A Fixed Point Theorem for a General Contractive Condition of Integral Type in Modular Spaces

Introduction: Orlicz and Birnbaum considered spaces of functions with growth properties different from the power type growth control, provided by the p L -norm. This generalization found many applications in differential and integral equations with kernels of non-power types. A more abstract approach was given by Nakano in connection with the theory of order spaces. His point of view was generalized by Musielak and Orlicz (1950) who de ned what is now commonly known as modular spaces. Aim: Banach's fixed point theorem is the first and famous fact in the fixed point theory. As an application, one can constructive a solution of an ordinary differential equations. The propose of this paper is to establish some analogues of Banach’s xed point theorem for contractive mappings and mention an application in modular spaces. Material and Method: the existence of a xed point for mappings satisfying a contractive condition of integral type in modular spaces is studied. Moreover, we extend the contractive conditions of integral type to solve an integral equation in Musielak-Orlicz space. Our method is based on constructing a contractive operator of integral type in such way, it has a fixed point. This fixed point can be considered as a solution of the integral equation. Results: Some xed point theorems in modular spaces are presented. Moreover, as an application of these theorems, the existence of a solution of an integral equation in MusielakOrlicz space, is proved. Conclusion: one can consider an integral equation in the modular spaces and can find a solution of it. This shows that one can solve integral equations in a space where members has growth properties different from the power type growth control, provided by the p L -norm.