Convergence and Best Proximity Points for Generalized Contraction Pairs

Slah Sahmim,Abdelbasset Felhi,Hassen Aydi

doi:10.3390/math7020176

Abstract

This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq Basha (Basha, S., Best proximity points: global optimal approximate solutions, J. Glob. Optim. 2011, 49, 15–21) As an application, we give a result for nonexpansive mappings in normed vector spaces.

Highlights

Introduction and PreliminariesLet ( X, d) be a metric space
Best proximity point theorems furnish sufficient conditions yielding the existence of approximate solutions, which are optimal, as well
We give conditions ensuring the existence of best proximity points via contraction pairs

Summary

Introduction

Introduction and PreliminariesLet ( X, d) be a metric space. Consider two nonempty subsets P and Q of X. Given P and Q two nonempty subsets of X, : (i ) Every cyclical Cauchy sequence is bounded. The pair ( P, Q) is called proximally complete if, for every cyclically Cauchy sequence (ξ n ) ⊆ P ∪ Q, (ξ 2n ) and (ξ 2n+1 ) have convergent subsequences in P and Q, respectively.

Results

Conclusion