Solution Approaches to Nonlinear Volterra Integral Equations

The goal of this paper is to discuss in detail the solvability in the space of Lebesgue integrable functions in the unbounded interval of a nonlinear Volterra integral equation and use as far as possible the fact that a given integro-differential equation can be reduced to an equivalent nonlinear integral functional equation. In the present paper, using Schauder\'s fixed-point theorem along with a weak measure of noncompactness defined by De Blasi, we establish sufficient conditions which guarantee the existence of solutions. We shall thus avoid the problem of noncompactness which appears in a natural way when working in infinite-dimensional spaces since only for additional compactness assumptions can the classical fixed-point results be applied. In the present paper, we apply the De Blasi\'s measure and extend the classical existence results to include a bigger class of nonlinear integral equations which need not be compact. Also an illustrative example is given for showing the application of our existence theorem by which conditions from the paper could be fulfilled. This example represents the practical relevance of our theoretical findings and points out the versatility of the proposed approach to real-world problems modeled by nonlinear Volterra integral equations. The obtained results represent a contribution to the ongoing effort of mathematical analysis of integral equations, bringing some novelties in discussing their solvability on unbounded domains and enlarging the scope of applicable mathematical tools. The present work establishes an improvement in the theoretical and practical foundation of nonlinear integral equations, providing at the same time a sound framework for treating various applications of mathematical physics and engineering disciplines.