On Ω Class of Mappings in a p-Cyclic Complete Metric Space

Erdal Karapınar,Prabavathy Magadevan,Boyan Zlatanov,Saravanan Karpagam

doi:10.3390/sym11040534

Abstract

In this manuscript, we introduce the concept of Ω -class of self mappings on a metric space and a notion of p-cyclic complete metric space for a natural number ( p ≥ 2 ) . We not only give sufficient conditions for the existence of best proximity points for the Ω -class self-mappings that are defined on p-cyclic complete metric space, but also discuss the convergence of best proximity points for those mappings.

Highlights

In the classical Banach fixed point theorem, the undertaking operator is necessarily continuous due to contraction inequality
One of the significant results was constructed by Bryant [1] who proved the following result: In a complete metric space, if, for some positive integer n ≥ 2, the nth iteration of the given mapping forms a contraction, it possess a unique fixed point. Another outstanding approach was proposed by Kirk, Srinivasan and Veeramani [2] by introducing the notion of cyclic contraction
We introduce a notion of p-cyclic strict contraction, which is a generalization of strict contraction in the usual sense

Summary

Introduction

In the classical Banach fixed point theorem, the undertaking operator is necessarily continuous due to contraction inequality. One of the significant results was constructed by Bryant [1] who proved the following result: In a complete metric space, if, for some positive integer n ≥ 2, the nth iteration of the given mapping forms a contraction, it possess a unique fixed point. Another outstanding approach was proposed by Kirk, Srinivasan and Veeramani [2] by introducing the notion of cyclic contraction. We mainly follow the notations defined in [9]

Motivation

Preliminaries

Ω Class of Mappings

Best Proximity Point Results of Ω Class of Mappings

Conclusions