Ample Spectrum Contractions in Branciari Distance Spaces

Abstract

Very recently, the notion of ample spectrum contraction has been introduced in order to unify, under the same axioms, a large number of contractive mappings that have had great success in the field of Fixed Point Theory in recent years and that have been used in a wide variety of applications in Nonlinear Analysis (Meir-Keeler contractions, Geragthy contractions, contractions under simulation functions, contractions under R-functions, etc.) However, the subtle conditions that define ample spectrum contractions cannot be extended as they are to new kinds of abstract metric spaces because they involve key properties that are only fulfilled in metric spaces. In this paper, based on a very recent work in which the authors unravel the essential properties of the topology in Branciari spaces, we investigate the reasons why the proposed axiomatic fails in Branciari spaces and we illustrate how to overcome such drawbacks. As a consequence, we characterize the notion of ample spectrum contraction in the setting of Branciari distance spaces and we also investigate the existence and uniqueness of fixed points for such family of contractions in the context of complete Branciari distance spaces.