The stability of solutions to delay differential equations in Banach spaces

Abstract

Background. The work is devoted to the analysis of stability in the sense Lyapunov of steady-state solutions of nonlinear differential equations in Banach spaces with time-dependent operators and with time-dependent delays. Delay differential equations model dynamic processes in many problems of physics, natural science, and technology, and methods for constructing sufficient conditions for the stability of their solutions are needed. Existing methods for finding sufficient conditions for the stability of solutions to nonlinear differential equations in Banach spaces are complex enough to be used in solving specific physical and technical problems. Of current interest is the development of methods for constructing sufficient conditions for stability, asymptotic stability, and boundedness of solutions of differential equations in Banach spaces. Materials and methods. The mathematical apparatus used in this work is the logarithmic norm and its properties. When studying the stability of solutions to nonlinear differential equations with delays in Banach spaces, a comparison is made between the norm and the logarithmic norm of the operators in the equation. The proofs of the statements formulated in the paper are carried out by the method of contradiction. Results. Algorithms are proposed that make it possible to obtain sufficient conditions for stability, asymptotic stability, and boundedness of solutions of nonlinear differential equations in Banach spaces with operators and with time-dependent delays. Sufficient stability conditions are expressed in terms of the norms and logarithmic norms of the operators entering the equations. Conclusions. A method is proposed for constructing sufficient conditions for stability, asymptotic stability, and boundedness of solutions of nonlinear differential equations in Banach spaces with time-dependent coefficients and delays. The method can be used in the study of non-stationary dynamic systems described by nonlinear differential equations with time-dependent delays.

Keywords

• lyapunov stability
• differential equations in banach spaces
• delays
• logarithmic norm