Best Proximity Points for Some Classes of Proximal Contractions

Given a self-mapping g : A -> A and a non-self-mapping T : A -> B, the aim of this work is to provide sufficient conditions for the existence of a unique point x is an element of A, called g-best proximity point, which satisfies d(gx, Tx) = d(A, B). In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function x -> d(gx, Tx), 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.