The Convex (δ, L) Weak Contraction Mapping Theorem and its Non Self Counterpart in Graphic Language

Abstract

Let be a metric space. A map is said to be a weak contraction [1] if there exists and such that the following inequality holds for all

On the other hand, the idea of convex contractions appeared in [2] and [3]. In the first part of this paper, motivated by [1]-[3], we introduce a concept of convex  weak contraction, and obtain a fixed point theorem associated with this mapping. In the second part of this paper, we consider the map is a non-self map, and obtain a best proximity point theorem. Finally, we leave the reader with some open problems.