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 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.
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