On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

M De La Sen

doi:10.1155/2016/4186960

Abstract

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-calledr-weaker Meir-Keeler orr,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.

Highlights

The background literature on best proximity points and associated convergence properties in cyclic contractions and proximal contractions in the framework of fixed point theory is abundant
On the other hand, fixed point theory has a wide amount of applications, for instance, in the study of stability of dynamic systems and differential and difference equations
The relevance of cyclic contractions and cyclic nonexpansive mappings is of interest when strips of the solutions of dynamic systems or difference equations have to lie in different time intervals or due to control actions or external events in distinct defined sets

Summary

Introduction

The background literature on best proximity points and associated convergence properties in cyclic contractions and proximal contractions in the framework of fixed point theory is abundant. The paper has a section of preliminaries where the concepts of Meir-Keeler functions, weaker Meir-Keeler functions, and stronger Meir-Keeler functions are generalized “ad hoc” to be used to define weak generalized contractive mappings involving subsets of a generalized metric space which do not intersect in general In this context, appropriate nondecreasing functions φ : [0, ∞) → [0, ∞), generalized weaker Meir-Keeler functions φ : [D, ∞) → [D, ∞), and stronger Meir-Keeler functions ψ : [0, ∞) → [0, 1) are used to define the generalized (φ − φ)- and (φ − ψ)-weak p-cyclic contraction mappings defined and studied in this paper. Some illustrative examples adapted to the stated and proved results are discussed

Preliminaries

Best Proximity Point and Fixed Point Theorems