Regularization of a Singularly Perturbed Integro-differential Equation with Several Unstable Spectral Values of Exponential Inhomogeneity

The article considers Cauchy’s integro-differential problem with two exponential inhomogeneities, the spectral values of which vanish at various isolated points on a segment of the independent variable. The problem belongs to the class of singularly perturbed equations with an unstable spectrum. Previously, a similar problem was considered for the case when there is a single point of spectral values instability. A study of such problems is especially difficult in the vicinity of zeros of the spectral inhomogeneity value. In the case considered, the application of the well-known procedure of Lomov's regularization method does not seem to be possible. Therefore, it was decided to apply a method for constructing the asymptotics of the solution to the original problem based on the use of a regularized asymptotic solution to the fundamental solution of the corresponding homogeneous equation, the construction of which has not been considered until now from the standpoint of the regularization method. In the first part of the study, an algorithm for the regularization method is developed for constructing the asymptotics (of any order in the parameter) of the fundamental solution of the corresponding homogeneous integro-differential equation. The second part is devoted to constructing the asymptotics of the solution to the original problem. The main term of the asymptotics is constructed in detail, and the possibility of constructing its higher terms is pointed out.