Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times
Бекiтiлмеген уақыт мезетiндегi импульстi жүйенiң ось бойындағы екiжақты, шектелген шешiмдерi және орташалау әдiсi
Метод усреднения и двусторонние, ограниченные на оси решения импульсных систем с нефиксированными моментами времени
Stanzhytskyi O.N. Assanova A.T. Mukash M.A.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#104Issue 4142 - 150 pp.
The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.
averaging method , equilibrium position , impulsive effects , small parameter , stability
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Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
K.Zhubanov Aktobe Regional State University, Aktobe, Kazakhstan
Taras Shevchenko National University of Kyiv
Institute of Mathematics and Mathematical Modeling
K.Zhubanov Aktobe Regional State University
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