A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.
On the asymptotic solutions of a class of ordinary differential equations of the fourth order, with special reference to an equation of hydrodynamics
