An Advanced Generalized Fixed Point Theorem for Solving Ordinary Differential Equations

Abstract

This study examines the use of the generalized fixed point (FP) theorems in handling ordinary differential equations (ODEs), and specifically in the framework of fundamental principles and applications. Their major findings such as Banach's Contraction Principle and Schauder's Theorem are used to prove the existence and uniqueness of solutions in diverse mathematical systems. The paper generalizes fixed point theories to include more general classes of contraction mappings and boundary value problems, thus providing new methods of analysis to solve complicated ODEs. Through a critical analysis of metric spaces and iterative procedures, the study presents some of the historical and theoretical breakthroughs and puts into perspective practical applications in the engineering, physics, and computational mathematics sectors, among others. The major contribution of the work is the formulation of generalized contraction principles, that allow extending the applicability of FP theorems to non-traditional spaces and open up new methods of solving differential equations.