J. K. Khusanov¹, A. E. Kaxxorov²

¹Sambhram University (Jizzakh, Uzbekistan)

²Academic lyceum of Tashkent State Technical University named after I. Karimov (Tashkent, Uzbekistan)

Abstract. The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

Key Words: nonlinear scalar differential equation, variable delay, stability, attraction of solutions, Lyapunov functional

For citation: J. K. Khusanov, A. E. Kaxxorov. On the stability of a nonlinear nonautonomous scalar equation with variable delay. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:4(2023), 299–312. DOI: https://doi.org/10.15507/2079-6900.25.202304.299-312

Information about the authors:

Jumanazar Kh. Khusanov, Professor, Sambhram University (3, H. Nosirov Street, Jizzakh 130100, Uzbekistan), Dr. Sc. (Physics and Mathematics), ORCID: https://orcid.org/0000-0001-9444-9324, d.khusanov1952@mail.ru

Azizbeck E. Kaxxorov, Graduate Student, Academic lyceum of Tashkent State Technical University named after I.Karimov (2, University street, Tashkent 100095, Uzbekistan), ORCID: https://orcid.org/0000-0001-5723-8640, azizqahhorov@gmail.com