Best Proximity Point Theorem in Quasi-Pseudometric Spaces

Abstract

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the error $\inf \{d(x,y):y\in T(x)\}$, and hence the existence of a consummate approximate solution to the equation $T(X)=x$.