On Stability of Solutions to One Class of Nonlinear Difference Systems

We consider some class of systems of nonlinear ordinary differential equations. We adjust the difference schemes corresponding to the equations under study in order to guarantee agreement between differential and difference systems in the sense of stability of the zero solution. We obtain conditions under which perturbations do not violate the asymptotic stability of solutions to difference systems.     

Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

Asymptotic stability of solutions of a class of systems of nonlinear differential equations with delay

We study systems of differential equations with delay whose right-hand sides are represented as sums of potential and gyroscopic components of vector fields. We assume that in the absence of a delay zero solutions of considered systems are asymptotically stable. By the Lyapunov direct method, using the Razumikhin approach, we prove that in the case of essentially nonlinear equations the asymptotic stability of zero solutions is preserved for any value of the delay.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: II

We consider systems of nonautonomous nonlinear differential equations with the infinite delay. We study the stability properties and the limiting equations whose right-hand sides are defined as the limit points of some sequence in the introduced function space. By using the method of limiting equations, we obtain new sufficient conditions for the asymptotic stability of the zero solution of the considered class of equations.     

带非线性边界条件的反应扩散方程的数值方法

１引言近年来关于非线性抛物型方程数值解法的研究取得了许多好的结果,其中以Ｃ．Ｖ．Ｐａｏ为主的研究者们利用上、下解方法对带线性边界条件的半线性抛物型方程的有限差分系统进行了广泛的研究,提出了一系列有效的迭代算法（见［１］、［２］、［３］、［４］）．但对带非线性边界条件的半线性抛物型方程初边值问题,作者至今尚未见到有研究者将上、下解方法用在相应的差分系统上,求得数值解．其主要原因是由于边界上函数的非线性,解在边界网格点上的值未知且无法用内部网格点上的值直接表示,相应的差分系统表示形式受到影响,边界网…     

On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.     

ON EXPONENTIAL STABILITY OF NON-AUTONOMOUS STOCHASTIC SEMILINEAR EVOLUTION EQUATIONS

11MroductlonThe purpose ofthls paper Is to Investigate eWone尬lal stability of*theity mild solutions forcenain Hilbert space－Mued stochastlc evoMlon eqll砒ions,Roughy spe出0ng;we cons讪r山efollowing equation:I 伏I＝*x,＋风Il加L十从L,剧dWn,c〔瓜＋咖。(””””“”(11)D 人n 二x．Where A Is the Infinlteslmalgener砒or ofa certain几semigroup S(t),t＞0;on H and F(t;、)and B(t;·)are In general nonlinear mappings from H to H and H to L(x,H),the family ofall bounded linear operators from …     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

Stability of solutions of monotone systems of impulsive differential equations

We obtain sufficient conditions for the stability of solutions of nonlinear systems of impulsive differential equations in a cone. The main tool in our study is the comparison principle suggested in the paper. This approach permits one to study critical cases; it is illustrated by examples of nonlinear systems for which the trivial orbit of the continuous and discrete components is simultaneously unstable.     

On the use of a general quadratic Lyapunov function for studying the stability of Takagi–Sugeno systems

We study the stability of the zero solution to a nonlinear system of ordinary differential equations on the base of its Takagi–Sugeno (TS) representation. As is known, the most constructive stability and stabilization conditions for TS systems stated as linear matrix inequalities are established with the help of a general quadratic Lyapunov function (GQLF). However, such conditions are often too rigid. Using a modification of the Lyapunov direct method, we propose asymptotic stability conditions with weaker requirements to GQLF. They allow an application to a wider class of systems. We also give some illustrative examples.     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Stability analysis based on nonlinear inhomogeneous approximation

The asymptotic stability of zero solutions for essentially nonlinear systems of differential equations in triangular inhomogeneous approximation is studied. Conditions under which perturbations do not affect the asymptotic stability of the zero solution are determined by using the direct Lyapunov method. Stability criteria are stated in the form of inequalities between perturbation orders and the orders of homogeneity of functions involved in the nonlinear approximation system under consideration.     

Numerical methods for nonlinear first‐order partial differential equations with deviated variables

In the article classical solutions of initial problems for nonlinear differential equations with deviated variables are approximated by solutions of quasilinear systems of difference equations. Interpolating operators on the Haar pyramid are used. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. This new approach to solving nonlinear equations with deviated variables numerically is based on a method of linearization for initial problems. Numerical examples are given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Stability analysis of nonlinear functional differential and functional equations

Sufficient conditions for the stability and asymptotic stability of the theoretical solutions to nonlinear systems of functional differential and functional equations are derived.     

Stability of solutions to impulsive differential equations in critical cases

We propose a new approach to constructing a piecewise differentiable Lyapunov function for some classes of nonlinear nonstationary systems of impulsive differential equations in the critical case. This approach allows us to obtain new sufficient conditions for the Lyapunov stability of solutions to this class of systems.     

Some estimates of the solutions to time-delay differential equations with parameters

Under study are the systems of quasilinear time-delay differential equations with parameters and periodic coefficients. Some sufficient conditions are derived for asymptotic stability of the zero solution, and the estimates of solutions are obtained that characterize the decay rate at the infinity.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

Linearized stability in periodic functional differential equations with state-dependent delays

In this paper, we study stability of periodic solutions of a class of nonlinear functional differential equations (FDEs) with state-dependent delays using the method of linearization. We show that a periodic solution of the nonlinear FDE is exponentially stable, if the zero solution of an associated linear periodic linear homogeneous FDE is exponentially stable.     

On the exponential stability of solutions of periodic systems of the neutral type with several delays

We obtain conditions for the exponential stability of the zero solution of linear periodic systems of differential equations of the neutral type with several constant delays, which are stated in terms of a Lyapunov–Krasovskii functional of a special form. We derive estimates that specify the decay rate of solutions at infinity.