Singularly Perturbed Integro-Differential Systems with Kernels Depending on Solutions of Differential Equations

We consider integro-differential equations (IDEs) with a rapidly oscillating inhomogeneity and with a Volterra-type integral operator whose kernels can contain both a classical rapidly decreasing exponential (the simplest case) and fundamental solutions of differential systems (the general case). Difficulty in constructing a regularized (according to S.A. Lomov) asymptotics in the general case is due to the complex asymptotic structure of the fundamental solution matrix (Cauchy matrix) of the homogeneous differential system. In the present paper, we first construct a regularized asymptotics of the Cauchy matrix, which is then used to construct a regularized asymptotics of the solution of the IDE.