A new contribution to discontinuity at fixed point

TL;DR

This paper introduces new fixed point results for discontinuous mappings using a modified contractive condition, with applications to discontinuous activation functions in neural networks.

Contribution

It provides novel fixed point theorems that do not require continuity at the fixed point, expanding the scope of contractive mappings.

Findings

Established new fixed point theorems for discontinuous maps.

Applied results to discontinuous activation functions.

Extended classical inequalities to broader contexts.

Abstract

The aim of this paper is to obtain new solutions to the open question on the existence of a contractive condition which is strong enough to generate a fixed point but which does not force the map to be continuous at the fixed point. To do this, we use the right-hand side of the classical Rhoades' inequality and the number $M(x,y)$ given in the definition of an $(\alpha ,\beta )$-Geraghty type-$I$ rational contractive mapping. Also we give an application of these new results to discontinuous activation functions.