Generalized solution to the Volterra equations with piecewise continuous kernels

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution’s regular part refinement is proposed using the successive approximations method.