Regularized asymptotic solutions of integro-differential equations with fast and slow variables

The paper considers a nonlinear integro-differential system with fast and slow variables. Such systems have not been considered previously from the point of view of constructing regularized (according to Lomov) asymptotic solutions. Known works were mainly devoted to the construction of the asymptotics of the Butuzov-Vasil'eva boundary layer type, which, as is known, can be applied only if the spectrum of the matrix of the first variation (on the degenerate solution) is located strictly in the open left half-plane of a complex variable. In the case when the spectrum of the indicated matrix falls on the imaginary axis, the method of regularization by S.A. Lomov. However, this method was developed mainly for singularly perturbed differential systems that do not contain integral terms, or for integro-differential problems without slow variables. In this paper, the regularization method is generalized to two-dimensional integro-differential equations with fast and slow variables.