On the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces
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Research Article On the Existence and Convergence of Approximate Solutions for Equilibrium Problems in Banach Spaces Nan-Jing Huang, Heng-You Lan, and Kok Lay Teo Received 15 June 2006; Revised 11 February 2007; Accepted 20 February 2007 Recommended by Charles Ejike Chidume
We introduce and study a new class of auxiliary problems for solving the equilibrium problem in Banach spaces. Not only the existence of approximate solutions of the equilibrium problem is proven, but also the strong convergence of approximate solutions to an exact solution of the equilibrium problem is shown. Furthermore, we give some iterative schemes for solving some generalized mixed variational-like inequalities to illuminate our results. Copyright © 2007 Nan-Jing Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let X be a real Banach space with dual X ∗ , let K ⊂ X be a nonempty subset, and let f : K × K → R = (−∞,+∞) be a given bifunction. By the equilibrium problem introduced by Blum and Oettli in [1], we can formulate the following equilibrium problem of finding an x ∈ K such that f (x, y) ≥ 0,
∀ y ∈ K,
(1.1)
where f (x,x) = 0 for all x ∈ K. The following is a list of special cases of problem (1.1). (1) If f (x, y) = N(x,x),η(y,x) + b(x, y) − b(x,x) for all x, y ∈ K, where N : K × K → X ∗ , η : K × K → X, and b : K × K → R, then the problem of finding an x ∈ K such that
N(x, x),η(y, x) + b(x, y) − b(x, x) ≥ 0,
∀ y ∈ K,
(1.2)
2
Journal of Inequalities and Applications
is a special case of problem (1.1). This problem is known as the generalized mixed variational-like inequality. Problem (1.2) was considered by Huang and Deng [2] in the Hilbert space setting with set-valued mappings. (2) If X = X ∗ = H is a Hilbert space, N(x, y) = Tx − Ay, and b(x, y) = m(y) for all x, y ∈ K, where T,A,m : K → X ∗ , then problem (1.2) reduces to the following mixed variational-like inequality problem, which is to find an x ∈ K such that
T(x) − A(x),η(y, x) + m(y) − m(x) ≥ 0,
∀ y ∈ K.
(1.3)
This problem was introduced and studied by Ansari and Yao [3] and Ding [4]. Remark 1.1. Through appropriate choices of the mappings f , N, η, and b, it can be easily shown that problem (1.1) covers many known problems as special cases. For example, see [1–8] and the references therein. It is well known that many interesting and complicated problems in nonlinear analysis, such as nonlinear programming, optimization, Nash equilibria, saddle points, fixed points, variational inequalities, and complementarity problems (see [1, 9–12] and the references therein), can all be cast as equilibrium problems in the form of problem (1.1). There are several papers available in the literature which are devoted to the development of iterative procedures for solving some of these equilibrium problems in finite as well as infinite-dimensional spaces. For exa
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