INTEGRO-DIFFERENTIAL PROBLEM ABOUT PARAMETRIC AMPLIFICATION AND ITS ASYMPTOTICAL INTEGRATION

Asymptotic integration of differential systems of equations with fast oscillating coefficients has been carried out by the Feschenko-Shkil-Nikolenko splitting method and the Lomov regularization method. Equations of this type are often encountered in study of various questions related to dynamic stability, to properties of media with a periodic structure and other applied problems. In the monograph by Yu.L. Daletski and M.G. Krein an asymptotic analysis is given for one of these problems the problem on parametric amplification. In the present paper, we generalize this problem to integro-differential equations, the differential part of which coincides with the parametric amplification problem. The main purpose of the research is to identify the influence of the integral term in the asymptotic behavior of the solution. It is considered the general case, i.e. the case of both the lack of resonance (when the integer linear combination of frequencies of the fast oscillating cosine does not coincide with the spectrum frequency of the limit operator), and its presence (when such coincidence takes place). The developed algorithm is obviously generalized to systems of equations with an arbitrary matrix of the differential part, Received: November 22, 2019 c © 2020 Academic Publications §Correspondence author 332 A.A. Bobodzhanov, B.T. Kalimbetov, V.F. Safonov the pure imaginary spectrum, and with an arbitrary number of fast oscillating coefficients (such as the cosine considered in the paper). AMS Subject Classification: 34K26, 45J05