Cyclic $(\alpha ,\beta )$-Admissible Mappings in Modular Spaces and Applications to Integral Equations

Abstract

The main concern of this study is to present a generalization of Banach's fixed point theorem in some classes of modular spaces, where the modular is convex and satisfying the $\Delta _{2}$-condition. In this work, the existence and uniqueness of fixed point for $(\alpha ,\beta )-(\psi ,\varphi )-$ contractive mapping and $\alpha -\beta -\psi -$weak rational contraction in modular spaces are proved. Some examples are supplied to support the usability of our results. As an application, the existence of a solution for an integral equation of Lipschitz type in a Musielak-Orlicz space is presented.