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Vladikavkazskii Matematicheskii Zhurnal, 2022, Volume 24, Number 4, Pages 105–116
DOI: https://doi.org/10.46698/w0398-0994-2990-z
(Mi vmj840)
 

Total Poisson boundedness and total oscillability of solutions of systems of differential equations

K. S. Lapin

Mordovian State Pedagogical University named after M. E. Evseviev, 11 A Studencheskaya St., Saransk 430007, Russia
References:
Abstract: In the works of the author, the study of a special form of boundedness of solutions of systems of differential equations, namely, their Poisson boundedness, has started. The concept of Poisson boundedness of a solution generalizes the classical concept of boundedness of a solution and means that there is a ball in the phase space and there is a countable system of disjoint intervals on the time semiaxis such that the sequence of right ends of intervals tends to plus infinity and the solution for all values of time from these intervals is contained in the ball. Further, in the author's papers, on the basis of methods of Lyapunov functions, Lyapunov vector functions, and higher-order derivatives of Lyapunov functions, sufficient conditions for various types of Poisson boundedness of all solutions were obtained. In particular, sufficient conditions were obtained for total Poisson boundedness (Poisson boundedness under small perturbations), partial total Poisson boundedness, and also partial total Poisson boundedness of solutions with partially controlled initial conditions. In this paper, we obtaine an asymptotic or, in other words, final characterization of the concept of Poisson boundedness of a solution, which made it possible to establish a connection between the concept of a Poisson bounded solution and the concept of an oscillating solution. Further, the concepts of total oscillating of solutions, partial total oscillating of solutions, and partial total oscillating of solutions with partially controlled initial conditions are introduced. Based on the above final characterization of the concept of Poisson boundedness of a solution, and also on the basis of the method of Lyapunov vector functions with comparison systems, we obtain sufficient conditions for total oscillating, partial total oscillating, and partial total oscillating of solutions with partially controlled initial conditions. As a consequence, sufficient conditions for the above types of total oscillating of solutions are obtained in terms of Lyapunov functions.
Key words: total boundedness of solutions, unboundedness of solutions, Lyapunov vector functions, Poisson boundedness of solutions, oscillating of solutions, partial oscillating of solutions.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation МК-211.2020.1
Received: 04.10.2021
English version:
Sib. Math. J., 2023, Volume 64, Issue 4, Pages 988–995
DOI: https://doi.org/10.1134/S0037446623040201
Bibliographic databases:
Document Type: Article
UDC: 517.925.54
MSC: 34C11, 34D20
Language: Russian
Citation: K. S. Lapin, “Total Poisson boundedness and total oscillability of solutions of systems of differential equations”, Vladikavkaz. Mat. Zh., 24:4 (2022), 105–116; Sib. Math. J., 64:4 (2023), 988–995
Citation in format AMSBIB
\Bibitem{Lap22}
\by K.~S.~Lapin
\paper Total Poisson boundedness and total oscillability of solutions of systems of differential equations
\jour Vladikavkaz. Mat. Zh.
\yr 2022
\vol 24
\issue 4
\pages 105--116
\mathnet{http://mi.mathnet.ru/vmj840}
\crossref{https://doi.org/10.46698/w0398-0994-2990-z}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4527683}
\transl
\jour Sib. Math. J.
\yr 2023
\vol 64
\issue 4
\pages 988--995
\crossref{https://doi.org/10.1134/S0037446623040201}
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