A new extension to the controlled metric type spaces endowed with a graph

Abstract

In this paper, we initiate a new extension of b-metric spaces, called controlled metric-like spaces, by changing the condition [℘(s,r)=0⇔s=r]by [℘(s,r)=0⇒s=r] and that means basically we may have a non-zero self-distance. We prove some fixed point theorems which generalize many results in the literature. Also, we present an interesting application to illustrate our results by considering controlled metric-like spaces endowed with a graph.