Báo cáo hóa học: Research Article A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces
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Báo cáo hóa học: " Research Article A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces"Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2011, Article ID 572156, 14 pagesdoi:10.1155/2011/572156Research ArticleA New Strong Convergence Theorem forEquilibrium Problems and Fixed Point Problems inBanach Spaces Weerayuth Nilsrakoo Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand Correspondence should be addressed to Weerayuth Nilsrakoo, nilsrakoo@hotmail.com Received 5 June 2010; Revised 28 December 2010; Accepted 20 January 2011 Academic Editor: Fabio Zanolin Copyright q 2011 Weerayuth Nilsrakoo. 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 introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.1. IntroductionThroughout this paper, we denote by Æ and Ê the sets of positive integers and real numbers,respectively. Let E be a Banach space, E∗ the dual space of E and C a closed convex subsetsof E. Let F : C × C → Ê be a bifunction. The equilibrium problem is to find x ∈ C such that F x, y ≥ 0, ∀y ∈ C. 1.1The set of solutions of 1.1 is denoted by EP F . The equilibrium problems includefixed point problems, optimization problems, variational inequality problems, and Nashequilibrium problems as special cases. Let E be a smooth Banach space and J the normalized duality mapping from E to E∗ .Alber 1 considered the following functional ϕ : E × E → 0, ∞ defined by 2 2 ϕ x, y x − 2 x, Jy y x, y ∈ E . 1.22 Fixed Point Theory and ApplicationsUsing this functional, Matsushita and Takahashi 2, 3 studied and investigated the followingmappings in Banach spaces. A mapping S : C → E is relatively nonexpansive if the followingproperties are satisfied: R1 F S / , R2 ϕ p, Sx ≤ ϕ p, x for all p ∈ F S and x ∈ C, R3 F S FS,where F S and F S denote the set of fixed points of S and the set of asymptotic fixed pointsof S, respectively. It is known that S satisfies condition R3 if and only if I − S is demiclosedat zero, where I is the identity mapping; that is, whenever a sequence {xn } in C convergesweakly to p and {xn − Sxn } converges strongly to 0, it follows that p ∈ F S . In a Hilbert space x − y 2 for all x, y ∈ H .H , the duality mapping J is an identity mapping and ϕ x, yHence, if S : C → H is nonexpansive i.e., Sx − Sy ≤ x − y for all x, y ∈ C , then it isrelatively nonexpansive. Recently, many authors studied the problems of finding a common element of the setof fixed points for a mapping and the set of solutions of equilibrium problem in the setting ofHilbert space and uniformly smooth and uniformly convex Banach space, respectively see,e.g., 4–21 and the references therein . In a Hilbert space H , S. Takahashi and W. Takahashi 17 introduced the iteration as follows: sequence {xn } generated by u, x1 ∈ C, 1 F zn , y y − zn , zn − xn ≥ 0, ∀y ∈ C, rn 1.3 xn βn xn 1 − βn S αn u 1 − αn zn , 1for every n ∈ Æ , where S is nonexpansive, {αn } and {βn } are appropriate sequences in 0, 1 ,and {rn } is an appropriate positive real sequence. They proved that {xn } converges stronglyto some element in F S ∩ EP F . In 2009, Takahashi and Zembayashi 19 proposed theiteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence{xn } generated by u1 ∈ E, 1 xn ∈ C such that F xn , y y − xn , Jxn − Jun ≥ 0, ∀y ∈ C, rn 1.4 −1 un J αn Jxn 1 − αn J Sxn , 1for every n ∈ Æ , S is relatively nonexpansive, {αn } is an appropriate sequence in 0, 1 , and{rn } is an appropriate positive real sequence. They proved that if J is weakly sequentiallycontinuous, then {xn } converges weakly to some element in F S ∩ EP F . Motivated by S. Takahashi and W. Takahashi 17 and Takahashi and Zembayashi 19 ,we prove a strong convergence theorem for finding a common element of the fixed points setof a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in auniformly smooth and uniformly convex Banach space.Fixed Point Theory and Applications 32. PreliminariesWe collect ...