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Open Access Research

Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Buthinah A Bin Dehaish1* and WM Kozlowski2

Author Affiliations

1 Department of Mathematics, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi Arabia

2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

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Fixed Point Theory and Applications 2012, 2012:118  doi:10.1186/1687-1812-2012-118

Published: 20 July 2012

Abstract

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a> be a uniformly convex modular function space with a strong Opial property. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M2">View MathML</a> be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/118/mathml/M1">View MathML</a>. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.

MSC: 47H09, 47H10.

Keywords:
fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity