We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense.

It is proved that the solutions of Tikhonov systems, in addition to having the property of a limiting transition, are pseudoholomorphic under certain conditions additional to the conditions of A.N. Tikhonov’s theorem. At the same time, the number of fast and slow variables can be anything. Both initial and boundary value problems for systems of this type are considered.

A method for constructing pseudo-holomorphic solutions to strongly nonlinear singularly perturbed systems of differential equations, which a logical continuation of the Lomov regularization method, is proposed. The existence of integrals of such systems, holomorphic in the small parameter, is proven, and sufficient conditions for the convergence of expansion of solutions to these systems in powers of the small parameter in the usual sense are obtained.

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

For nonlinear evolution equations in a Banach space that depend in two ways—regularly and singularly—on a small parameter, we construct \(\varepsilon \)-pseudoregular solutions of the Cauchy problem, i.e., its formal solutions representable as series in powers of the small parameter with coefficients that depend on it in a singular way and converging in a certain neighborhood of the zero value of the parameter uniformly over a given time interval. Sufficient conditions are obtained under which the sum of such a series is an exact, and hence pseudoholomorphic, solution of this problem.

To investigate Tikhonov systems arising in problems of control theory we apply the method of holomorphic regularization, which allows one to obtain solutions to singularly perturbed problems in the...