Abstract

This paper is concerned with a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, which extends and improves that of Qin et al. (2010) and many others.

1. Introduction

Throughout this paper, we always assume that be a nonempty closed convex subset of a real Hilbert space with inner product and norm denoted by and , respectively. For a sequence in , we denote the strong convergence and the weak convergence of to by and , respectively. The domain of the function is the set Let be a proper extended real-valued function, and let be a bifunction from into such that , where is the set of real numbers. The so-called the mixed equilibrium problem is to find such that The set of solution of problem (1.2) is denoted by , that is, It is obvious that if is a solution of problem (1.2) then . As special cases of problem (1.2), we have the following. (i)If , then problem (1.2) is reduced to find such that We denote by the set of solutions of equilibrium problem, which problem (1.4) was studied by Blum and Oettli [1]. (ii)If for all where a mapping , then problem (1.4) is reduced to find such that We denote by the set of solutions of variational inequality problem, which problem (1.5) was studied by Hartman and Stampacchia [2]. (iii)If , then problem (1.2) is reduced to find such that We denote by the set of solutions of minimize problem.

Recall that is the metric projection of onto ; that is, for each there exists the unique point in such that . A mapping is called nonexpansive if for all , and uniformly L-Lipschitzian if there exists a constant such that for each , for all , and a mapping is called a contraction if there exists a constant such that for all . A point is a fixed point of provided . We denote by the set of fixed points of ; that is, . If is a nonempty bounded closed convex subset of and is a nonexpansive mapping of into itself, then is nonempty (see [3]).

Iterative methods are often used to solve the fixed point equation . The most well-known method is perhaps the Picard successive iteration method when is a contraction. Picard's method generates a sequence successively as for all with chosen arbitrarily, and this sequence converges in norm to the unique fixed point of . However, if is not a contraction (for instance, if is a nonexpansive), then Picard's successive iteration fails, in general, to converge. Instead, Mann's iteration method for a nonexpansive mapping (see [4]) prevails, generates a sequence recursively by where chosen arbitrarily and the sequence lies in the interval . Recall that a mapping is said to be (i)asymptotically pseudocontractive [5, 6] if there exists a sequence with such that it is easy to see that (1.8) is equivalent to for all , (ii)asymptotically pseudocontractive in the intermediate sense [7] if there exists a sequence with such that if we define then and it follows that (1.10) is reduced to it is easy to see that (1.12) is equivalent to for all ; it is obvious that if for all , then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.

The Mannâ€™s algorithm for nonexpansive mappings has been extensively investigated (see [8â€“10] and the references therein). One of the well-known results is proven by Reich [10] for a nonexpansive mapping on , which asserts the weak convergence of the sequence generated by (1.7) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition . It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [11]). The strong convergence guaranteed has been proposed by Nakajo and Takahashi [12], they modified the Mann's iteration method (1.7) which is to find a fixed point of a nonexpansive mapping by a hybrid method, which called that the shrinking projection method (or the CQ method) as the following theorem.

Theorem NT. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping of into itself such that . Suppose that chosen arbitrarily and the sequence defined by where . Then converges strongly to .

Subsequently, Schu [5] modified Ishikawa's iteration method (see [13]) for the class of asymptotically pseudocontractive mappings as the following theorem.

Theorem S. Let be a nonempty bounded closed convex subset of a real Hilbert space . Let be a completely continuous uniformly such that and asymptotically pseudocontractive mapping defined as in (1.9) with the sequence such that . Let for all . Suppose that chosen arbitrarily and the sequence defined by where such that for some and and . Then converges strongly to some fixed point of .

Quite recently, Zhou [14] showed that every uniformly -Lipschitzian and asymptotically pseudocontractions which are also uniformly asymptotically regular has a fixed point and the fixed point set is closed and convex, and he also introduced iterative scheme to find a fixed point of a uniformly L-Lipschitzian and asymptotically pseudocontractive mapping as the following theorem.

Theorem Z. Let be a nonempty bounded closed convex subset of a real Hilbert space . Let be a uniformly L-Lipschitzian such that and asymptotically pseudocontractive mapping with a fixed point defined as in (1.8) with the sequence such that . Suppose that chosen arbitrarily and the sequence defined by where such that . Then converges strongly to .

To be more precisely, Qin et al. [7] showed in the framework of a real Hilbert spaces for the uniformly -Lipschitzian and asymptotically pseudocontractive mapping in the intermediate sense that the fixed point set is closed and convex (see Lemma 1.4 in [7]) and the demiclosedness principle holds (see Lemma 1.5 in [7]), and they also introduced an iterative scheme to find a fixed point of a uniformly -Lipschitzian such that and asymptotically pseudocontractive mapping in the intermediate sense on a nonempty bounded closed convex defined as in (1.13) with the sequences and such that and , and let for all as follows: where . They proved that under the sequences such that for some and , if is nonempty, then the sequence generated by (1.17) converges strongly to a fixed point of .

Inspired and motivated by the works mentioned above, in this paper, we introduce a general iterative scheme (3.1) below to find a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves Qin et al. [7] and many others.

2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . For solving the mixed equilibrium problem, let us assume that the bifunction , the function and the set satisfy the following conditions: â€‰ for all ; is monotone; that is, for all ; for each , for each is convex and lower semicontinuous;for each is weakly upper semicontinuous; for each and , there exists a bounded subset and such that for any , is a bounded set.

Lemma 2.1 (see [15]). Let be a Hilbert space. For any and , we have

Lemma 2.2 (see [3]). Let be a nonempty closed convex subset of a Hilbert space . Then the following inequality holds:

Lemma 2.3 (see [16]). Let be a nonempty closed convex subset of a Hilbert space , satisfying the conditions (A1)â€“(A5), and let be a proper lower semicontinuous and convex function. Assume that either or holds. For , define a mapping as follows: for all . Then, the following statement hold: (1)for each , ;(2) is single-valued; (3) is firmly nonexpansive; that is, for any , (4); (5) is closed and convex.

Lemma 2.4 (see [3]). Every Hilbert space has Radon-Riesz property or Kadec-Klee property, that is, for a sequence with and then .

Lemma 2.5 (see [7]). Let be a nonempty closed convex of a real Hilbert space , and let be a uniformly -Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that is nonempty. Then is demiclosed at zero. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .

3. Main Results

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space , a bifunction from into satisfying the conditions (A1)â€“(A5), and a proper lower semicontinuous and convex function with either or holds. Let be a uniformly L -Lipschitzian such that and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences and such that and . Let for all . Assume that be a nonempty bounded subset of . For chosen arbitrarily, suppose that , and are generated iteratively by where and satisfying the following conditions: (C1) such that for some and ; (C2) for some ;(C3). Then the sequences , and converge strongly to .

Proof. Pick . Therefore, by (3.1) and the definition of in Lemma 2.3, we have and by , and Lemma 2.3 (4), we have By (3.2), (3.3), and the nonexpansiveness of , we have By (3.3), Lemma 2.1, the uniformly -Lipschitzian of , and the asymptotically pseudocontractive mapping in the intermediate sense of , we have Substituting (3.6) and (3.7) into (3.5), and by the condition (C1) and (3.4), we have where and .
Firstly, we prove that is closed and convex for all . It is obvious that is closed, and by mathematical induction that is closed for all , that is is closed for all . Let . Since for any , is equivalent to for all . Therefore, for any and , we have for all , and we have for all . Since is convex, and by putting in (3.9), (3.10), and (3.11), we have is convex. Suppose that is given and is convex for some . It follows by putting in (3.9), (3.10), and (3.11) that is convex. Therefore, by mathematical induction, we have is convex for all , that is, is convex for all . Hence, we obtain that is closed and convex for all .
Next, we prove that for all . It is obvious that . Therefore, by (3.1) and (3.8), we have , and note that , and so . Hence, we have . Since is a nonempty closed convex subset of , there exists a unique element such that . Suppose that is given such that , and for some . Therefore, by (3.1) and (3.8), we have . Since , therefore, by Lemma 2.2, we have for all . Thus, by (3.1), we have , and so . Hence, we have . Since is a nonempty closed convex subset of , there exists a unique element such that . Therefore, by mathematical induction, we obtain for all , and so for all , and we can define for all . Hence, we obtain that the iteration (3.1) is well defined.
Next, we prove that is bounded. Since for all , we have for all . It follows by that for all . This implies that is bounded, and so are , and .
Next, we prove that and as . Since , therefore, by (3.13), we have for all . This implies that is a bounded nondecreasing sequence; there exists the limit of , that is for some . Since , therefore, by (3.1), we have It follows by (3.15) that Therefore, by (3.14), we obtain Indeed, from (3.1) we have substituting into (3.18) and into (3.19), we have Therefore, by the condition (A2), we get It follows that Thus, we have It follows by the condition (C2) that where . Therefore, by the condition (C3) and (3.17), we obtain
Next, we prove that and as . Since , by (3.1), we have it follows by the condition (C1) that Since and the condition (C1), we have . Therefore, from (3.27) by (3.17) and , we obtain By the uniformly -Lipschitzian of , we have Therefore, by (3.25) and (3.28), we obtain
Next, we prove that , and as . From (3.26), by the condition (C1), we have it follows that Therefore, by (3.17), , and , we obtain From (3.1), we have ; therefore, by (3.28), we obtain By (3.2), (3.3), and the firmly nonexpansiveness of , we have it follows that Therefore, from (3.8), we have it follows by the condition (C1) that Therefore, by (3.33), , and , we obtain Since is bounded, there exists a subsequence of which converges weakly to . Next, we prove that . From and as by (3.30), therefore, by Lemma 2.5, we obtain . From (3.1), we have it follows by the condition (A2) that Hence, Therefore, from (3.39) and by as , we obtain For a constant with and , let . Since , thus, . So, from (3.43), we have By (3.44), the conditions (A1) and (A4), and the convexity of , we have it follows that Therefore, by the condition (A3) and the weakly lower semicontinuity of , we have as for all , and hence, we obtain , and so .
Since is a nonempty closed convex subset of , there exists a unique such that . Next, we prove that as . Since , we have for all ; it follows that Since , therefore, by (3.13), we have Since by (3.39) and , we have as . Therefore, by (3.47), (3.48) and the weak lower semicontinuity of norm, we have It follows that Since as , therefore, we have Hence, from (3.50), (3.51), Kadec-Klee property, and the uniqueness of , we obtain It follows that converges strongly to and so are , and . This completes the proof.

Remark 3.2. The iteration (3.1) is the difference with the iterative scheme of Qin et al. [7] as follows.(1)The sequence is a projection sequence of onto for all such that (2) A solving of a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems by iteration is obtained.
We define the condition (B3) as the condition (B1) such that . If , then Theorem 3.1 is reduced immediately to the following result.

Corollary 3.3. Let be a nonempty closed convex subset of a real Hilbert space , and let a bifunction from into satisfying the conditions (A1)â€“(A5) with either (B2) or (B3) holds. Let be a uniformly -Lipschitzian such that and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences and such that and . Let for all . Assume that be a nonempty bounded subset of . For chosen arbitrarily, suppose that , and are generated iteratively by where and satisfying the following conditions: such that for some and ; for some ; . Then the sequences , and converge strongly to .

If , then Corollary 3.3 is reduced immediately to the following result.

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space . Let be a uniformly -Lipschitzian such that and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences and such that and . Let for all . Assume that be a nonempty bounded subset of . For chosen arbitrarily, suppose that , and are generated iteratively by where , and such that for some and . Then the sequences and converge strongly to .

We introduce the equilibrium problem to the optimization problem: where is a nonempty closed convex subset of a real Hilbert space and let is a proper convex and lower semicontinuous. We denote by the set of solution of problem (3.56). We define the condition (B4) as the condition (B3) such that is a bifunction defined by for all . Observe that . We obtain that Corollary 3.3 is reduced immediately to the following result.

Corollary 3.5. Let is a nonempty closed convex subset of a real Hilbert space , and be a proper lower semicontinuous and convex function with either (B2) or (B4) holds. Let be a uniformly -Lipschitzian such that and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences and such that and . Let for all . Assume that be a nonempty bounded subset of . For chosen arbitrarily, suppose that and are generated iteratively by