Abstract. In this paper, we study and prove the existence of fixed point theorems for fourcompatible, weakly compatible and occasionally weakly compatible mapping of integral type in modular metric spaces and generalized the result of Azadifar et al. [2] Rahimpoor et al. [16] and Rashwan and Hammad [17].

Abstract. The notion of a modular metric spaces were introduced by Chistyakov [5,6]. Abdou and Khamsi [1] gave the analog of Banach contraction principle in modular metric spaces. More recently, Alfuraidan [3] gave generalization of Banach contraction principle on a modular metric space endowed with a graph which is the modular metric version of Jachymski [8] fixed point results.In this paper, we generalize and prove some fixed point results for Kannan contraction and weakly contractive mappings in a modular metric space endowed with a graph. The result of this paper is new and improving the previously known result in modular metric spaces endowed with a graph.