Fixed point results for Pata contraction on a metric space with a graph and its application

Abstract

Let $(X,d)$ be a metric space endowed with a graph $G$ such that the set $V(G)$ of vertices of $G$ coincides with $X$‎. ‎We define the notion of Pata-$G$-contraction type maps and obtain some fixed point theorems for such mappings‎. ‎This extends and subsumes many recent results which were obtained for other contractive type mappings on a partially ordered metric space‎. ‎As an application‎, ‎we present theorem on the convergence of successive approximations for some linear operators on a Banach space‎.