Integrated semigroups and linear partial differential equations with delay

We study existence and uniqueness of solutions for linear partial differential equations with delay in L^p-spaces using an approach of Batkai and Piazzera and a recent perturbation result for integrated semigroups. We apply our result to an equation with delay in the highest-order derivatives.     

New nonoscillation criteria for delay differential equations

This paper is concerned with the nonoscillation of first order delay differential equations of the form

     

On the behavior of the solutions to periodic linear delay differential and difference equations

Some new results on the behavior of the solutions to periodic linear delay differential equations as well as to periodic linear delay difference equations are given. These results are obtained by the use of two distinct roots of the corresponding (so called) characteristic equation.     

Stability caused by impulses for delay differential equations

In the present paper sufficient conditions are obtained for the stability of the zero solution of a nonlinear delay differential equation caused by impulses. This work is partially supported by NNSF of China and EYTF of National Educational Committee     

Positive periodic solutions for a class of delay differential equations

In this paper, we employ fixed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equation

x^′(t)=α(t)x(t)-β(t)x²(t)+γ(t)x(t-τ(t))x(t).     

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

Uniform stability theorems for delay differential equations with impulses

In the paper, we obtain sufficient conditions for the uniform stability of the zero solution of the delay differential equation with impulses

     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

Oscillation of first order delay differential equations with distributed delay

In this paper, we obtain some oscillation criteria for the first order delay differential equation with distributed delay

     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

Asymptotic behavior of certain delay differential equations with forcing term

We study the asymptotic behavior of solutions of the following forced delay differential equation:

Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results obtained are applied to some well-known delay differential ecological equations with forcing term.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Asymptotic behavior of delay differential equations with instantaneously terms

This paper deals with scalar delay differential equation with instantaneously term

     

Impulsive stabilization of second-order delay differential equations

In this paper, we prove that the non-impulsive equations can be stabilized by the imposition of proper impulsive control. Some recent results are extended and improved. We give two examples to illustrate the efficiency of our results.     

Perron type theorems for functional differential equations with infinite delay in a Banach space

For functional differential equations in a Banach space, we obtain some results on the asymptotic behavior of solutions, which correspond to Perron type theorems for differential equations without delay or with finite delay.