Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations

Abstract

This paper proposes some iterative constructions of fixed points for showing the existence and uniqueness of solutions for functional equations and fractional differential equations (FDEs) in the framework of CAT (0) spaces. Our new approach is based on the $M^{\ast }$ -iterative scheme and the class of mappings with the KSC condition. We first obtain some $∆$ and strong convergence theorems using $M^{\ast }$ -iterative scheme. Using one of our main results, we solve a FDE from a broad class of fractional calculus. Eventually, we support our main results with a numerical example. A comparative numerical experiment shows that the $M^{\ast }$ -iterative scheme produces high accurate numerical results corresponding to the other schemes in the literature. Our results are new and generalize several comparable results in fixed point theory and applications.