*

* Proof. *We have that and are closed convex subsets of and for every . We only prove that for every and that a sequence is well-defined. We have and . Assume that and for some . Since , there exists a unique element , and hence
This implies that
That is, . Therefore, we prove that .

On the other hand, is a Meir-Keeler contraction on , there exists a unique element by Lemma 7. Let for each . Since , it follows from Lemma 9 that . We also have