A new contribution to discontinuity at fixed point

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 $(α,β)$-Geraghty type-$I$ rational contractive mapping. Also we give an application of these new results to discontinuous activation functions.