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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 414718, 26 pages
http://dx.doi.org/10.1155/2012/414718
Research Article

A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization Problems

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bang Mod, Bangkok 10140, Thailand

Received 9 August 2012; Accepted 24 September 2012

Academic Editor: RuDong Chen

Copyright © 2012 Pongsakorn Sunthrayuth and Poom Kumam. 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 a new general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by many others.

1. Introduction

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed and convex subset of . Let be a real-valued function, where is the set of real numbers. Let be two nonlinear mappings and be an equilibrium-like function, that is, for all . We consider the following new generalized equilibrium problem: find such that The set of solutions of the problem (1.1) is denoted by . As special cases of the problem (1.1), we have the following results. (1)If , then problem (1.1) reduces to the following generalized equilibrium problem: find such that which was considered by Cho et al. [1] for more details. The set of solutions of the problem (1.1) is denoted by .(2)If and , where is an equilibrium bifunction, then problem (1.1) reduces to the following mixed equilibrium problem: find such that which was considered by Ceng and Yao [2] for more details. The set of solutions of the problem (1.3) is denoted by .(3)If , and where is an equilibrium bifunction, then problem (1.1) reduces to the following equilibrium problem: find such that The set of solutions of problem (1.4) is denoted by .(4)If , , then problem (1.1) reduces to the following classical variational inequality problem: find such that The set of solutions of the problem (1.5) is denoted by .

In brief, for an appropriate choice of the mapping , the function , and the convex set , one can obtain a number of the various classes of equilibrium problems as special cases. In particular, the equilibrium problems (1.4) which were introduced by Blum and Oettli [3] and Noor and Oettli [4] in 1994 have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. In [3, 4], it has been shown that equilibrium problems include variational inequalities, fixed point, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. This means that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Hence collectively, equilibrium problems cover a vast range of applications. Related to the equilibrium problems, we also have the problems of finding the fixed points of the nonlinear mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of nonlinear mappings (e.g., see [518] and the references therein).

Recall the following definitions.

Definition 1.1. The mapping is said to be (1)nonexpansive if (2)-Lipschitzian if there exists a constant such that (3)-strict pseudocontraction [19] if there exists a constant such that (4)pseudocontractive if
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. It is easy to see that (1.8) is equivalent to that is, is -inverse-strongly monotone. From [19], we know that if is a -strictly pseudocontractive mapping, then is Lipschitz continuous with constant , that is, , for all .
In this paper, we use to denote the set of fixed points of .

Definition 1.2. A countable family of mapping is called a family of -strict pseudocontraction mappings if there exists a constant such that On the other hand, let be a nonempty closed and convex subset of a real Hilbert space . Let be three bifunctions and let be six nonlinear mappings and let be three functions. We consider the following problem of finding such that which is called a new general system of generalized nonlinear mixed composite-type equilibria, where for all . Next, we present some special cases of problem (1.12) as follows. (1)If , , , and for all , then the problem (1.12) reduces to the following new general system of generalized nonlinear mixed composite-type equilibria: find such that where for all .(2)If , , , and , then the problem (1.12) reduces to the following general system of generalized nonlinear mixed composite-type equilibria: find such that which was introduced and considered by Ceng et al. [20], where for all .(3)If , , , and for all and , then the problem (1.12) reduces to the following a general system of generalized equilibria: find such that which was introduced and considered by Ceng and Yao [21], where for all .(4)If , , and , , for all , then the problem (1.12) reduces to the following generalized mixed equilibrium problem with perturbed mapping: find such that which was introduced and considered by Hu and Ceng [22].(5)If , , and for all , then the problem (1.12) reduces to the following general system of variational inequalities: find such that which was introduced and considered by Kumam et al. [23], where for all . (6)If , , for all , and , then the problem (1.12) reduces to the following general system of variational inequalities: find such that which was introduced and considered by Ceng et al. [24], where for all .

In 2010, Cho et al. [1] introduced an iterative method for finding a common element of the set of solutions of generalized equilibrium problems (1.2), the set of solutions for a systems of nonlinear variational inequalities problems (1.18), and the set of fixed points of nonexpansive mappings in Hilbert spaces. Ceng and Yao [21] introduced and considered a relaxed extragradient-like method for finding a common element of the set of solutions of a system of generalized equilibria, the set of fixed points of a strictly pseudocontractive mapping, and the set of solutions of a equilibrium problem in a real Hilbert space and obtained a strong convergence theorem. The result of Ceng and Yao [21] included, as special cases, the corresponding ones of S. Takahashi and W. Takahashi [10], Ceng et al. [24], Peng and Yao [25], and Yao et al. [26].

Motivated and inspired by the works in the literature, we introduce a new general system of generalized nonlinear mixed composite-type equilibria (1.12) and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. The results presented in this paper extend the recent results of Cho et al. [1], Ceng and Yao [21], Ceng et al.[20], and many authors.

2. Preliminaries

A bounded linear operator is said to be strongly positive, if there exists a constant such that Recall that, a mapping is said to be contractive if there exists a constant such that A mapping is called -inverse-strongly monotone if there exists a constant such that Let be a nonempty closed convex subset of a real Hilbert space . 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 nonexpansive (see [27]) and for , Let be a real-valued function, be a mapping and be an equilibrium-like function. Let be a positive real number. For all , we consider the following problem. Find such that which is known as the auxiliary generalized equilibrium problem.

Let be the mapping such that, for all , is the solution set of the auxiliary problem (2.6), that is, Then, we will assume the Condition [28] as follows: (a) is single-valued;(b) is nonexpansive;(c).

Notice that the examples of showing the sufficient conditions for the existence of the condition can be found in [6].

Throughout this paper, we assume that a bifunction and is a lower semicontinuous and convex function satisfy the following conditions: (H1), ;(H2) is monotone, that is, , ;(H3)for all ,   is weakly upper semicontinuous;(H4)for all ,   is convex and lower semicontinuous;(A1)for all and , there exist a bounded subset and such that for all , (A2) is a bounded set.

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

Lemma 2.1 (see [29]). Let be a nonempty closed and convex subset of a real Hilbert sapce . Let be a bifunction satisfying condition and let be a lower semicontinuous and convex function. For and define a mapping follows Assume that either or holds, then the following statements hold (i) for all and is single-valued;(ii) is firmly nonexpansive, that is, for all , (iii);(iv) is closed and convex.

Remark 2.2. If , then is rewritten as (see [21, Lemma 2.1] for more details).

Lemma 2.3 (see [30]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.4 (see [31]). Let be a real Hilbert space. Then the following inequalities hold. (i),  and .(ii).

Definition 2.5 (see [32]). Let be a sequence of mappings from a subset of a real Hilbert space into itself. We say that satisfies the condition if where , for all .

Lemma 2.6 (see [32]). Suppose that satisfies the -condition such that (i)for each , is converse strongly to some point in (ii)let the mapping defined by for all . Then, .

Lemma 2.7 (see [33]). Let be a closed and convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for all is well defined, nonexpansive, and holds.

Lemma 2.8 (see [19]). Let be a -strict pseudocontraction. Define by for each . Then, as , S is nonexpansive such that .

Lemma 2.9 (see [34]). Let be a closed and convex subset of a real Hilbert space and let be a nonexpansive mapping. then, the mapping is demiclosed. That is, if is a sequence in such that and , then .

Lemma 2.10 (see [35]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i);(ii) or .
then, .

Lemma 2.11. Let be a nonempty closed and convex subset of a real Hilbert space . Let mappings be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Then, we have where . In particular, if , then is nonexpansive.

Proof. From Lemma 2.4, for all , we have It is clear that, if , then is nonexpansive. This completes the proof.

Lemma 2.12. Let be a nonempty closed and convex subset of a real Hilbert space . Let mappings be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be the mapping defined by If , then is nonexpansive.

Proof. From Lemma 2.11, for all , we have which implies that is nonexpansive. This completes the proof.

Lemma 2.13. Let be a nonempty closed and convex subset of a real Hilbert space . Let    be a bifunction satisfying conditions and let be a nonlinear mapping. Suppose that be a real positive number. Let be a lower semicontinuous and convex function. Assume that either condition or holds. Then, for is a solution of the problem (1.12) if and only if , and , where is the mapping defined as in Lemma 2.12.

Proof. Let be a solution of the problem (1.12). Then, we have This completes the proof.

Corollary 2.14 (see [20]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a bifunction satisfying conditions and let be a nonlinear mapping. Suppose that be a real positive number. Let be a lower semicontinuous and convex function. Assume that either condition or holds. Then, for is a solution of the problem (1.14) if and only if , , where is the mapping defined by

Corollary 2.15 (see [21]). Let be a nonempty closed and convex subset of a real Hilbert space . Let be a bifunction satisfying conditions and let be a nonlinear mapping. Suppose that is a real positive number. Assume that either condition or holds. Then, for is a solution of the problem (1.15) if and only if , , where is the mapping defined by

Corollary 2.16 (see [23]). Let be a nonempty closed and convex subset of a real Hilbert space . For given is a solution of the problem (1.17) if and only if , and , where is the mapping defined by

Corollary 2.17 (see [24]). Let be a nonempty closed and convex subset of a real Hilbert space . For given is a solution of the problem (1.18) if and only if , , where is the mapping defined by

3. Main Results

We are now in a position to prove the main result of this paper.

Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space such that . Let be lower semicontinuous and convex functionals, be an equilibrium-like function, be a mapping, and be a bifunction satisfying conditions . Assume that either condition or holds. Let be -inverse-strongly monotone, be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be a family of -strict pseudocontraction mappings. Define a mapping , for all , and . Assume that the condition is satisfied and , where is defined as in Lemma 2.13. Let , , and be three constants. Let be a contraction mapping with a coefficient and let be a strongly positive bounded linear operator on with a coefficient such that . For , let the sequence defined by where such that , , , , and are two sequences in . Suppose that satisfies the -condition. Let be the mapping defined by for all and suppose that . Assume the following conditions are satisfied: (C1) and , (C2).
Then the sequence defined by (3.1) converges strongly to , where is the unique solution of the variational inequality or equivalently, , where is a metric projection mapping from onto , and is a solution of the problem (1.12), where and .

Proof. Note that from the conditions and , we may assume, without loss of generality, that for all . Since is a linear bounded self-adjoint operator on , by (2.2), we have Observe that This show that is positive. It follows that First, we show that is bounded. Taking , it follows from Lemma 2.13 that Putting and , we obtain . Notice that . Since is nonexpansive and is -inverse-strongly monotone, we have and hence We observe that Setting . By Lemma 2.8, we have is a nonexpansive mapping such that for all . Then, we have It follows that By induction, we have Hence, is bounded, so are , , , and . From definition of and for all , it follows that By our assumption, satisfies the PT-condition, we obtain that that is satisfies the PT-condition.
Next, we show that . Since , we have It follows from (3.15) that Let for all . Then, we have Combining (3.16) and (3.17), we have Since satisfies the PT-condition, we can define a mapping by We observe that Be Lemma 2.6, we have Consequently, it follows from the conditions , , and (3.18) that Hence, by Lemma 2.3, we obtain that Consequently, we have On the other hand, we observe that It follows that From the conditions