Further generalized contraction mapping principle and best proximity theorem in metric spaces

Yongfu Su,Jen-Chih Yao

doi:10.1186/s13663-015-0373-7

Abstract

AbstractThe aim of this paper is to prove a more generalized contraction mapping principle. By using this more generalized contraction mapping principle, a further generalized best proximity theorem was established. Some concrete results have been derived by using the above two theorems. The results of this paper improve many important results published recently in the literature.

Highlights

The Banach contraction mapping principle is a classical and powerful tool in nonlinear analysis
Weak contractions are generalizations of Banach contraction mapping which have been studied by several authors, and in particular some types of weak contractions in complete metric spaces were introduced in [ ]
A mapping T : X → X is said to be a Geraghtycontraction if there exists β ∈ Γ such that for any x, y ∈ X, d(Tx, Ty) ≤ β d(x, y) d(x, y), Su and Yao Fixed Point Theory and Applications (2015) 2015:120 where the class Γ denotes those functions β : [, +∞) → [, +∞) satisfying the following condition: β(tn) → ⇒ tn →

Summary

Introduction

The Banach contraction mapping principle is a classical and powerful tool in nonlinear analysis. Let T be a self-map of a metric space (X, d) and φ : [ , +∞) → [ , +∞) be a function.

Objectives

Results

Conclusion