$(\epsilon-\delta)$ conditions and fixed point theorems

Abstract

In this paper, we prove some fixed point theorems under generalized $(\epsilon-\delta)$ type rational contractions in which the fixed point may be a point of discontinuity. Our results generalize and improve a host of well-known fixed point theorems existing in the literature. In addition to it we give a fixed point theorem for $(\epsilon-\delta)$ non-expansive mappings in metric spaces. Several examples are given to illustrate our results.