Special symmetric periodic solutions of differential systems with distributed delay

We discuss the existence of periodic solutions to a system of differential equations with distributed delay which shows a certain type of symmetry. For this, such solutions are related to the solutions of a system of second-order ordinary differential equations.     

Fixed points and stability of neutral stochastic delay differential equations

In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized.     

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.     

The fixed sign property of solutions and stability of linear differential equations with varying distributed delay

We study one class of linear differential equations with varying distributed delay. We obtain an effective (in terms of parameters of the initial problem) criterion for the positivity of the Cauchy function of this class of equations. Based on this result, we establish effective criteria for the exponential stability of equations under consideration.     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

Neutral delay differential equations with positive and negative coefficients

Consider the neutral delay differential equation [display math001] where [display math002] We studied the asymptotic behavior of the nonoscillatory solutions of Eq. (1) and we obtained sufficient conditions for the oscillation of all solutions, all bounded solutions, and all unbounded solutions of Eq. (1)     

On the Instability of Linear Nonautonomous Delay Systems

The unstable properties of the linear nonautonomous delay system x(t) = A(t)x(t) + B(t)x(t – r(t)), with nonconstant delay r(t), are studied. It is assumed that the linear system y(t) = (A(t) + B(t))y(t) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r(t) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r(t) and the results depending on the asymptotic properties of the delay function.     

Degenerate non-stationary differential equations with delay in Banach spaces

A generalization of the operator method by Grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in Banach spaces, whose operator coefficients are time depending. Some applications to ordinary and partial differential equations with delay are described.     

Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u₁,u₂,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=T(Δu₁,Δu₂,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.     

Delay-dependent dissipativity of nonlinear delay differential equations

In this paper, the delay-dependent dissipativity of nonlinear delay differential equations is studied. A new dissipativity criterion is derived, which is less conservative than those in the existing literature in some cases, especially for equations with small delays.     

Positive solutions for a scalar differential equation with several delays

For a scalar delay differential equation , we obtain new explicit conditions for the existence of a positive solution.     

Attractor minimal sets for cooperative and strongly convex delay differential systems

We study the skew-product semiflow induced by a family of convex and cooperative delay differential systems. Under some monotonicity assumptions, we obtain an ergodic representation for the upper Lyapunov exponent of a minimal subset. In addition, when eventually strong convexity at one point is assumed and there exist two completely strongly ordered minimal subsets K₁?_CK₂, we show that K₁ is an attractor subset which is a copy of the base. The long-time behaviour of every trajectory strongly ordered with K₂ is then deduced. Some examples of application of the theory are shown.     

The exact boundaries of the stability domains of linear differential equations with distributed delay

For a differential equation with a distributed varying delay, sufficient criteria for the asymptotic and uniform stability of solutions are obtained. The constructed examples demonstrate exactness of the boundary of the obtained stability domain.     

一类滞后型退化系统的周期解

本文考虑一类滞后型退化高维周期系统B(t)x′(t)=A(t)x(t) f(t,xt),利用线性系统指数型二分性理论和Krasnoselskii不动点定理得到了保证其周期解存在的充分性条件.     

On the stability of impulsive functional differential equations with infinite delays

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

Waveform relaxation method for stochastic differential equations with constant delay

This paper extends the waveform relaxation method to stochastic differential equations with constant delay terms, gives sufficient conditions for the mean square convergence of the method. A lot of attention is paid to the rate of convergence of the method. The conditions of the superlinear convergence for a special case, which bases on the special splitting functions, are given. The theory is applied to a one-dimensional model problem and checked against results obtained by numerical experiments.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

Khasminskii-type theorems for stochastic functional differential equations with infinite delay

The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for stochastic functional equations with infinite delay. The main aim of this paper is to establish the existence-and-uniqueness theorems of global solutions for stochastic functional differential equations with infinite delay.