On solvability of a nonlinear Volterra integral equation

The goal of this article is to provide a detailed discussion on the solveability of nonlinear Volterra integral equations in the space of integral functions. Unlimited le bass and using the fact that the given integral function equation can be reduced to a nonlinear integral function equation as much as possible. Applying Schauder\'s fixed point theorem to the weak incompatibility measure defined by de Blasi in this paper. We have created sufficient conditions that guarantee the existence of solutions, so we avoid the inconsistency problem that is naturally encountered when working on infinite spaces. This is because the classical fixed point results can only be used for the additional compactness assumption. In this paper we use de Blasi\'s measure and extend the classical existence results to include larger classes of nonlinear integral equations. It happens without the need for brevity. An example is also provided to show an application of our existence theorem that can satisfy the conditions from this paper. This example demonstrates the practical relevance of our theoretical findings, and indicates the versatility of the proposed approach to real-world problems. It is modeled by the Yanti nonlinear Volterra integral equation. The present work improves the theoretical and practical foundation of the nonlinear integral equation. At the same time it provides a good framework for treating various applications of physics, mathematics and engineering.\r\nKeywords: Mathematical physics, caratheodory conditions, Schauder fixed point theorem, integral functional equation, various applications.