Open Access Research

One-local retract and common fixed point in modular function spaces

Saleh A Al-Mezel1*, Abdullah Al-Roqi1 and Mohamed A Khamsi2

Author Affiliations

1 Department of Mathematics, King Abdulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA

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Fixed Point Theory and Applications 2012, 2012:109 doi:10.1186/1687-1812-2012-109

Published: 5 July 2012

Abstract

In this paper, we introduce and study the concept of one-local retract in modular function spaces. In particular, we prove that any commutative family of ρ-nonexpansive mappings defined on a nonempty, ρ-closed and ρ-bounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact and normal.

MSC: Primary 47H09; Secondary 46B20, 47H10.

Keywords:
convexity structure; fixed point; modular function space; nonexpansive mappings; normal structure; retract