Abstract

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.

中文翻译：

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Banach空间中时滞微分方程解的稳定性

摘要

分析了巴拿赫空间中含时变算子和时变时滞的非线性微分方程稳态解的李亚普诺夫稳定性。时滞微分方程模拟物理、自然科学和技术中许多问题的动态过程，因此需要构造足够条件以保证其解的稳定性的技术。可用于寻找巴拿赫空间中非线性微分方程解的足够稳定性条件的方法很难用于解决特定的物理和应用问题。因此，开发构建巴拿赫空间中微分方程解的稳定性、渐近稳定性和有界性的充分条件的方法是一个具有挑战性的问题。本研究中使用的数学工具的核心在于对数范数及其属性。在研究Banach空间中时滞非线性微分方程解的稳定性时，对进入方程的算子的范数和对数范数进行了比较。文章中的陈述已通过反证法得到证明。已经提出的算法可以导出具有瞬态算子和延迟的 Banach 空间中非线性微分方程解的稳定性、渐近稳定性和有界性的充分条件。通过代入方程的算子范数和对数范数来表达稳定性的充分条件。提出了一种方法来构造具有瞬态系数和延迟的 Banach 空间中非线性微分方程解的稳定性、渐近稳定性和有界性的充分条件。该方法可用于研究由具有时滞的非线性微分方程描述的非平稳动态系统。