The Initialization Problem and Passage to the Limit in a System of Nonlinear Integro-Differential Equations with Rapidly Varying Kernels
The problem of passage to the limit in a system of nonlinear singularly perturbed integro-differential equations with a rapidly decreasing kernel is addressed. In contrast to publications dealing with equations with a small parameter at the derivative and with the limiting operator's spectrum lying strictly to the left of the imaginary axis, the existence of purely imaginary points of the spectrum is assumed in this study. In this case, passage to the limit in solving the original problem (with the small parameter tending to zero) to solving the singular system in the uniform metric is impossible in the general case. The purpose of this study is to isolate a class of initial data (the initialization class) with which passage to the limit in the metric of the space of continuous functions is possible. Information about the principal term of a regularized (in the Lomov sense) asymptotic solution was used in studying this matter. However, since the system of differential equations that is consistent with the coefficients of this asymptotic solution is a nonlinear one, the solvability of this system as a whole in a specified finite interval of time remains questionable. In the earlier works of the authors it was shown that this differential system is a normal form; that is, it contains only nonlinear resonant monomials on its right-hand side, due to which its order can be decreased. However, this circumstance does not remove the problem of its solvability as a whole. The situation becomes very difficult in the presence of purely imaginary points in the spectrum. These points generate rapidly oscillating terms in the solution of the initial problem, which prevent the passage to the limit in a uniform metric. The authors succeeded in showing that the subsystem corresponding to purely imaginary eigenvalues will be a closed one. Owing to this, it was possible to distinguish a class of initial vectors for the initial problem with which rapidly oscillating components in the solution vanish, and a uniform transition becomes possible.