Solution Stability of Delay Differential Equations in Banach Spaces

• Published in: Technical physics 2023-09, Vol.68 (9), p.245-249
• Main Author: Boykov, I. V.
• Format: Article
• Language: English
• Subjects: Algorithms; Asymptotic methods; Asymptotic properties; Banach spaces; Classical and Continuum Physics; Delay; Differential equations; Dynamical systems; Logarithms; Mathematical analysis; Nonlinear differential equations; Norms; Operators (mathematics); Physics; Physics and Astronomy; Stability; Time dependence
• ISSN: 1063-7842; 1090-6525
• DOI: 10.1134/S1063784223700159

The Lyapunov stability of steady-state solutions to nonlinear differential equations with time-dependent operators and time-dependent delay in Banach spaces has been analyzed. Delay differential equations simulate dynamic processes in many problems of physics, natural science, and technology, so that techniques to construct sufficient conditions for the stability of their solutions are necessary. Methods available for finding sufficient stability conditions for solutions to nonlinear differential equations in Banach spaces are difficult to use in solving specific physical and applied problems. Therefore, the development of ways for constructing sufficient conditions for stability, asymptotic stability, and boundedness of solutions to differential equations in Banach spaces is a challenging issue. The core of the mathematical apparatus used in this study consists in the logarithmic norm and its properties. In studying the stability of solutions to delay nonlinear differential equations in Banach spaces, the norm and logarithmic norm of operators entering into the equations have been compared. Statements formulated in the article have been proved by contradiction. Algorithms have been suggested that make it possible to derive sufficient conditions for the stability, asymptotic stability, and boundedness of solutions to nonlinear differential equations in Banach spaces with time-dependent operators and delays. The sufficient conditions for stability have been expressed through the norms and logarithmic norms of operators entering into the equations. A method has been suggested to construct sufficient conditions for the stability, asymptotic stability, and boundedness of solutions to nonlinear differential equations in Banach spaces with time-dependent coefficients and delays. This method can be used to study nonstationary dynamic systems described by nonlinear differential equations with time-dependent delays.