A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasi-Ï•-asymptotically nonexpansive mappings in Banach spaces

Zhao Jing, He, He Songnian

Description

In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed generalized quasi- Ï• -asymptotically nonexpansive mappings and the set of solutions of equilibrium problem in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others. Mathematics Subject Classification (2000) : 47H09; 47H10; 47J05; 54H25.

Sujets

• Strong convergence
• Banach space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 3
• Langue: English

Extrait

Zhao and HeFixed Point Theory and Applications2012,2012:33 http://www.fixedpointtheoryandapplications.com/content/2012/1/33

R E S E A R C H

Open Access

A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasijasymptotically nonexpansive mappings in Banach spaces * Jing Zhao and Songnian He

* Correspondence: [e-mail] College of Science, Civil Aviation University of China, Tianjin 300300, P.R. China

Abstract In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed generalized quasij asymptotically nonexpansive mappings and the set of solutions of equilibrium problem in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others. Mathematics Subject Classification (2000): 47H09; 47H10; 47J05; 54H25. Keywords:equilibrium problem, generalized quasijasymptotically nonexpansive mapping, strong convergence, common fixed point, Banach space

1. Introduction and preliminary LetEbe a Banach space with the dualE*. LetCbe a nonempty closed convex subset ofEandf:C×C®ℝa bifunction, whereℝis the set of real numbers. The equili brium problem forfis to find ˆ x such that f  x,y ≥ (1:1)

for allyÎC. The set of solutions of (1.1) is denoted byEP(f). Given a mappingT:C ®E*, letf(x, y) =〈Tx, yx〉for allx,yÎC. Then x ˆ ∈ EP f if and only if Tx ˆ ,y − ˆ x≥0 for allyÎC, i.e., ˆ is a solution of the variational inequality. Numerous problems in physics, optimization, engineering and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for example, BlumOettli [1] and Moudafi [2]. For solving the equilibrium problem, let us assume thatfsatisfies the following conditions: (A1)f(x, x) = 0 for allxÎC; (A2)fis monotone, that is,f(x, y) +f(y, x)≤0 for allx, yÎC; (A3) for eachx, y, zÎC, lim t®0 f(tz+ (1 t)x, y)≤f(x, y); (A4) for eachxÎC, the functiony↦f(x, y) is convex and lower semicontinuous. LetEbe a Banach space with the dualE*. We denote byJthe normalized duality E mapping fromEto defined by

© 2012 Zhao and He; 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.