Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces

Kamraksa Uthai, Wangkeeree, Wangkeeree Rabian

Description

We first prove the existence of a solution of the generalized equilibrium problem (GEP) using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings. AMS Subject Classification : 47H09, 47H10

Sujets

• Banach space
• Fixed point
• Metric map

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 7
• Langue: English

Extrait

Kamraksa and WangkeereeFixed Point Theory and Applications2011,2011:11 http://www.fixedpointtheoryandapplications.com/content/2011/1/11

R E S E A R C HOpen Access Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in banach spaces 1 1,2* Uthai Kamraksaand Rabian Wangkeeree

* Correspondence: [e-mail] 1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Full list of author information is available at the end of the article

Abstract We first prove the existence of a solution of the generalized equilibrium problem (GEP) using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solution set of a GEP and the fixed point set of countable family of nonexpansive mappings in the frameworks of Banach spaces. By means of a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solution set of GEP and common fixed point set of nonexpansive mappings. AMS Subject Classification: 47H09, 47H10 Keywords:Banach space, Fixed point, Metric projection, Generalized equilibrium pro blem, Nonexpansive mapping

1. Introduction LetEbe a real Banach space with the dualE* andCbe a nonempty closed convex subset ofEand thesets of positive integers and real numbers,. We denote by E* respectively. Also, we denote byJthe normalized duality mapping fromEto 2 defined by ∗ ∗∗2∗2 Jx=x∈E:x,x=x=x,∀x∈E where〈∙,∙〉denotes the generalized duality pairing. We know that ifEis smooth, then Jis singlevalued and ifEis uniformly smooth, thenJis uniformly normtonorm con tinuous on bounded subsets ofE. We shall still denote byJthe singlevalued duality mapping. Let :C×C→ be a bifunction andA:C®E* be a nonlinear mapping. We consider the following generalized equilibrium problem (GEP): Findu∈Csuch thatf u,y+Au,y−u ≥0,∀y∈C (1:1) The set of suchuÎCis denoted byGEP(f), i.e., GEP f={u∈C:f u,y+Au,y−u ≥0,∀y∈C}

© 2011 Kamraksa and Wangkeeree; 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.