Exploring Contraction Mappings and Fixed Point Existence in Banach Spaces: A Theoretical and Applied Framework

Abstract

This work is a detailed research about the fixed-point theory with respect to Banach spaces with reference to different types of contraction mappings Banach, Kannan, Caristi, and Zamfirescu mappings. The Banach Fixed-Point Theorem forms the very heart of it, being not only theoretically elegant but also practical in terms of computational efficiency. The paper explores the generalizations and extensions of the fixed-point theorems through intensive mathematical research and examines the behavior of the convergence of the various methods as far as the iterative techniques are concerned. The various types of contractions are compared in details where their convergence speed, stability, and use in real life issues are highlighted. The theoretical constructs are confirmed by way of numerical simulations and tabulated results, and the patterns of convergence are explained by visual representations. In addition, the thesis discusses the various uses of the fixed-point theory in digital image processing, differential equations, operator theory and computational mathematics. Special emphasis is put on how compact linear operators, partial metric spaces and simulation functions are used to make the fixed-point results more robust. This combined methodology mediates between abstract mathematics and applied fieldwork, and illustrates the extensiveness of the use of fixed-point analysis in both theoretical and computational science.