Certain Novel Best Proximity Theorems with Applications to Complex Function Theory and Integral Equations

Abstract

Let E and F be nonempty disjoint subsets of a metric space (M,d). For a non-self-mapping φ:E→F, which is fixed-point free, a point ϰ∈E is said to be a best proximity point for the mapping φ whenever the distance of the point ϰ to its image under φ is equal to the distance between the sets, E and F. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations.