时不变和时变的Caputo分数阶时滞微分系统的常数变易公式

This paper studies the variation of constant formulae for linear Caputo fractional delay differential systems. We discuss the exponential estimates of the solutions for linear time invariant fractional delay differential systems by using the Gronwall＇s integral inequality. The variation of constant formula for linear time invariant fractional delay differential systems is obtained by using the Laplace transform method. In terms of the superposition principle of linear systems and fundamental solution matrix, we also establish the variation of constant formula for linear time varying fractional delay differential systems. The obtained results generalize the corresponding ones of integer-order delayed differential equations.     

Oscillatory and Periodic Solutions of Impulsive Differential Equations with Piecewise Constant Argument

We study the existence of oscillatory and periodic solutions of a class of first order scalar impulsive delay differential equations with piecewise constant argument.     

General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

Variation of parameters formula and Gronwall inequality for differential equations with a general piecewise constant argument

A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.     

Stable invariant manifolds for delay equations with piecewise constant argument

We construct smooth stable invariant manifolds for a class of delay equations with piecewise constant delay, for any sufficiently small perturbation of a nonuniform exponential dichotomy. We build on former work for perturbations of a uniform exponential dichotomy, also for delay equations with piecewise constant delay. These equations can be described as delay equations with an impulsive behavior of the derivative, such that at certain times the derivative changes abruptly.     

一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程积分解的存在性

本文讨论了一类具有无穷时滞中立型非稠定脉冲随机泛函微分方程,利用Sadovskii不动点原理等工具得到了其积分解的存在性,给出其在一类二阶无穷时滞中立型非稠定脉冲随机偏微分方程积分解的存在性中的应用.     

Integral manifolds for uncertain impulsive differential–difference equations with variable impulsive perturbations

In the present paper sufficient conditions for the existence of integral manifolds of uncertain impulsive differential–difference equations with variable impulsive perturbations are obtained. The investigations are carried out by means the concepts of uniformly positive definite matrix functions, Hamilton–Jacobi–Riccati inequalities and piecewise continuous Lyapunov’s functions.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.     

Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments

We consider nonlinear differential equations with piecewise constant arguments in the general case. This is based in the study of an equivalent integral equation, and in a solution of an integral inequality of Gronwall type. We establish the existence, uniqueness and the asymptotic behavior of the solutions of the equations. Equivalences, including unbounded solutions, with simpler equations are obtained.     

Boundary value problems for higher order linear impulsive differential equations

In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered.     

具分段常数变元的脉冲微分方程周期解的存在性

通过构造差分方程的周期数列解，研究了一类具有分段常数变元的脉冲微分方程周期解的存在性．     

On numerical approximation using differential equations with piecewise-constant arguments

In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments (EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and oscillation of a class of linear delay equations with continuous and piecewise constant arguments. This research was partially supported by Hungarian NFSR Grant No. T046929.     

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions

Picone type formula for half‐linear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions is derived. Employing the formula, Leighton and Sturm–Picone type comparison theorems as well as several oscillation criteria for impulsive differential equations are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

一类交替型脉冲微分系统的解

主要讨论一类超前型与滞后型交替的脉冲微分系统.首先给出具常系数的脉冲微分系统解存在的充分条件以及解唯一的表达形式;对于变系数的微分系统也作了相应的讨论.     

Approximation of Solutions of Fractional-Order Delayed Cellular Neural Network on $$\varvec{[0,\infty )}$$

In this paper, we study a class of fractional-order cellular neural network containing delay. We prove the existence and uniqueness of the equilibrium solution followed by boundedness. Based on the theory of fractional calculus, we approximate the solution of the corresponding neural network model over the interval \([0,\infty )\) using discretization method with piecewise constant arguments and variation of constants formula for fractional differential equations. Furthermore, we conclude that the solution of the fractional-delayed system can be approximated for large t by the solution of the equation with piecewise constant arguments, if the corresponding linear system is exponentially stable. At the end, we give two numerical examples to validate our theoretical findings.     

一维交替型脉冲微分系统

讨论一维空间中超前型与滞后型交替的脉冲微分系统.首先考虑具常系数的脉冲微分系统平凡解稳定的充分条件;其次研究了具变系数的脉冲微分系统的振动性,并给出了其解的表示式.     

Stepanov ergodic perturbations for some neutral partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of μ?pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ?pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential Stability and Periodic Solutions of Impulsive Neural Network Models with Piecewise Constant Argument

In this paper we introduce an impulsive cellular neural network models with piecewise alternately advanced and retarded argument. The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique periodic solution. The approaches are based on employing Banach’s fixed point theorem and a new integral inequality of Gronwall type with impulses and deviating arguments. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides flexibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.     

Sufficient Conditions for the Oscillation of Bounded Solutions of a Class of Impulsive Differential Equations of Second Order with a Constant Delay

Sufficient conditions are found for oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay. Some asymptotic properties are studied for the bounded solutions.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.