Solution to the Volterra Operator Equations of the 1st kind with Piecewise Continuous Kernels

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to Volterra operator equations of the first kind with piecewise continuous kernels, and proposes an iterative algorithm for solution improvement.

Contribution

It provides new sufficient conditions for solutions' existence and uniqueness, and introduces an asymptotic approximation and iterative method for non-unique solutions.

Findings

Conditions for existence and uniqueness derived

Asymptotic approximation for non-unique solutions constructed

An iterative algorithm for solution refinement proposed

Abstract

The sufficient conditions for existence and uniqueness of continuous solutions of the Volterra operator equations of the first kind with piecewise continuous kernel are derived. The asymptotic approximation of the parametric family of solutions are constructed in case of non-unique solution. The algorithm for the solution's improvement is proposed using the successive approximations method.