An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems
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Research Article An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems Qing-you Liu,1 Wei-you Zeng,2 and Nan-jing Huang2 1
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Nan-jing Huang, [email protected] Received 11 January 2009; Accepted 28 May 2009 Recommended by Fabio Zanolin We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. 2007. Copyright q 2009 Qing-you Liu 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 C be a nonempty closed convex subset of a real Hilbert space H and let Φ : C × C → R be a bifunction, where R is the set of real numbers. Let Ψ : C → H be a nonlinear mapping. The generalized equilibrium problem GEP for Φ : C × C → R and Ψ : C → H is to find u ∈ C such that Φu, v Ψu, v − u ≥ 0
∀v ∈ C.
1.1
The set of solutions for the problem 1.1 is denoted by Ω, that is, Ω {u ∈ C : Φu, v Ψu, v − u ≥ 0, ∀v ∈ C}.
1.2
2
Fixed Point Theory and Applications
If Ψ 0 in 1.1, then GEP1.1 reduces to the classical equilibrium problem EP and Ω is denoted by EPΦ, that is, EPΦ {u ∈ C : Φu, v ≥ 0, ∀v ∈ C}.
1.3
If Φ 0 in 1.1, then GEP1.1 reduces to the classical variational inequality and Ω is denoted by VIΨ, C, that is, VIΨ, C {u∗ ∈ C : Ψu∗ , v − u∗ ≥ 0, ∀v ∈ C}.
1.4
It is well known that GEP1.1 contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities see, e.g., 1–6 and the reference therein. A mapping A : C → H is called α-inverse-strongly monotone 7 , if there exists a positive real number α such that 2 Ax − Ay, x − y ≥ αAx − Ay
1.5
for all x, y ∈ C. It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S : C → C is called nonexpansive if Sx − Sy ≤ x − y
1.6
for all x, y ∈ C. We denote by FS the set of fixed points of S, that is, FS {x ∈ C : x Sx}. If C ⊂ H is bounded, closed and convex and S is a nonexpansive mappings of C into itself, then FS is nonempty see 8 . In 1997, Fl˚am and Antipin 9 introduced an iterative scheme of finding the
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