Integro-differentiated singularly perturbed equations with fast oscillating coefficients In the study of various issues related to dynamic stability, with the properties of media with a periodic structure, in the study of other applied problems, one has to deal with differential equations with rapidly oscillating coefficients. Asymptotic integration of differential systems of equations with such coefficients was carried out by the splitting method and the regularization method. In this paper, a system of integrodifferential equations is considered. The main objective of the study is to identify the influence of the integral term on the asymptotics of the solution to the original problem. The case of the absence of resonance is considered, i.e. the case when the integer linear combination of frequencies of the rapidly oscillating coefficient does not coincide with the frequency of the spectrum of the limit operator.

The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.

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In the paper, we consider distribution of eigenvalues of a third-order differential operator with nonlocal boundary conditions on a segment. The characteristic operator of the considered spectral problem is an entire function that has an integral representation. Countability and asymptotics of zeros for an entire function on the complex plane in sectors with a small span along the imaginary axis is proved.

In the work, the Cauchy problem is considered for a singularly perturbed system with zero points of the spectrum. It is proved normal solvability of general iterative problems. Unique solvability and convergence of formal solutions to the exact one are formulated.