Regularized Asymptotics of Solutions of Integro-Differential Equations with Zero Operator in the Differential Part and with Slowly and Rapidly Varying Kernels

摘要

We consider the Cauchy problem for a linear integro-differential equation whose differential part contains only the first derivative multiplied by a small positive parameter and whose integral part is the sum of two Volterra integral operators, one with slowly and one with rapidly varying kernel. Earlier, this Cauchy problem has only been considered for the case in which the integral part of the equation does not contain an operator with slowly varying kernel. We develop the Lomov regularization method for the new class of problems and use it to construct the asymptotics of the solution and obtain a convergence estimate.