Existence and Stability Analysis of Nonlinear Volterra-Fredholm Integral Equations

Abstract

The aim of this paper is to prove the existence and uniqueness of a solution for a Volterra-Fredholm nonlinear integral equation in two variables under certain conditions. For instance, by using the Lipschitz condition with Banach's contraction principle under assumptions (1)-(4), a unique solution exists in Banach space Z. Moreover, we examine some fundamental characteristics of the solutions for a Volterra-Fredholm nonlinear integral equation in two variables which occur in applications, using the inequality established in [6, Theorem 1]. It has also been demonstrated that the functions and parameters included in the equation under investigation continuously influence the behavior of solutions under perturbations in parameters. This investigation may allow for the extension of findings, and the last section contains an illustrative example for the validity of the obtained results, making the method much more attractive for practical applications. The examples show the method is straightforward and effective, and the approach can also be extended to other nonlinear integral equation problems.