Báo cáo hóa học: Research Article New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces

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Báo cáo hóa học: " Research Article New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces"Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2011, Article ID 372975, 18 pagesdoi:10.1155/2011/372975Research ArticleNew Iterative Scheme for Finite Families ofEquilibrium, Variational Inequality, and Fixed PointProblems in Banach Spaces Shenghua Wang1, 2 and Caili Zhou3 1 School of Applied Mathematics and Physics, North China Electric Power University, Baoding 071003, China 2 Department of Mathematics, Gyeongsang National University, Jinju 660-714, Republic of Korea 3 College of Mathematics and Computer, Hebei University, Baoding 071002, China Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com Received 6 December 2010; Accepted 30 January 2011 Academic Editor: S. Al-Homidan Copyright q 2011 S. Wang and C. Zhou. 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. We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-φ-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.1. IntroductionThroughout this paper, let R denote the set of all real numbers. Let E be a smooth Banachspace and E∗ the dual space of E. The function φ : E × E → R is defined by 2 2 φ x, y x − y, Jx y , ∀x, y ∈ E, 1.1where J is the normalized dual mapping from E to E∗ defined by x∗ ∈ E∗ : x, x∗ x∗ 2 2 Jx x , ∀x ∈ E. 1.22 Fixed Point Theory and ApplicationsLet C be a nonempty closed and convex subset of E. The generalized projection Π : E → C isa mapping that assigns to an arbitrary point x ∈ E the minimum point of the function φ x, y ,that is, ΠC x x, where x is the solution to the minimization problem φ x, x inf φ z, x . 1.3 z∈C x − y 2 and ΠCIn Hilbert spaces, φ x, y PC , where PC is the metric projection. It isobvious from the definition of function φ that 2 2 y−x ≤ φ y, x ≤ y x , ∀x, y ∈ E. 1.4We remark that if E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E,φ x, y 0 if and only if x y. For more details on φ and Π, the readers are referred to 1–4 . Let T be a mapping from C into itself. We denote the set of fixed points of T by F T . Tis called to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C and quasi-nonexpansiveif F T / ∅ and x − T y ≤ x − y for all x ∈ F T and y ∈ C. A point p ∈ C is called to bean asymptotic fixed point of T 5 if C contains a sequence {xn } which converges weakly to psuch that limn → ∞ xn − T xn 0. The set of asymptotic fixed points of T is denoted by F T .The mapping T is said to be relatively nonexpansive 6–8 if F T F T and φ p, T x ≤φ p, x for all x ∈ C and p ∈ F T . The mapping T is said to be φ-nonexpansive if φ T x, T y ≤φ x, y for all x, y ∈ C. T is called to be quasi-φ-nonexpansive 9 if F T / ∅ and φ p, T x ≤φ p, x for all x ∈ C and p ∈ F T . In 2005, Matsushita and Takahashi 10 introduced the following algorithm: x0 x ∈ C, J −1 αn Jxn yn 1 − αn J T xn , Cn z ∈ C : φ z, yn ≤ φ z, xn , 1.5 Qn {z ∈ C : xn − z, Jx − Jxn ≥ 0}, xn PCn ∩Qn x, ∀n ≥ 0, 1where J is the duality m ...