Existence and uniqueness of solutions for third-kind linear Volterra integral equations

Abstract

In this paper, there are studied third-kind linear Volterra integral equations with smooth data and the operator of multiplying by a smooth function that degenerates at the initial point of the integration interval. A theorem of existence, uniqueness, and continuity of a solution is proved. Conditions for smoothness and the degree of smoothness of a solution are obtained. Additionally, the existence and uniqueness of the solution in the \(L^p\) space are proved.