The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

Salvador Romaguera,Pedro Tirado

doi:10.3390/sym11060741

Abstract

We obtain quasi-metric versions of the famous Meir–Keeler fixed point theorem from which we deduce quasi-metric generalizations of Boyd–Wong’s fixed point theorem. In fact, one of these generalizations provides a solution for a question recently raised in the paper “On the fixed point theory in bicomplete quasi-metric spaces”, J. Nonlinear Sci. Appl. 2016, 9, 5245–5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equations.

Highlights

Throughout this paper, we denote by N the set of all positive integer numbers
We show that T is a d–Meir–Keeler map on ( X, d)
To prove the existence of solution for many recurrence equations typically associated to well-known algorithms as Quicksort, Hanoi, Largetwo, Divide and Conquer, etc., via the fixed point theory in quasi-metric spaces

Summary

Introduction

Throughout this paper, we denote by N the set of all positive integer numbers. if T is a self-map of a set X and x ∈ X, we will write Tx instead of T ( x ) if no confusion arises. The only exception to this approach was the famous Boyd and Wong fixed point theorem [2] (Theorem 1) It was given in [1] (Example 2.14) an easy example of a self-map T of a bicomplete quasi-metric space ( X, d) for which there exists a right upper semicontinuous function φ : [0, ∞) → [0, ∞) satisfying φ(t) < t for all t > 0, and d( Tx, Ty) ≤ φ(d( x, y)) for all x, y ∈ X, but T has no fixed points. Every bicomplete quasi-metric space is sequentially complete, and it is well known that the converse is not true in general not even for compact Hausdorff quasi-metric spaces We illustrate this fact with the following easy example. (1/n)n∈N is a Cauchy sequence in the metric space ( X, ds ) which does not converge for the (metric) topology τds

Results

An Application

Conclusions