Existence of a Solution for a Nonlinear Integral Equation via Fixed Point Theory

Integral equations serve as a central tool in modelling phenomena across physics, engineering, and applied sciences. Proving existence and uniqueness of solutions for nonlinear integral equations remains a core challenge, especially when kernels depend nonlinearly on the unknown function. In this work we establish novel, easily verifiable conditions under which a nonlinear Fredholm integral equation admits a unique solution by constructing a contraction mapping in a Banach space. The novelty of our approach lies in refining the classical Lipschitz framework to accommodate variable kernels and providing explicit convergence estimates. We also present a completely verified example and relate our method to recent advances in fixed point techniques for integral equations. This combination of rigorous theory and concrete verification makes the paper suitable for applications in mathematical modelling.