General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems
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General iterative algorithms for mixed equilibrium problems, variational inequalities and fixed point problems Xiao-Jie Wang, Lu-Chuan Ceng* , Hui-Ying Hu and Shi-Xiu Li * Correspondence: [email protected] Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
Abstract In this paper, we introduce and analyze a general iterative algorithm for finding a common solution of a mixed equilibrium problem, a general system of variational inequalities and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some mild conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution, which also solves some optimization problem. The result presented in this paper improves and extends some corresponding ones in the earlier and recent literature. MSC: 49J30; 47H10; 47H15 Keywords: mixed equilibrium problem; nonexpansive mapping; variational inequality; fixed point; strongly positive bounded linear operator; inverse strongly monotone mapping
1 Introduction Let H be a real Hilbert space with the inner product ·, · and the norm · . Let C be a nonempty, closed and convex subset of H, and let T : C → C be a nonlinear mapping. Throughout this paper, we use F(T) to denote the fixed point set of T. A mapping T : C → C is said to be nonexpansive if Tx – Ty ≤ x – y,
∀x, y ∈ C.
(.)
Let F : C × C → R be a real-valued bifunction and ϕ : C → R be a real-valued function, where R is the set of real numbers. The so-called mixed equilibrium problem (MEP) is to find x ∈ C such that F(x, y) + ϕ(y) – ϕ(x) ≥ ,
∀y ∈ C,
(.)
which was considered and studied in [, ]. The set of solutions of MEP (.) is denoted by MEP(F, ϕ). In particular, whenever ϕ ≡ , MEP (.) reduces to the equilibrium problem (EP) of finding x ∈ C such that F(x, y) ≥ ,
∀y ∈ C,
which was considered and studied in [–]. The set of solutions of the EP is denoted by EP(F). Given a mapping A : C → H, let F(x, y) = Ax, y – x for all x, y ∈ C. Then x ∈ EP(F) ©2014 Wang et al.; 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.
Wang et al. Fixed Point Theory and Applications 2014, 2014:80 http://www.fixedpointtheoryandapplications.com/content/2014/1/80
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if and only if Ax, y – x ≥ for all y ∈ C. Numerous problems in physics, optimization and economics reduce to finding a solution of the EP. Throughout this paper, assume that F : C × C → R is a bifunction satisfying conditions (A)-(A) and that ϕ : C → R is a lower semicontinuous and convex function with restriction (B) or (B), where (A) F(x, x) = for all x ∈ C; (A) F is monotone, i.e., F(x, y) + F(y, x) ≤ for any x, y ∈ C; (A) F is upper-hemicontinuous, i.e., for each x, y, z ∈ C, lim sup F tz + ( – t)x, y ≤
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