Open Access Research

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

Uthai Kamraksa1 and Rabian Wangkeeree12*

Author Affiliations

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2 Centre of Excellence in Mathematics, Che, Si Ayutthaya Road, Bangkok 10400, Thailand

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Fixed Point Theory and Applications 2011, 2011:11  doi:10.1186/1687-1812-2011-11

Published: 28 June 2011

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 problem; Nonexpansive mapping