A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems a
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RESEARCH
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A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems Thanyarat Jitpeera1 and Poom Kumam1,2* *
Correspondence: [email protected] 1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok, 10140, Thailand 2 Computational Science and Engineering Research Cluster (CSEC), King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok, 10140, Thailand
Abstract In this article, we introduce a new general iterative method for solving a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solutions of generalized mixed equilibrium problems and the set of solutions of the variational inclusion for a β -inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006), Su et al. (Nonlinear Anal. 69:2709-2719, 2008), Tan and Chang (Fixed Point Theory Appl. 2011:915629, 2011) and some authors. MSC: 46C05; 47H09; 47H10 Keywords: nonexpansive mapping; inverse-strongly monotone mapping; generalized mixed equilibrium problem; variational inclusion
1 Introduction Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product ·, · and the norm · , respectively. A mapping S : C → C is said to be nonexpansive if Sx – Sy ≤ x – y, ∀x, y ∈ C. If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then F(S) := {x ∈ C : Sx = x} is nonempty []. A mapping S : C → C is said to be a k-strictly pseudo-contraction [] if there exists ≤ k < such that Sx – Sy ≤ x – y + k(I – S)x – (I – S)y , ∀x, y ∈ C, where I denotes the identity operator on C. We denote weak convergence and strong convergence by notations and →, respectively. A mapping A of C into H is called monotone if Ax – Ay, x – y ≥ , ∀x, y ∈ C. A mapping A is called α-inverse-strongly monotone if there exists a positive real number α such that Ax – Ay, x – y ≥ αAx – Ay , ∀x, y ∈ C. A mapping A is called α-strongly monotone if there exists a positive real number α such that Ax – Ay, x – y ≥ αx – y , ∀x, y ∈ C. It is obvious that any α-inverse-strongly monotone mappings A is a monotone and α -Lipschitz continuous mapping. A linear bounded operator A is called strongly positive if there exists a constant γ¯ > with the property Ax, x ≥ γ¯ x , ∀x ∈ H. A self © 2012 Jitpeera and Kumam; 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.
Jitpeera and Kumam F
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