Common fixed point and invariant approximation in hyperbolic ordered metric spaces

Mujahid Abbas,Abdul Rahim Khan,Mohamed Amine Khamsi

doi:10.1186/1687-1812-2011-25

Abstract

Abstract We prove a common fixed point theorem for four mappings defined on an ordered metric space and apply it to find new common fixed point results. The existence of common fixed points is established for two or three noncommuting mappings where T is either ordered S-contraction or ordered asymptotically S-nonexpansive on a nonempty ordered starshaped subset of a hyperbolic ordered metric space. As applications, related invariant approximation results are derived. Our results unify, generalize, and complement various known comparable results from the current literature. 2010 Mathematics Subject Classification: 47H09, 47H10, 47H19, 54H25.

Highlights

Metric fixed point theory has primary applications in functional analysis
Geometric conditions on underlying spaces play a crucial role for finding solution of metric fixed point problems
Khamsi and Khan [5] studied some inequalities in hyperbolic metric spaces, which lay foundation for a new mathematical field: the application of geometric theory of Banach spaces to fixed point theory

Summary

Introduction

Metric fixed point theory has primary applications in functional analysis. The interplay between geometry of Banach spaces and fixed point theory has been very strong and fruitful. Several results regarding existence and approximation of a fixed point of a mapping rely on convexity hypotheses and geometric properties of the Banach spaces. In 2009, Dorić [12] proved some fixed point theorems for generalized (ψ, )-weakly contractive mappings in ordered metric spaces. Some results on invariant approximation for these mappings are established which in turn extend and strengthen various known results Such metric spaces are usually called convex metric spaces The set Y is said to be an ordered q-starshaped if there exists q in Y such that Y includes every metric segment joining any of its point comparable with q.

Then x

Since h

Tn is a selfmap on

Then for