Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

Stability of solutions to differential equations with several variable delays. III

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. We treat the coefficients and maximum admissible values of delays as parameters that define a family of equations of the considered class. Using the necessary and sufficient stability conditions established in preceding papers, we obtain an analytic form and a geometric interpretation of boundaries of stability domains for families of equations with a small number of independent parameters.     

Asymptotics of solutions of difference equations with delays

We consider a linear scalar difference equation with several variable delays and constant coefficients. The coefficients and maximum admissible values of delays are supposed to be the set of parameters that define a family of equations of the investigated class. We obtain effective necessary and sufficient conditions of the uniform and exponential stability of solutions to all equations of the family, as well as the conditions of the sign-definiteness and monotonicity of stable solutions.     

On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.     

Asymptotic behavior of a linear difference equation with continuous time

We consider a class of linear homogeneous difference equations with constant coefficients and commensurate delays. We prove an asymptotic formula for the solutions as t → ∞ under the assumption of the existence of a simple “dominant” real characteristic value. Research supported by FONDECYT Grant, Chile, No. 1.070.980.     

Well-posed solvability of functional-differential equations with unbounded operator coefficients

We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.     

Asymptotic behaviour and stability to linear non-autonomous neutral delay difference equations

This paper is concerned with the asymptotic behaviour and the stability of a class of linear neutral delay difference equations with variable coefficients and constant delays. Via an appropriate solution of the so-called generalized characteristic equation, an asymptotic criterion and a stability result are established.     

On Some Problems of the Asymptotic Theory of Linear Differential Equations of the nth Order

We investigate smoothness properties of the roots of algebraic equations with almost constant coefficients and construct a transformation, which may be efficiently used for the investigation of the asymptotic behavior of a fundamental family of solutions of a broad class of nonautonomous linear differential equations of the nth order.     

具有正负系数的二阶脉冲中立型方程的振动性

考虑一类具有正负系数的多时滞二阶线性脉冲中立型微分方程,证明了该方程解的振动性等价于一类非脉冲中立型方程解的振动性,得到了方程所有解存在的充分条件.     

On a Boundary Value Problem for a System of Differential–Difference Equations of Retarded Type

Differential Equations - We consider a control system described by a system of differential equations of retarded type with variable matrix coefficients and several delays. The relationship between...     

A remark on the ODE with two discrete delays

The aim of this paper is to outline a formal framework for the analytical analysis of the Hopf bifurcations in the delay differential equations with two independent time delays. Some results for the differential-difference equations with two delays, when the both of the coefficients of linearized equation are negative were obtained in [X. Li, S. Ruan, J. Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999) 254-280]. In the paper we present some remarks on the case studied in [X. Li, S. Ruan, J. Wei, Stability and bifurcation in delay-differential equations with two delays, J. Math. Anal. Appl. 236 (1999) 254-280] and also two other cases, namely when the coefficients of linearized equation have different signs and when coefficients are both positive.     

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [T. Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007) 191-205] and Wan and Duan [L. Wan, J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett. 78 (5) (2008) 490-498] to cover a class of more general stochastic partial differential equations with jumps. Finally, an illustrative example is established to demonstrate our established theory.     

A simple direct method to find equivalence transformations of a generalized nonlinear Schrödinger equation and a generalized KdV equation

A simple direct method is presented to find equivalence transformations of nonlinear mathematical physics equations. By using the direct method, we obtain the continuous equivalence transformations of a class of nonlinear Schröequations with variable coefficients and a family of nonlinear KdV equations with variable coefficients. For the nonlinear Schrödinger equations with variable coefficients, the equivalence transformations obtained by the direct method coincide, in nature, with those obtained via the infinitesimal Lie criterion, but our computation is much simpler.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Diagonal dominance and harmless off-diagonal delays

Systems of linear differential equations with constant coefficients, as well as Lotka-Volterra equations, with delays in the off-diagonal terms are considered. Such systems are shown to be asymptotically stable for any choice of delays if and only if the matrix has a negative weakly dominant diagonal.

     

Uniform persistence and periodic solutions for a discrete predator-prey system with delays

In this paper, we deal with a discrete predator-prey system with delay. We first give a sufficient condition for the uniform persistence of the system. Assuming that the coefficients in the system are periodic, by generalizing the Yoshizawa's theorem on the existence of periodic solution for ordinary differential equations to the difference equations with delays, we obtain the existence of a periodic solution basing on the uniform persistence result.     

中立型方程d/(dt)[x(t)+px(t-τ)-rx(t-ρ)]+qx(t-s)-hx(t-v)=0的振动性

本文研究了既具正系数又具负系数的一阶中立型微分方程ｄｄｔ［ｘ（ｔ）＋ｐｘ（ｔ－τ）－ｒｘ（ｔ－ρ）］＋ｑｘ（ｔ－ｓ）－ｈｘ（ｔ－ｖ）＝０其中ｐ，ｒ，τ，ρ，ｑ，ｈ，ｓ，ｖ都是正的常数，建立了一切解振动的带有若干个可调参数的一个充要条件和一系列充分性判据及某种形式的一个充要条件．有些充要条件和充分性判据包含或改进了前人的相应结果．     

Estimates of solutions to the linear differential equations of neutral type with several delays of the argument

Under study is a systemof linear differential equations of neutral type with several delays of the argument. We obtain the conditions on the matrix coefficients of the system under which all solutions decrease with an exponential rate at infinity. Using some functionals of Lyapunov-Krasovskii type, the uniform estimates of solutions are established.     

Oscillation analysis of neutral difference equations with delays

In this paper, we obtain some oscillation criteria for a class of neutral difference equations with time delays. We also investigate the behavior of the eventually positive solutions of these equations. To verify our results we give various numerical simulations by using the MATLAB programming.     

Regularization of linear difference equations with analytic coefficients and their applications

We propose a regularization method for four-element linear difference equations with analytic coefficients. We study these equations in the class of functions which are holomorphic in the complex plane with a cruciform cut and vanish at infinity. We give several examples illustrating the dependence of the solvability properties of equations on the choice of periodic coefficients. We describe various applications.