An Iterative Algorithm for Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities
© Yeong-Cheng Liou. 2010
Received: 13 September 2009
Accepted: 10 January 2010
Published: 21 January 2010
We present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithms strongly converge to which solves some variational inequality.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonlinear mapping, let be a function, and let be a bifunction of into . Now we consider the following mixed equilibrium problem:
The set of solution of problem (1.1) is denoted by EP.
The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, [3–8]. Some methods have been proposed to solve the mixed equilibrium problem and the equilibrium problem. In 1997, Flaim and Antipen  introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi  introduced another iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.2) and the set of fixed point points of a nonexpansive mapping. Furthermore, Yao et al.  introduced some new iterative schemes for finding a common element of the set of solutions of the equilibrium problem (1.2) and the set of common fixed points of finitely (infinitely) nonexpansive mappings. Very recently, Ceng and Yao  considered a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings. Peng and Yao  developed a CQ method. They obtained some strong convergence results for finding a common element of the set of solutions of the mixed equilibrium problem (1.1) and the set of the variational inequality and the set of fixed points of a nonexpansive mapping. Their results extend and improve the corresponding results in [1, 9, 12].
where is the zero vector in . The set of solutions of problem (1.6) is denoted by . If , then problem (1.6) becomes the generalized equation introduced by Robinson . If , then problem (1.6) becomes the inclusion problem introduced by Rockafellar . It is known that (1.6) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized. Recently, Zhang et al.  introduced a new iterative scheme for finding a common element of the set of solutions to the problem (1.6) and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al.  introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. For some related works, please see [1, 2, 9–11, 13–34] and the references therein.
Inspired and motivated by the works in the literature, in this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithms strongly converge to which solves some variational inequality.
A set-valued mapping is called monotone if, for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies .
where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse strongly monotone and that a solution of problem (1.6) is a fixed point of the operator for all , see, for instance, .
Lemma (see ).
Lemma (see ).
Lemma (see ).
3. Main Results
In this section, we will prove our main result. First, we give some assumptions on the operators and the parameters. Subsequently, we introduce our iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of a variational inclusion. Finally, we will show that the proposed algorithm has strong convergence.
Let be a nonempty closed convex subset of a real Hilbert space . Let be a lower semicontinuous and convex function and let be a bifunction satisfying conditions (H1)–(H5). Let be a strongly positive bounded linear operator with coefficient and let be a maximal monotone mapping. Let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Let and be two constants such that and .
Now we introduce the following iteration algorithm.
Now we study the strong convergence of the algorithm (3.1).
We divide our proofs into the following five steps:
From (3.1), we deduce that
Substituting (3.14) into (3.13), we get
Next, we show that . In fact, since is -inverse strongly monotone, is Lipschitz continuous monotone mapping. It follows from Lemma 2.2 that is maximal monotone. Let , that is, . Again since , we have , that is, . By virtue of the maximal monotonicity of , we have
From (3.1), we have
The author was supported in part by NSC 98-2622-E-230-006-CC3 and NSC 98-2923-E-110-003-MY3.
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