Some fixed point theorems in generalized modular metric space

In this paper, we discuss some properties of generalized modular metric spaces. This space was first introduced by Turkoglu and Manav in 2018. Later, we extend the concepts about contraction mappings in generalized modular metric spaces. We also develop a convexity structure and defined the N normal structure in this space.The results show us that if X is a complete modular metric space and T^N is a contraction mapping then T has a unique fixed point.Moreover, we need the constant of generalized modular metric axiom should be less than or equal to 1 to guarantee the ball is closed. The closeness property of the ball is necessary to generate normal structure in generalized modular metric space. This normal structure property assures the existence of the fixed point of a nonexpansive mapping.