#### For citation:

Kobilzoda M. M., Naimov A. N. On the Positive Solutions of a Model System of Nonlinear Ordinary Differential Equations. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2020, vol. 20, iss. 2, pp. 161-171. DOI: 10.18500/1816-9791-2020-20-2-161-171, EDN: BJNLYY

# On the Positive Solutions of a Model System of Nonlinear Ordinary Differential Equations

This article investigates the properties of positive solutions of a model system of two nonlinear ordinary differential equations with variable coefficients. We found the new conditions on coefficients for which an arbitrary solution (x(t), y(t)) with positive initial values x(0) and y(0) is positive, nonlocally continued and bounded at t > 0. For this conditions we investigated the question of global stability of positive solutions via method of constructing the guiding function and the method of limit equations. Via the method of constructing the guide function we proved that if the system of equations has a positive constant solution (x∗, y∗), then any positive solution (x(t), y(t)) at t → +∞ approaches (x∗, y∗). And in the case when the coefficients of the system of equations have finite limits at t → +∞ and the limit system of equations has a positive constant solution (x∞, y∞), via method of limit equations we proved that any positive solution (x(t), y(t)) at t → +∞ approaches (x∞, y∞). The results obtained can be generalized for the multidimensional analog of the investigated system of equations.

- Gorsky A. A., Lokshin B. Y., Rozov N. Kh. The regime intensified in one system of nonlinear equations. Differential Equations, 1999, vol. 35, no. 11, pp. 1571–1581 (in Russian).
- Gorsky A. A., Lokshin B. Y. A mathematical model of goods production and sale for production supervision and planning. Fundamentalnaya i Prikladnaya Matematika, 2002, vol. 8, no. 1, pp. 39–45 (in Russian).
- Mukhamadiev E., Naimov A. N., Sobirov M. K. Research positive solutions of dynamic model of production and sale goods. Sovremennye metody prikladnoi matematiki, teorii upravleniya i komp’yuternykh tekhnologii: sbornik trudov X mezhdunarodnoi konferentsii (“PMTUKT-2017”) [Modern methods of applied mathematics, control theory and computer technology: proceedings of X Int. Conf. (“PMTUKT-2017”)]. Voronezh, Nauchnaia kniga, 2017, pp. 268 271 (in Russian).
- Kobilzoda M. M., Naimov A. N. On positive and periodic solutions of one class of systems of nonlinear ordinary differential equations on a plane. Vestnik Voronezhskogo gosudarstvennogo universiteta. Ser. Fizika. Matematika [Proccedings of Voronezh State University. Ser. Physics. Mathematics], 2019, no. 1, pp. 117–127 (in Russian).
- Pliss V. A. Nonlocal problems of the theory of oscillations. New York, London, Acad. Press, 1966. 306 p. (Rus. ed.: Moscow, Nauka, 1964. 367 p.).
- Krasnosel’skii M. A., Zabreiko P. P. Geometrical Methods of Nonlinear Analysis. Berlin, Heidelberg, New York, Tokyo, Springer Verlag, 1984. 409 p. (Russ. ed.: Moscow, Nauka, 1975. 11 p.).
- Mukhamadiev E. On the theory of bounded solutions of ordinary differential equations. Differ. Uravn., 1974, vol. 10, no. 4, pp. 635–646 (in Russian).
- Mukhamadiev E. Research on the theory of periodic and bounded solutions of differential equations. Mathematical Notes of the Academy of Sciences of the USSR, 1981, vol. 30, no. 3, pp. 713–722. DOI: https://doi.org/10.1007/BF01141630
- Hartman P. Ordinary Differential Equations. New York, John Wiley and Sons, 1964. 612 p. (Rus. ed.: Moscow, Mir, 1970. 720 p.).

- 1186 reads