The existence and convergence of best proximity points for generalized proximal contraction mappings

Wutiphol Sintunavarat,Poom Kumam

doi:10.1186/1687-1812-2014-228

Abstract

In 2011, Sadiq Basha (Nonlinear Anal. 74:5844-5850, 2011) studied and established best proximity point theorems for proximal contractions of the first and the second kinds which are more general than the fixed point theorems of self-contractions. The purpose of this paper is to extend the notion of proximal contraction mappings of the first and the second kinds. We also establish the existence and convergence of best proximity point theorems for these classes and give an example to validate our main results. MSC:47H10, 47H09.

Highlights

Let X be an arbitrary nonempty set
It states that if (X, d) is a complete metric space and T : X → X is a contraction mapping (i.e., d(Tx, Ty) ≤ αd(x, y) for all x, y ∈ X, where α ∈ [, )), T has a unique fixed point. It has been generalized in different ways by mathematicians over the years. Almost all such results relate to the existence of a fixed point for self-mappings
We prove the existence and convergence as regards best proximity point theorems for these classes and give some illustrative examples of our main results

Summary

Introduction

Let X be an arbitrary nonempty set. A fixed point for a self-mapping T : X → X is a point x ∈ X such that Tx = x. Definition A mapping S : A → B is said to be a generalized proximal contraction of the first kind with respect to g : A → A ∪ B if there exists a mapping ξS ∈ ΞS(g) such that d(a, Sx) = d(b, Sy) = d(A, B) ⇒ d(a, b) ≤ ξS(x)d(x, y) for all a, b, x, y ∈ A.

Objectives

Results

Conclusion