Proximal algorithms for a class of mixed equilibrium problems
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RESEARCH
Open Access
Proximal algorithms for a class of mixed equilibrium problems Yisheng Song1,2* and Qingnian Zhang3 *
Correspondence: [email protected] 1 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, P.R. China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China Full list of author information is available at the end of the article
Abstract We present two proximal algorithms for solving the mixed equilibrium problems. Under some simpler framework, the strong and weak convergence of the sequences defined by two general algorithms is respectively obtained. In particular, we deal with several iterative schemes in a united way and apply our algorithms for solving the classical equilibrium problem, the minimization problem, the classical variational inequality problem and the generalized variational inequality problem. Our results properly include some corresponding results in this field as a special case. MSC: 47H06; 47J05; 47J25; 47H10; 90C33; 90C25; 49M45; 65C25; 49J40; 65J15; 47H09 Keywords: mixed equilibrium problem; proximal algorithm; variational inequality
1 Introduction Throughout the paper, H is a real Hilbert space with inner product ·, · and induced norm · . Let K be a nonempty closed convex subset of H, F : K × K → R be a bifunction and ϕ : K → R ∪ {+∞} be a proper generalized real valued function, where R is the set of real numbers. Our interest is in finding a solution to the following problem which is referred to as the mixed equilibrium problem (for short, MEP) for F, ϕ find x ∈ K such that F(x, y) + ϕ(y) – ϕ(x) ≥ ,
∀y ∈ K.
(.)
We denote the set of solutions for MEP by MEP(F, ϕ) = x ∈ K; F(x, y) + ϕ(y) – ϕ(x) ≥ , ∀y ∈ K . Obviously, MEP (.) is a classical equilibrium problem (for short, EP) for F when ϕ ≡ find x ∈ K such that F(x, y) ≥ ,
∀y ∈ K.
(.)
The set MEP(F, ) of solutions of EP (.) is denoted by EP(F). If F ≡ , then MEP (.) becomes the minimization problem (for short, MP) for a function ϕ find x ∈ K such that ϕ(y) ≥ ϕ(x),
∀y ∈ K,
(.)
and MEP(, ϕ) is denoted by Argmin(ϕ). © 2012 Song and Zhang; 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.
Song and Zhang Fixed Point Theory and Applications 2012, 2012:166 http://www.fixedpointtheoryandapplications.com/content/2012/1/166
Page 2 of 15
Given a mapping T : K → H, let F(x, y) = Tx, y – x for all x, y ∈ K . Then z ∈ EP(F) if and only if Tz, y – z ≥ for all y ∈ K , i.e., EP (.) turns into a classical variational inequality problem (for short, VIP) for T find x ∈ K such that Tx, y – x ≥ ,
∀y ∈ K.
(.)
At the same time, MEP (.) also reduces a generalized variational inequality problem (for short, GVIP) for a mapping T and a function ϕ fi
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