Convergence to common solutions of various problems for nonexpansive mappings in Hilbert spaces
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RESEARCH
Open Access
Convergence to common solutions of various problems for nonexpansive mappings in Hilbert spaces Kyung Soo Kim* *
Correspondence: [email protected] Graduate School of Education, Mathematics Education, Kyungnam University, Changwon, Kyungnam 631-701, Republic of Korea
Abstract In this paper, motivated and inspired by Ceng and Yao (J. Comput. Appl. Math. 214(1):186-201, 2008), Iiduka and Takahashi (Nonlinear Anal. 61(3):341-350, 2005), Jaiboon and Kumam (Nonlinear Anal. 73(5):1180-1202, 2010), Kim (Nonlinear Anal. 73:3413-3419, 2010), Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006) and Saeidi (Nonlinear Anal. 70:4195-4208, 2009), we introduce a new iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem, and the set of fixed points of a left amenable semigroup {Tt : t ∈ S} of nonexpansive mappings with respect to W-mappings and a left regular sequence {μn } of means defined on an appropriate space of bounded real-valued functions of the semigroup S. Furthermore, we prove that the iterative scheme converges strongly to a common element of the above four sets. Our results extend and improve the corresponding results of many others. MSC: 43A65; 47H05; 47H09; 47H10; 47J20; 47J25; 74G40 Keywords: mixed equilibrium problem; variational inequality problem; reversible semigroup; mean; common fixed point
1 Introduction Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C. Let ϕ : C → R be a real-valued function and θ : C × C → R be an equilibrium bifunction with θ (u, u) = for each u ∈ C. We consider the mixed equilibrium problem (for short, MEP) is to find x* ∈ C such that MEP : θ x* , y + ϕ(y) – ϕ x* ≥ ,
∀y ∈ C.
In particular, if ϕ ≡ , this problem reduces to the equilibrium problem (for short, EP), which is to find x* ∈ C such that EP : θ x* , y ≥ ,
∀y ∈ C.
Denote the set of solutions of MEP by . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems and the equilibrium problems as special cases. © 2012 Kim; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kim Fixed Point Theory and Applications 2012, 2012:185 http://www.fixedpointtheoryandapplications.com/content/2012/1/185
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A mapping T of C into itself is called nonexpansive if Tx – Ty ≤ x – y, for all x, y ∈ C. We denote by F(T) the set of fixed points of T. It is well known that F(T) is closed convex. Recall that a mapping f : C → C is called contractive if there exists a constant α ∈ (, ) such that f
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