On the solutions of the homogeneous mutually conjugated Volterra integral equations

Abstract

The paper deals with a pair of conjugate integral equations of Volterra type 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. By using the Carleman-Vekua method of regularization the investigated equation is reduced to the Abel equation of the second kind. The existence of the non-trivial solutions is established for homogeneous Volterra integral equation of the second kind with a special kernel and its adjoint equation in weight Lebesgue classes.