Best Proximity Points for Some Classes of Proximal Contractions

Abstract

Given a self-mapping $g:A\to A$ and a non-self-mapping $T:A\to B$, the aim of this work is to provide sufficient conditions for the existence of a unique point $x\in A$, called g-best proximity point, which satisfies $d\left(gx,Tx\right)=d\left(A,B\right)$. In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function $x\to d\left(gx,Tx\right)$, thereby getting an optimal approximate solution to the equation $Tx=gx$. An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach's contraction principle to the case of non-self-mappings.