Abstract

We propose a new strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence theorem of proposed algorithms is investigated without the Lipschitz condition for the bifunctions. Our results complement many known recent results in the literature.

1. Introduction

Let be a real Hilbert space endowed with an inner product and a norm associated with this inner product, respectively. Let be a nonempty closed convex subset of , and let be a bifunction from to such that for all . An equilibrium problem in the sense of Blum and Oettli [1] is stated as follows: Problem of the form (1) on one hand covers many important problems in optimization as well as in nonlinear analysis such as (generalized) variational inequality, nonlinear complementary problem, and nonlinear optimization problem, just to name a few. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. On the other hand, it is rather convenient for reformulating many practical problems in economics, transportation, and engineering (see [1, 2] and the references quoted therein). We denote the set of solutions of the problem (1) by .

The existence of solution and its characterizations can be found, for example, in [3], while the methods for solving problem (1) have been developed by many researchers [4, 5]. On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems [6]. The problem of finding a common point in the solution set of problem and the set of fixed points of a nonexpansive mapping recently becomes an attractive subject, and various methods have been developed for solving this problem. Most of the existing algorithms for this problem are based on the proximal point method applying to equilibrium problem combining with a Mann’s iteration to the problem of finding a fixed point of .

In 2006, S. Takahashi and W. Takahashi [7] proposed an iterative scheme under the name viscosity approximation methods for finding a common element of set of solutions of (1) and the set of fixed points of nonexpansive mapping in a real Hilbert space . This method generated an iteration sequence starting from a given initial point and computed as where is a contraction of into itself and the sequences of parameters and were chosen appropriately. Under certain choice of and , the authors showed that two iterative sequences and converge strongly to , where denotes the projection onto .

Recently, Anh in [8] proposed to use the extragradient-type iteration instead of the proximal point iteration given in [9] for solving problem . More precisely, given , the proximal point iteration given in [9] is replaced by the two following mathematical programs, which seems numerically easier than previous ones. More precisely, the following algorithm is suggested in [8]: It was proved that if is pseudomonotone and satisfies the Lipschitz-type condition, there are Lipschitz constants and if then the sequence strongly converges to a solution of problem . Recently, Anh and Muu [10] emphasized that the Lipschitz-type condition (4), in general, is not satisfied, and if yes, finding the constants and is not an easy task. Furthermore, solving the strongly convex programs (3) is expensive except special cases when has a simple structure. They suggested and studied a new algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. More precisely, they introduced the following algorithm: where stands for -subdifferential of the convex function at and , , , and were chosen appropriately. Under the certain conditions, converse strongly to a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space.

On the other hand, the problem of finding a common fixed point element of a finite family of self-mappings () is expressed as follows: where is the set of the fixed points of the mapping (). Let us denote by the solution sets of the fixed-point problem (6). Problem of finding a fixed point of a mapping or a family of mappings is a classical problem in nonlinear analysis. The theory and solution methods of this problem can be found in many research papers and monographs (see [11]). The problem of finding a common fixed point of a finite sequence of mappings has been studied by many researchers. For instance, in 2005, Blum and Oettli [1] proposed an iterative algorithm for finding a common fixed point of strict pseudocontraction mapping (). The method computed a sequence starting from and taking where the sequence of parameters was chosen in a specific way to ensure the convergence of the iterative sequence . The authors proved that the sequence converges weakly to a point . Very recently, Anh et al. [12] suggested and analyzed an algorithm for finding a common solution of two problems (1) and (6). Typically, this problem can be written as follows: They presented an algorithm for finding a solution of problem (9). More precisely, they suggested the following algorithm: where is pseudomonotone and continuous on and Lipschitz-type continuous on , , for all , and are the sequences of parameters which were chosen appropriately. Under certain conditions on the parameters and , they proved that a sequence converges weakly to a solution of (9).

In this paper, motivated by the idea of Anh and Muu [10] and Anh et al. [12], we propose a new algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence of proposed algorithms is investigated under certain assumptions. Our results complement many known recent results in the literature.

2. Preliminaries

Let be a nonempty convex subset of a Hilbert space . We write to indicate that the sequence converges weakly to as and to indicate that the sequence converges strongly to as . Since is closed, convex, for any , there exists a unique point in , denoted by satisfying is called the metric projection of to . It is well known that satisfies the following properties: The concept of strict pseudocontraction is considered in [13], defined as follows.

Definition 1. Let be a nonempty closed convex subset of a real Hilbert space . A mapping is said to be a strict pseudocontraction if there exists a constant such that where is the identity mapping on . If , then is called nonexpansive on .

The following proposition lists some useful properties of a strict pseudocontraction mapping.

Proposition 2 (see [13]). Let be a nonempty closed convex subset of a real Hilbert space ; let be a -strict pseudocontraction, and, for each , is a -strict pseudocontraction for some . Then, (i)satisfies the following Lipschitz condition: (ii) is demiclosed at . That is, if the sequence contains in such that and , then ;(iii)the set of fixed points is closed and convex;(iv)if () and , then is a -strict pseudocontraction with ;(v)if is chosen as in (iv) and has a common fixed point, then

Lemma 3 (see [14]). Suppose that and are two sequences of nonnegative real numbers such that where . Then, the sequence is convergent.

The following idea of the -subdifferential of convex functions can be found in the work of Brøndsted and Rockafellar [15] but the theory of -subdifferential calculus was given by Hiriart-Urruty [16].

Definition 4. Consider a proper convex function . For a given , the -subdifferential of at is given by

Remark 5. It is known that if the function is proper lower semicontinuous convex, then, for every , the -subdifferential is a nonempty closed convex set (see [17]).

3. Main Results

Now, we are in a position to state and prove the main strong convergence theorem for the given iterative scheme.

Assumption 6. Let the bifunction be satisfied by the following conditions: (i)for each , and is lower semicontinuous convex on ;(ii)if is bounded and as , then the sequence with is bounded;(iii) is pseudomonotone on with respect to every solution of and satisfies the following condition, called strict paramonotonicity property: (iv)for each , is weakly upper semicontinuous on the open set .

Assumption 7. Consider the following:
(i) for each , is -strict pseudocontraction for some ;
(ii) the solution set of the problem is nonempty; that is,

Assumption 8. Suppose that the sequences , , , , and of nonnegative numbers satisfy the following conditions: (i), , and , where ;(ii), , and ;(iii);(iv) for all .

Algorithm 9. Now, the iteration scheme for finding a common point in the set of solutions of problem and the set of common fixed points of -strict pseudocontraction, for each , can be written as follows:

Remark 10 (see [10], Remark 2.1). Consider the following:(i)if is pseudomonotone on with respect to the solution set of the problem , then, under Assumptions 6 (i) and (iv), the set is convex;(ii)Assumption 6 (ii) is true if whenever Assumption 6 (i) is satisfied and the bifunction is continuous on ;(iii)Assumption 6 (iii) is true if is pseudomonotone on and satisfies the paramonotone property as (iv)since is lower semicontinuous convex on , applying Remark 5, we conclude that . Thus, Algorithm 9 is well defined.

Theorem 11. Suppose that Assumptions 6–8 are satisfied. Then, the sequences and generated by Algorithm 9 converge strongly to the same point , where .

Proof. The proof is divided into several steps as follows.
Step 1. For every and every , we show that and there exists the limit For each , let By Proposition 2, we see that is a -strict pseudocontraction on and the sequence generated by Algorithm 9 can be rewritten as Then, for all , we have Since and , we have Combining this inequality with (30) yields Using again and , we have
which gives that . Consequently,
This together with (32) implies that Since , and for all , we have Combining (35) and (36), we obtain that On the other hand, since , that is, for all , by pseudomonotonicity of with respect to , we have for all . Replacing by , we get . Then, from (37), it follows that By Assumptions 8 (ii) and (iii), we found that . Using Lemma 3 and (38), we arrive at the existence of
Step 2. For every , we show that Since is pseudomonotone on and , we have . By Step  1, We have Summing up the above inequalities for every , we obtain that It follows from the boundedness of the sequences and that we can assume that for a constant . Thus, which together with (43) implies Thus, Then, by and , we can deduce that as desired.
Step 3. For any , suppose that is the subsequence of such that and, without loss of generality, we may assume that as for some . We show that solves . To this end, since is weakly upper semicontinuous, we have On the other hand, since is pseudomonotone with respect to and , we have From (49) and (50), we can conclude that . By Assumption 6, we can deduce that is a solution of as well.
Step  4. We prove that any weakly cluster point of the sequence is a common fixed point of -strict pseudocontraction, for each . In particular, . Let be any weakly cluster point of and let be a subsequence of weakly converging to . By convexity and the closedness of , is weakly closed. Hence, . We first show that It follows from (29) that which gives that Consequently, By Proposition 2 (i), we arrive at the following: Thus, we obtain For each , we suppose that converges to as such that . Then, for each and , we have It follows from (56) that We obtain that By Proposition 2 (ii), we have It then follows from Proposition 2 (v) that we have In particular, we conclude that .
Step  5. Finally, we prove that It follows from (38) that, for all , where for all and . Now, using property (14) of the metric projection, we have Since , , and as , we obtain that For the simplicity of notation, let for each . Then, for all , since is convex, we have , and therefore Replacing with in (63), we can obtain the following: Combining this inequality with (66), we have which gives that which implies also that is a Cauchy sequence. Hence, strongly converges to some point . However, since , letting , we obtain in the limit that Therefore, . Then, from (65), we can conclude that . Finally, since , we have .