ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 529407, 17 pagesdoi:10.5402/2011/529407

Research Article

Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

Pattanapong Tianchai

Faculty of Science, Maejo University, Chiangmai 50290, Thailand

Received 13 July 2011; Accepted 18 August 2011

Academic Editor: A. Levy

Copyright © 2011 Pattanapong Tianchai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup 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 the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

1. Introduction

Throughout this paper, we always assume that is a nonempty closed convex subset of a real Hilbert space with inner product and norm denoted 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 mixed equilibrium problem is to find such that The set of solutions 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, problem (1.4) which 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, problem (1.5) which 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 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 and generates a sequence recursively by where chosen arbitrarily and the sequence lies in the interval . Recall that a mapping is said to be as follows.(i)-strictly pseudocontractive (see [5]) if there exists a constant such that in brief, we use -SPC to denote the -strictly pseudocontractive, it is obvious that is a nonexpansive if and only if is a 0-SPC.(ii)Asymptotically -SPC (see [6]) if there exists a constant and a sequence of nonnegative real numbers with such that for all , if then is an asymptotically nonexpansive with for all ; that is, is an asymptotically nonexpansive (see [7]) if there exists a sequence with such that for all , it is known that the class of -SPC mappings, and the classes of asymptotically -SPC mappings are independent (see [8]).

The Mann's algorithm for nonexpansive mappings has been extensively investigated (see [5, 9, 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 . Recently, Marino and Xu [11] devoloped and extended Reich's result to SPC mapping in Hilbert space setting. More precisely, they proved the weak convergence of the Mann's iteration process (1.7) for a -SPC mapping on , and, subsequently, this result was improved and carried over the class of asymptotically -SPC mappings by Kim and Xu [12].

It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [13]). The strong convergence is guaranteed and has been proposed by Nakajo and Takahashi [14], 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 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 let be the sequence defined by where . Then converges strongly to . Subsequently, Marino and Xu [15] introduced an iterative scheme for finding a fixed point of a -SPC mapping as the following theorem.

Theorem MX. Let be a closed convex subset of a Hilbert space , and, be a -SPC mapping for some . Assume that . Suppose that chosen arbitrarily, and let be the sequence defined by where . Then the sequence converges strongly to . Quite recently, Kim and Xu [12] has improved and carried Theorem MX over the more wider class of asymptotically -SPC mappings as the following theorem.

Theorem KX. Let be a closed convex subset of a Hilbert space , and, be an asymptotically -SPC mapping for some and a bounded sequence such that . Assume that is a nonempty bounded subset of . Suppose that chosen arbitrarily, and let be the sequence defined by where as , , and such that . Then the sequence converges strongly to .

Recall that a discrete family of self-mappings of is said to be a Lipschitzian semigroup on if the following conditions are satisfied. (1) where denotes the identity operator on . (2). (3)There exists a sequence of nonnegative real numbers such that A discrete Lipschitzian semigroup is called nonexpansive semigroup if for all , contraction semigroup if for all and, asymptotically nonexpansive semigroup if , respectively. We use to denote the common fixed point set of the semigroup ; that is, .

Very recently, Kim [16] introduced asymptotically -SPC semigroup, a discrete family of self-mappings of which is said to be asymptotically -SPC semigroup on if, in addition to (1), (2) and the following condition are satisfied. ( 3′)There exists a constant and a bounded sequence of nonnegative real numbers with such that

Note that for both discrete asymptotically nonexpansive semigroups and discrete asymptotically -SPC semigroups, we can always assume that the Lipschitzian constants are such that for all and ; otherwise, we replace for all with . Therefore, for a single asymptotically -SPC mapping note that (1.15) immediately reduces to (1.9) by taking and such that for all and .

To be more precise, Kim also showed in the framework of Hilbert spaces for the asymptotically -SPC semigroups that is continuous on for all and that is closed and convex (see Lemma  3.2 in [16]), and the demiclosedness principle (see Theorem  3.3 in [16]) holds in the sense that if is a sequence in such that and then, , and he also introduced an iterative scheme to find a common fixed point of a discrete asymptotically -SPC semigroup and a bounded sequence such that as follows: where . He proved that under the parameter for all , if is a nonempty bounded subset of , then the sequence generated by (1.16) converges strongly to .

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 common fixed point for a discrete asymptotically -SPC semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained based on the shrinking projection method which extend and improve the corresponding ones due to Kim [16].

2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . For a sequence in , we denote the strong convergence and the weak convergence of to by and , respectively, and the weak -limit set of by .

For solving the mixed equilibrium problem, let us assume that the bifunction , the function and the set satisfy the following conditions. (A1) for all .(A2) is monotone; that is, for all .(A3)For each , (A4)For each is convex and lower semicontinuous. (A5)For each is weakly upper semicontinuous. (B1)For each and , there exists a bounded subset and such that for any , (B2) is a bounded set.

Lemma 2.1 (see [17]). Let be a Hilbert space. For any and , one has

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 [18]). Let be a nonempty closed convex subset of a Hilbert space , satisfying the conditions , 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 [16]). Let be a nonempty closed convex subset of a Hilbert space , and let be an asymptotically -strictly pseudocontractive semigroup on . Let be a sequence in such that and . Then .

3. Main Results

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from into satisfying the conditions (A1)–(A5), and let be a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let be an asymptotically -SPC semigroup on for some and a bounded sequence such that . Assume that is a nonempty bounded subset of . For chosen arbitrarily, suppose that , and are generated iteratively by where satisfying the following conditions: (C1) such that ,(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), (3.4), Lemma 2.1, and the asymptotically -SPC semigroupness of , we have where for all .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 (3.6), (3.7), and (3.8), we have that is convex. Suppose that is given and is convex for some . It follows by putting in (3.6), (3.7), and (3.8) that is convex. Therefore, by mathematical induction, we have that 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 (3.5), 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 (3.5), 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 (3.10), 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 (3.12) that Therefore, by (3.11), we obtain Indeed, from (3.1), we have Substituting into (3.15) and into (3.16), 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 (3.14), we obtain Next, we prove that , , and as . Since , by (3.1), we have It follows that Therefore, by the condition (C1), (3.14), and , we obtain From (3.23) and the condition (C1), we have Therefore, Hence, by (3.14) and , we obtain By (3.2), (3.3), and the firmly nonexpansiveness of , we have It follows that Therefore, by the condition (C1) and (3.5), we have It follows that Hence, by (3.28) and , we obtain Since is bounded, there exists a subsequence of which converges weakly to . Next, we prove that . From (3.22) and (3.25), we have , and as ; therefore, by Lemma 2.5, we obtain . From (3.1), we have It follows by the condition (A2) that Hence, Therefore, from (3.33) and by as , we obtain For a constant with and , let . Since , thus, . So, from (3.37), we have By (3.38), 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 (3.10), we have Since by (3.33) and , we have as . Therefore, by (3.41), (3.42), and the weak lower semicontinuity of norm, we have It follows that Since as ; therefore, we have Hence, from (3.44), (3.45), the 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 Kim [16] as the following. (1)The sequence is a projection sequence of onto for all such that(2)A solving of a common element of the set of common fixed point for a discrete asymptotically -SPC semigroup and the set of solutions of the mixed equilibrium problems by iteration is obtained.

For solving the equilibrium problem, let us define the condition as the condition such that . We have the following result.

Corollary 3.3. Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction from into satisfying the conditions (A1)–(A5) with either or holds. Let be an asymptotically -SPC semigroup on for some and a bounded sequence such that . Assume that is a nonempty bounded subset of . For chosen arbitrarily, suppose that , and are generated iteratively by where satisfying the following conditions: (C1) such that ,(C2) for some ,(C3).Then the sequences , and converge strongly to .

Proof. It is concluded from Theorem 3.1 immediately, by putting .

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space . Let be an asymptotically -SPC semigroup on for some and a bounded sequence such that . Assume that is a nonempty bounded subset of . For chosen arbitrarily, suppose that and are generated iteratively by where and such that . Then the sequences and converge strongly to .

Proof. It is concluded from Corollary 3.3 immediately, by putting .

4. Applications

We introduce the equilibrium problem to the optimization problem: where is a nonempty closed convex subset of a real Hilbert space and is a proper convex and lower semicontinuous. We denote by the set of solutions of problem (4.1). We define the condition as the condition such that is a bifunction defined by for all . Observe that . We give the interesting result as the following theorem.

Theorem 4.1. Let be a nonempty closed convex subset of a real Hilbert space , and let be a proper lower semicontinuous and convex function with either or holds. Let be an asymptotically -SPC semigroup on for some and a bounded sequence such that . Assume that is a nonempty bounded subset of . For chosen arbitrarily, suppose that and are generated iteratively by where satisfying the following conditions: (C1) such that , (C2) for some ,(C3). Then the sequences , and converge strongly to .

Proof. It is concluded from Corollary 3.3 immediately, by defining for all .

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