Stability of linear neutral differential equations with delays and impulses established by the fixed points method

In this paper, we consider the impulsive effects on the stability of the zero solution of the linear neutral differential equations with variable delays. By transforming the equations into ones without impulses and using fixed point theory, some sufficient conditions for asymptotic stability and exponential stability of the zero solution are obtained. The paper extends and improves results on sufficient conditions obtained by Jin and Luo (2008) [17], and Ardjouni and Djoudi (2011) [18], which is shown clearly in Example 1. This paper also shows that the impulse intensity and the impulse time both influence the decay rate of the convergence to zero of the solutions. Finally, two examples are given to show applications of some results obtained.     

带脉冲项的一类二阶微分方程边值问题两个正解的存在性

研究一类二阶Neumann边值问题在含有脉冲项的条件下两个正解的存在性,所使用的工具是锥拉伸与压缩不动点定理,将相应的一些结果推广到脉冲问题.     

Banach空间中非线性脉冲积分微分方程的正解(英)

借助于锥理论，本文讨论Banach空间中非线性脉冲积分微分方程的解．给出一阶脉冲微分方程存在唯一正解的条件及混合型脉冲积分微分方程至少具有两解的条件．     

Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

具阶段结构害虫防治模型的脉冲效应

对于用微分方程描述的种群生态动力系统，其研究结果已十分丰富，但自然界中的许多变化规律都呈现出脉冲效应，因此用脉冲微分方程描述某些运动状态在固定或不固定时刻的快速变化或跳跃更切合实际，尤其在刻画种群生长和流行病动力学行为方面，脉冲微分方程的描述显得更科学更真实，具有脉冲效应的种群动力学模型的研究目前还处于刚刚起步阶段，本对符合实际的有脉冲效应的具阶段结构的常系数害早防治模型进行了研究，得到了系统存在周期解的充分条件，系统存在唯一周期解的充分条件，系统周期解轨道渐近稳定的充分条件。     

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse

In this letter we propose a class of linear fractional difference equations with discrete-time delay and impulse effects. The exact solutions are obtained by use of a discrete Mittag-Leffler function with delay and impulse. Besides, we provide comparison principle, stability results and numerical illustration.     

Lyapunov's second method in problems of the stability of solutions of systems with impulse effect

A system of ordinary differential equations with impulse effect at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse effects are obtained under which the uniform asymptotic stability of the zero solution of the ‘unperturbed’ system implies the uniform asymptotic stability of the zero solution of the ‘perturbed’ system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects

This paper deals with a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments by employing a fixed point theorem in cones. Easily verifiable sufficient criteria are established. The paper extends some previous results and reports some new results about impulsive functional differential equations.     

Generalized solutions of impulse systems and the phenomenon of pulsations

For systems of differential equations with impulse action at nonfixed moments of time, the concept of a generalized solution is presented. On its basis a classification of impulse systems is proposed and conditions are indicated sufficient for the impulse system to belong to one class or another.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 657–663, May, 1991.     

Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-lipschitz conditions

In the paper, stochastic differential equations with random impulses and Markovian switching are brought forward, where the so-called random impulse means that impulse ranges are driven by a series of random variables and impulse times are a random sequence, so these equations extend stochastic differential equations with jumps and Markovian switching. Then the existence and uniqueness of solutions to such equations are investigated by employing the Bihari inequality under non-Lipschtiz conditions.     

Asymptotic Weighted-Periodicity of the Impulsive Parabolic Equation with Time Delay

The main aim of this paper is to investigate the effects of the impulse and time delay on a typeof parabolic equations.In view of the characteristics of the equation,a particular iteration scheme is adopted.The results show that Under certain conditions on the coefficients of the equation and the impulse,the solutionoscillates in a particular manner—called"asymptotic weighted-periodicity".     

A new result on impulsive differential equations involving non-absolutely convergent integrals

In this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations.     

Zero-sum differential games involving impulse controls

In this paper we are concerned with zero-sum differential games with impulse controls, as well as continuous and switching controls. The motivation is optimal impulse control problems with disturbances. The main result is the existence of the value function of the game. Our approach is the theory of viscosity solutions for Hamilton-Jacobi equations.Part of this paper was done while the author was a visiting scholar at INRIA, Sophia-Antipolis, France. This work was also partially supported by the Chinese NSF, the Chinese State Education Commission Science Foundation, and the Fok Ying Tung Education Foundation.     

Singular Dirichlet second-order BVPs with impulses

The paper deals with the existence of solutions to singular second-order differential equations with impulse effects and with the Dirichlet boundary conditions. The right-hand side of the differential equation can be singular in its phase variable.     

具有不依赖于状态脉冲的双曲型偏微分方程的振动准则

本文研究脉冲双曲型偏微分方程解的振动性质．得到了两类具有不依赖于状态脉冲的双曲边值问题的若干振动准则．     

On the stability of invariant sets of systems with impulse effect

A system of differential equations with impulse effect is considered. It is assumed that this system has an invariant set M$M$. By means of the direct Lyapunov method, the necessary and sufficient conditions of its uniform asymptotic stability are obtained. The conditions on the perturbations of right hand sides of differential equations and impulse effects, under which the uniform asymptotic stability of the invariant set M$M$ of the “nonperturbed” system implies the uniform asymptotic stability of the invariant set of the “perturbed” system, are obtained. The stability properties of invariant sets of periodic systems are also studied.     

The reducibility of differential equations with impulses in the space of bounded numerical sequences

The analogue of Erugin's theorem is considered for differential equations with impulses in the space of bounded numerical sequences. Sufficient conditions are given for the reduction of the problem regarding the reducibility of the equations of the indicated form with periodic coefficients to the case of finite-dimensional systems of periodic impulse equations of increasing dimensions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1376–1382, October, 1990.     

随机脉冲随机微分方程解的存在唯一性

提出了随机脉冲随机微分方程模型,其中所谓的随机脉冲是指脉冲幅度由随机变量序列驱动,并且脉冲发生的时间也是一个随机变量序列.因此,随机脉冲随机微分方程是对带跳的随机微分方程模型的推广.利用Gronwall不等式、Lipschtiz条件和随机分析技巧,得到了随机脉冲随机微分方程的解的存在唯一性条件.