Generalized Extensions of the Banach Fixed Point Theorem and Their Computational Applications in Nonlinear Analysis

Abstract

This paper discusses Banach Fixed Point Theorems with generalizations and its use in nonlinear analysis and control systems. The Banach Fixed Point Theorem is considered one of the foundations of mathematical analysis, which gives assurances of existence and uniqueness to solutions to other problems, particularly in metric spaces. But the practical use of this theorem often has to be more general, and extensions to include a more general form of contraction mappings have been made, e.g. the Kannan, Caristi, and other general mappings. These more generalized forms allow the study of fixed-point problems in more general contexts, such as multivalued and fuzzy systems, and in higher-order or perturbed metric spaces. This paper examines the convergence behavior of these generalized contraction mappings offering a theoretical understanding and practical uses of the contraction mappings in various applications including image processing, economics and in control systems. Numerical simulations and graphical results which indicate the effectiveness of these generalized mappings in stabilizing systems and solutions of integral equations are also incorporated in the paper. In the end, the research contributes to the applicability of the theory of fixed-point, which can help to solve various complicated nonlinear problems and contribute to the overall mathematical community.