Fixed-point Theorems in Metric Spaces and Its Applications for Nonlinear Equations

We provide a comprehensive review of key fixed-point theorems, including Banach's Contraction Principle Kalaiarasi and Jain (2022), Schauder's Fixed-Point Theorem Jain et al. (2021), and their generalizations Liew et al. (2015). Emphasis is placed on the conditions under which these theorems can be applied to nonlinear integral equations. Examples demonstrate the practical implementation of these theorems to guarantee the existence and uniqueness of solutions. The results highlight the interplay between the structure of metric spaces, operator properties Lu et al. (2021), and the formulation of integral equations Lusch et al. (2018), offering a robust framework for tackling nonlinear problems across diverse applications Lyu et al. (2023). This research investigates the applicability of these methods in addressing nonlinear integral equations, which are commonly found in the mathematical modeling of physical, biological, and engineering systems.