Geometric properties of discontinuous fixed point set of ($$\varvec{\epsilon -\delta }$$) contractions and applications to neural networks

Abstract

In this paper, we prove some fixed point theorems under a convex combination of generalized ( $$\epsilon -\delta $$ ) type rational contractions in which the fixed point may or may not be a point of discontinuity. As a by-product we explore some new answers to the open question posed by Rhoades (Contemp Math 72:233–245, 1988). Furthermore, we consider geometric properties of the fixed point set of a self-mapping on a metric space. We define a new kind of contractive mapping and prove that the fixed point set of this kind of contraction contains a circle (resp. a disc). Several non-trivial examples are given to illustrate our results. Apart from these, an application of discontinuous activation functions, frequently used in neural networks is also given.