Journal of Applied MathematicsVolume 2012 (2012), Article ID 414831, 35 pagesdoi:10.1155/2012/414831

Research Article

A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Poom Kumam1,2 and Phayap Katchang2,3

¹Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand³Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Received 17 November 2011; Accepted 23 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Poom Kumam and Phayap Katchang. 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.

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of . Recall that a mapping is nonexpansive if We denote the set of fixed points of by , that is . A mapping is said to be an -contraction if there exists a coefficient such that Let be a mapping. Then is called:(1)monotone if (2)-strongly monotone if there exists a positive real number such that for constant , this implies that that is, is -expansive and when , it is expansive;(3)-Lipschitz continuous if there exists a positive real number such that(4)-cocoercive [1, 2] if there exists a positive real number such that Clearly, every -cocoercive map is -Lipschitz continuous;(5)relaxed -cocoercive, if there exists a positive real number such that (6)relaxed -cocoercive, if there exists a positive real number such that for , is -strongly monotone. This class of mapping is more general than the class of strongly monotone mapping. It is easy to see that we have the following implication: -strongly monotonicity implying relaxed -cocoercivity,(7)-strictly pseudocontractive, if there exists a constant such that

Remark 1.1 (see [3, Remark  1.1 pages 135-136]). If is a -Lipschitz continuous and relaxed -cocoercive mapping with and , then satisfies the following: where .Similarly, if -Lipschitz continuous and relaxed -cocoercive mapping with and , then the mapping satisfies the following: where .

Let be a strongly positive linear bounded operator on if there is a constant with the property We recall optimization problem (for short, OP) as the following where are infinitely closed convex subsets of such that , , is a real number, is a strongly positive linear bounded operator on , and is a potential function for (i.e., for ). This kind of optimization problem has been studied extensively by many authors, see, for example, [4–7] when and , where is a given point in .

On the other hand, a family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions:(i) for all ;(ii) for all ;(iii) for all and ;(iv)for all is continuous.

We denote by the set of all common fixed points of , that is, . It is known that is closed and convex.

Let be a real-valued function and let be a finite family of equilibrium functions, that is, for each . The system of mixed equilibrium problems (for short, SMEP) for function is to find such that The set of solutions of (1.15) is denoted by , where is the set of solutions of the mixed equilibrium problem (for short, MEP), which is to find such that In particular, if , and , then the problem (1.15) reduces to the equilibrium problem (for short, EP), which is to find such that It is well known that the SMEP includes fixed point problem, optimization problem, variational inequality problem, and Nash equilibrium problem as its special cases (see [8–13] for more details).

For solving the solutions of a nonexpansive semigroup and the solutions of the system of mixed equilibrium problems were studied by many authors see [14–23] and reference therein. In 2010, Chang et al. [24] studied the following approximation method: where is the mapping defined by (2.22) below, is the mapping defined by (2.12), and is a nonexpansive semigroup. They proved that converges strongly to a fixed point of under control conditions on the parameters.

Let be two mappings. The general system of variational inequalities problem (see [25]) is to find such that where and are two positive real numbers. The set of solutions of the general system of variational inequalities problem is denoted by . In particular, if , then the problem (1.20) reduces to the following equation: which is defined by Verma [26] (see also Verma [27]), and is called the new system of variational inequalities. Further, if we set , then problem (1.20) reduces to the classical variational inequality is to find such that We denoted by the set of solutions of the variational inequality problem. The variational inequality problem has been extensively studied in literature, see, for example, [28–31] and references therein. In order to find the solutions of the general system of variational inequality problem (1.20), Wangkeeree and Kamraksa [32] considered the following iterative algorithm: where is a -Lipschitz continuous and relaxed -cocoercive mapping and -Lipschitz continuous and relaxed -cocoercive mapping, respectively. They proved that converges strongly to a fixed point of which is a solution of general system of variational inequality (1.20). Very recently, Jaiboon and Kumam [33] studied a new general iterative method for finding a common element of the set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces, which solves some optimization problems.

Inspired and motivated by Chang et al. [24], Jaiboon and Kumam [33], Kumam and Jaiboon [34] and Wangkeeree and Kamraksa [32], the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroup, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the strong convergence theorem in Hilbert spaces under control conditions on the parameters. Furthermore, we can apply our results for solving some optimization problems. Our results extend and improve the corresponding results in Chang et al. [24], Kumam and Jaiboon [34], Wangkeeree and Kamraksa [32], and many others.

2. Preliminaries

Let a real Hilbert space and a nonempty closed convex subset of . We denote strong convergence (weak convergence) by notation . In a real Hilbert space , it is well known that for all and .

Recall that for every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies for every . Obviously, this immediately implies that Moreover, is characterized by the following properties: and for all .

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [35]). Let be a -strict pseudo-contraction, then (1)the fixed point set of is closed convex so that the projection is well defined;(2)define a mapping by If , then is a nonexpansive mapping such that .

A family of mappings is called a family of uniformly -strict pseudo-contractions, if there exists a constant such that Let be a countable family of uniformly -strict pseudo-contractions. Let be the sequence of nonexpansive mappings defined by (2.9), that is,

Let be a sequence of nonexpansive mappings of into itself defined by (2.11) and let be a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows: Such a mapping is nonexpansive from to and it is called the -mapping generated by and .

For each , let the mapping be defined by (2.12). Then we can have the following crucial conclusions concerning . You can find them in [36]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [36]). Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, let be real numbers such that for every . Then, (1) is nonexpansive and , for all ;(2)for every and , the limit exists;(3)a mapping defined by is a nonexpansive mapping satisfying and it is called the -mapping generated by and .

Lemma 2.3 (see [37]). Let be a nonempty closed convex subset of a Hilbert space , a countable family of nonexpansive mappings with , a real sequence such that . If is any bounded subset of , then

Lemma 2.4 (see [38]). Each Hilbert space satisfies Opial’s condition, that is, for any sequence with , the inequality holds for each with .

Lemma 2.5 (see [39]). Assume is a strongly positive linear bounded operator on with coefficient and . Then, .

For solving the system of mixed equilibrium problems (1.15), let us assume that function satisfies the following conditions:(H1) is monotone, that is, , for all ;(H2) for each fixed , is convex and upper semicontinuous;(H3) for each is convex.

Let and be two mappings. is said to be(1)monotone if (2)-strongly monotone if there exists a positive real number such that (3)-Lipschitz continuous if there exists a constant such that

Let be a differentiable functional on , which is called:(1)-convex [40] if where is the Fréchet derivative of at ;(2)-strongly convex [41] if there exists a constant such that

In particular, if for all , then is said to be strongly convex.

Lemma 2.6 (see [42]). Let be a real Hilbert space and let be a lower semicontinuous and convex functional from to . Let be a bifunction from to satisfying (H1)–(H3). Assume that (i) is -Lipschitz continuous with constant such that(a), (b) is affine in the first variable,(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;(ii) is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology;(iii)for each , there exist bounded subsets and such that for any , For given , let be the mapping defined by for all . Then (1) is single-valued.(2), where is the set of solution of the mixed equilibrium problem, (3) is closed and convex.

Lemma 2.7 (see [43]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.8 (see [44]). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (1), (2) or .Then, .

Lemma 2.9 (see [45]). Let be a nonempty closed convex subset of a real Hilbert space and a proper lower-semicontinuous differentiable convex function. If is a solution to the minimization problem then In particular, if solves problem , then

Lemma 2.10 (see [46]). Let be a nonempty bounded closed convex subset of a Hilbert space and let be a nonexpansive semigroup on , then for any ,

Lemma 2.11 (see [47]). Let C be a nonempty bounded closed convex subset of , a sequence in C, and a nonexpansive semigroup on . If the following conditions are satisfied: (i);(ii), then .

Lemma 2.12 (see [25]). For given and is a solution of the problem (1.20) if and only if is a fixed point of the mapping is defined by where , and are positive constants and are two mappings.

Throughout this paper, the set of fixed points of the mapping is denoted by .

Lemma 2.13 (see [32]). Let be defined in Lemma 2.12. If is a -Lipschitzian and relaxed -cocoercive mapping and is a -Lipschitz and relaxed -cocoercive mapping where and , then G is nonexpansive.

Lemma 2.14 (demiclosedness principle [48]). Assume that is a nonexpansive self-mapping of a nonempty closed convex subset of a real Hilbert space . If has a fixed point, then is demiclosed; that is, whenever is a sequence in converging weakly to some (for short, ), and the sequence converges strongly to some (for short, ), it follows that . Here is the identity operator of .

3. Main Results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) for finding the solutions of a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space which and let be a contraction of into itself with . Let be a lower semicontinuous and convex functional from to and let be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudo-contractions, let be the countable family of nonexpansive mappings defined by , let be the -mapping defined by (2.12), and let be a mapping defined by (2.13) with . Let be a strongly positive linear bounded operator on with coefficient and let , be a -Lipschitz continuous and relaxed -cocoercive mapping with , and let be a -Lipschitz continuous and relaxed -cocoercive mapping with . Suppose that , where . Let , and , which are constants. For given arbitrarily and fixed , suppose , , and are the sequences generated iteratively by where is the mapping defined by (2.22) and and are two sequences in for all . Assume the following conditions are satisfied: (C1) is -Lipschitz continuous with constant such that(a), (b) is affine,(c)for each fixed , is sequentially continuous from the weak topology to the weak topology;(C2) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant such that ;(C3) for each and for all , there exist bounded subsets and such that for any , (C4) and ;(C5); (C6) and .Then, converges strongly to , which solves the following optimization problem (OP): and is a solution of the general system of variational inequality problem (1.20) such that .

Proof. By the condition (C4) and (C5), we may assume, without loss of generality, that for all . Indeed, is a strongly positive bounded linear operator on , we have Observe that so this shows that is positive. It follows that We shall divide the proofs into several steps.Step 1. We show that is bounded.Let . In fact, by the assumption that for each , is nonexpansive. Let and . Then, we have and . Since , then Putting , we have . Since and , therefore, we have Because and are nonexpansive mappings and from Remark 1.1, we have which yields that It follows from (3.11) and induction that Hence, is bounded, so are , , , , for all and , where .Step 2. We prove that and .Again, from Remark 1.1, we have the following estimates: On the other hand, since and are nonexpansive, we have where is a constant such that for all . It follows from (3.13) and (3.14) that we have It follows that where .Setting , for all , we have Then, we obtain It follows from (3.16) and (3.18) that By the conditions (C4), (C5) and from , and , we have Hence, by Lemma 2.7, we obtain It follows that Applying (3.22) into (3.13), we obtain that Step 3. We show that , , and , where .Since , we have that is By (C4), (C5), and (3.22) it follows that Since is firmly nonexpansive, , where and , we have and hence Observe that where It follows from condition (C4) that Putting (3.28) into (3.29) and using also (3.10), we have It follows that Therefore, by (3.22) and (3.31), we get Since and by (3.26) and (3.70), we have Since is a -Lipschitz continuous and relaxed -cocoercive mapping on and for any , we have Similarly, since is a -Lipschitz continuous and relaxed -cocoercive mapping on and , we also have Substituting (3.37) into (3.29), we have Again, substituting (3.38) into (3.29) and using also (3.10), we get Therefore, by (3.39) and (3.40), we have It follows from (3.22) and (3.31) that we obtain From (2.6), we have So, we obtain By (3.29), we get which implies that From (3.22), (3.31), and (3.43), we have Now, from (2.2) and (2.7), we observe that It follows from (3.36), (3.42), and (3.48) that we have since It follows from (3.36), (3.48) and (3.50), we get and from (3.26), and (3.52) that we have Since is a bounded sequence in , from Lemma 2.10 for all , we have and since it follows from (3.52) and (3.54) that we get On the other hand, since is firmly nonexpansive, and , we have and hence From (3.10), (3.29), and (3.58), for each , we have It follows that Therefore, by (3.22) and (3.31), we get Step 4. We prove that where is a solution of the optimization problem: To show this inequality, we can choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . From (3.53), we get .Next, we show that , where .(1) First, we prove that . Indeed, from Lemma 2.11 and (3.56), we get , that is, .(2) Next, we show that , where and . Assume that , then there exists a positive integer such that and so . Hence for any , that is, . This together with , shows ; therefore, we have . It follows from the Opial’s condition and (3.52) that which is a contradiction. Thus, we get .(3) Now, we prove that . Since , and , we have It follows that for all . From (3.61) and by conditions (C1)(c) and (C2), we get By the assumption that is lower semicontinuous, then it is weakly lower semicontinuous and by the condition (H2) that is lower semicontinuous, then it is weakly lower semicontinuous. Since , it follows from (3.36), (3.52), and (3.61) that for each . Taking the lower limit in (3.67), we have Therefore, .(4) Next, we show that . By (3.36) and (3.52), we have By Lemma 2.13 that is a nonexpansive, we obtain Thus, By Lemma 2.14, we obtain that . Hence is proved.Now, from Lemma 2.9, (3.64), and (3.53), we have By (3.52), (3.53), and (3.73), we obtain Step 5. Finally, we show that . From (3.1), we obtain Since , , and are bounded, there exist such that for all . It follows that where Applying Lemma 2.8 to (3.77), we conclude that . This completes the proof.

Remark 3.2. For example, of the control conditions (C4)–(C6), we set . We set is a 1-Lipschitz continuous and relaxed -cocoercive mapping, (i.e., and ).Then, we can choose and which satisfies the condition (C6) in Theorem 3.1.

Corollary 3.3. Let be a nonempty closed convex subset of a real Hilbert space which and let be a contraction of into itself with . Let be a lower semicontinuous and convex functional from to and let be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly - strict pseudo-contractions, let be the countable family of nonexpansive mappings defined by , let be the -mapping defined by (2.12), and let be a mapping defined by (2.13) with . Let be a strongly positive linear bounded operator on with coefficient and let , be a -Lipschitz continuous and relaxed -cocoercive mapping with , and let be a -Lipschitz continuous and relaxed -cocoercive mapping with . Suppose that . Let , and , which are constants. For given arbitrarily and fixed , suppose , ,, and are the sequences generated iteratively by where such that is the mapping defined by (2.22) and and are two sequences in for all . If the functions and satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then converges strongly to , which solves the following optimization problem (OP): and is a solution of the general system of variational inequality problem (1.20) such that .

Proof. Taking in Theorem 3.1. Hence, the conclusion follows. This completes the proof.

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space which and let be a contraction of into itself with . Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a countable family of uniformly -strict pseudo-contractions, let be the countable family of nonexpansive mappings defined by , let be the -mapping defined by (2.12), and let be a mapping defined by (2.13) with . Let be a strongly positive linear bounded operator on with coefficient and let , be a -Lipschitz continuous and relaxed -cocoercive mapping with , and let be a -Lipschitz continuous and relaxed -cocoercive mapping with . Suppose that . Let and , which are constants. For given arbitrarily and fixed , suppose , , and are the sequences generated iteratively by where and are two sequences in for all . If the sequence satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then converges strongly to , which solves the following optimization problem (OP): and is a solution of the general system of variational inequality problem (1.20) such that .

Proof. Put for all and . Take and , for all . Then, we get in Corollary 3.3. Hence, the conclusion follows. This completes the proof.

Corollary 3.5. Let be a nonempty closed convex subset of a real Hilbert space and let be a contraction of into itself with . Let be a nonexpansive semigroup on and let be a positive real divergent sequence. Let be a strongly positive linear bounded operator on with coefficient and let , be a -Lipschitz continuous and relaxed -cocoercive mapping with . Suppose that . Let and , which are constants. For given arbitrarily and fixed , suppose the , , and are the sequences generated iteratively by where and are two sequences in for all . If the sequence satisfy the conditions (C1)–(C6) as given in Theorem 3.1, then converges strongly to .

Proof. Setting , , and in Corollary 3.4, it follows from the proof of Theorem  4.1 in [25] that . Hence, the conclusion follows. This completes the proof.

Acknowledgments

The authors would like to thank the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand. Moreover, the authors are grateful to the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work.

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