On the Theory of the Existence of Bounded Solutions of Systems of Nonlinear Ordinary Differential Equations
Author(s):
Ergashboy Mukhamadiev
Doctor of Physical and Mathematical Sciences,Professor,
Professor of the Department of Mathematics,
Vologda State University.
Russia, 160000, Vologda, st. Lenina, 15.
emuhamadiev@rambler.ru
Alizhon Nabidjanovich Naimov
Doctor of Physical and Mathematical Sciences,Professor,
Professor of the Department of Mathematics,
Vologda State University.
Russia, 160000, Vologda, st. Lenina, 15.
naimovan@vogu35.ru
Abstract:
We formulate and prove necessary and
sufficient conditions that provide an a priori estimate
for bounded solutions for one class of systems
of nonlinear ordinary differential equations
with the main positively homogeneous part.
criterion for the existence of bounded solutions
is proved using the method of guiding functions
and Vazhevski's method under the condition
of an a priori estimate. These results refine
and generalize the previously obtained results
of the authors in the multidimensional case.
Keywords
- a priori estimate
- bounded solution
- homotopic functions
- method of guiding functions
- Vazhevski's method
References:
- Krasovsky N. N. Nekotoryye zadachi teorii ustoychivosti dvizheniya [Some problems of the theory of stability of motion]. Moscow: GIFML, 1959. 211 p
- Pliss V. A. Nelokal'nyye problemy teorii kolebaniy [Nonlocal problems of the theory of oscillations]. Moscow, Nauka Publ., 1964. 367 p
- Krasnosel'skii M. A. Operator sdviga po trayektoriyam differentsial'nykh uravneniy [Operator of Translation Along the Trajectories of Differential Equations]. Moscow, Nauka Publ., 1966. 331 p
- Krasnosel'skii M. A., Zabreiko P. P. Geometricheskiye metody nelineynogo analiza [Geometric methods of non-linear analysis]. Moscow, Nauka Publ., 1975. 512 p
- Obukhovskii V., Kornev S., Van Loi N., Zecca P. Method of guiding functions in problems of nonlinear analysis. Lecture notes in mathematics. GmbH, Springer-Verlag , 2076. 2013. Pp. 1-173
- Hartman P. Obyknovennyye differentsial'nyye uravneniya [Ordinary Differential Equations]. Moscow, Mir Publ., 1970. 720 p
- Mukhamadiyev E. O postroyenii pravil'noy napravlyayushchey funktsii dlya sistemy differentsial'nykh uravneniy [On the construction of a correct guiding function for a system of differential equations]. Doklady Akademii nauk SSSR, 1970; 190 (4): 777-779. (In Russ. )
- Mukhamadiyev E. Ogranichennyye resheniya i gomotopicheskiye nvarianty sistem nelineynykh differentsial'nykh uravneniy [Bounded solutions and homotopy invariants of systems of nonlinear differential equations]. Doklady Akademii nauk, 1996; 351 (5): 596-598. (In Russ. )
- Mukhamadiyev E., Naimov A. N. Kriterii sushchestvovaniya periodicheskikh i ogranichennykh resheniy dlya trekhmernykh sistem differentsial'nykh uravneniy [Criteria for the existence of periodic and bounded solutions for three-dimensional systems of differential equations]. Trudy IMM UrO RAN, 2021; 27 (1): 157-172. (In Russ. )
- Perov A. I., Kaverina V. K. Primeneniye idey metoda napravlyayushchikh funktsiy pri issledovanii neavtonomnoy sistemy obyknovennykh differentsial'nykh uravneniy [Application of the ideas of the method of guiding functions in the study of a non-autonomous system of ordinary differential equations]. Vestnik Tambovskogo univer. Seriya: Yestest. i tekhn. nauki, 2018; 23 (123): 510-516. (In Russ. )
- Perov A. I., Kaverina V. K. Ob odnoy zadache Vladimira Ivanovicha Zubova [On a problem posed by Vladimir Ivanovich Zubov]. Differents. uravneniya, 2019; 55 (2): 269-272. (In Russ. )
- Mukhamadiev E., Naimov A. N. On the homotopy classification of positively homogeneous functions of three variables. Petrozavodsk: J. Issues Anal., 2021; 10 (2): 67-78. (In Engl. )
