Iterative Methods in Hadamard Manifolds

Abstract

This study presents several theorems of Hadamard’s manifolds, initially introducing concepts of differential manifolds, including metrics and geodesics, and then examining concepts that determine Hadamard’s manifolds. The study later concerns Halpern and Mann’s iterations of non-expansive mappings on Hadamard’s manifolds, citing a numerical example to present how these methods can be applied, particularly concerning the methods of Halpern and Mann’s iterations. Additionally, several applications of convergence results for the origin point algorithm and Picard’s iterations are mentioned. The first case concerning a minimization problem is practiced by applying the convergence results on a generated minimal problem by finding a saddle point. This is followed by solving a variational inequality.