Best Proximity Point Results for Modified --Proximal Rational Contractions

N Hussain,P Salimi,M A Kutbi

doi:10.1155/2013/927457

N Hussain, P Salimi + Show 1 more

Open Access

https://doi.org/10.1155/2013/927457

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Abstract

We first introduce certain new concepts of --proximal admissible and ---rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.

Highlights

Introduction and PreliminariesLet (X, d) be a metric space and T be a self-mapping defined on a subset of X
If T is not a self-mapping, the equation Tx = x could have no solutions and, in this case, it is of a certain interest to determine an element x that is in some sense closest to Tx
The aim of this paper is to introduce certain new concepts of α-η-proximal admissible and α-η-ψ-rational proximal contractions of the first and second kinds

Summary

Introduction and Preliminaries

Let (X, d) be a metric space and T be a self-mapping defined on a subset of X. Best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. We establish certain best proximity point theorems for such rational proximal contractions. We deduce best proximity and fixed point results in partially ordered metric spaces. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach’s contraction principle and some of its generalizations as special cases. T : A → B is said to be a rational proximal contraction of the second kind if there exist nonnegative real numbers a, b, c, and d with a+b+2c+2d < 1, such that, for all x1, x2, u1, u2 ∈ A, d (u1, Tx1) = d (A, B) , d (u2, Tx2) = d (A, B). For the examples of α-admissible mappings with respect to η, we refer to [27] and the examples

Best Proximity and Fixed Point Results in Metric Spaces

Best Proximity and Fixed Point Results in Partially Ordered Metric Spaces