Fixed point results for $(\varphi,\psi)$-contractions in metric spaces endowed with a graph and applications

Abstract

In the present work, we introduce the concepts of$(G,\varphi,\psi)$-contraction and $(G,\varphi,\psi)$-graphiccontraction defined on metric spaces endowed with a graph $G$ andwe show that these two types of contractions generalize a largenumber of contractions. Subsequently, we investigate variousresults which assure the existence and uniqueness of fixed pointsfor such mappings. According to the applications of our mainresults, we further obtain a fixed point theorem for cyclicoperators and an existence theorem for the solution of a nonlinearintegral equation. Moreover, some illustrative examples areprovided to demonstrate our obtained results.