On a singular integral equation of Volterra and its adjoint one

Abstract

The paper deals solutions of mutually conjugate Volterra integral equations of the second kind with a variable limit of integration. Feature of equations consists of incompressibility of the corresponding kernels and, consequently, of the non-applicability of the successive approximation method. The non-zero solutions of singular homogeneous Volterra integral equations of the second kind were obtained. By direct verification it is established that the resulting functions are non-trivial solutions of the homogeneous Volterra integral equation of the second kind with a singular kernel and its adjoint equation.