Variational approaches to impulsive elastic beam equations of Kirchhoff type

In this paper, we study the existence of multiple solutions for impulsive fourth-order differential equations of Kirchhoff type. Using a variational method and some critical points theorems, we obtain some new criteria for guaranteeing that impulsive fourth-order differential equations of Kirchhoff type have three and infinitely many solutions. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.     

Stepanov-like pseudo almost periodic solutions for impulsive perturbed partial stochastic differential equations and its optimal control

This paper is mainly concerned with the Stepanov-like pseudo almost periodicity to a class of impulsive perturbed partial stochastic differential equations. Firstly, we prove the existence of $p$-mean piecewise Stepanov-like pseudo almost periodic mild solutions for the impulsive stochastic dynamical system in a Hilbert space under non-Lipschitz conditions. The results are obtained by using the fixed point techniques with fractional power arguments. Then the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, an example is provided to illustrate the developed theory.     

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.     

Variational approach to impulsive differential equations with periodic boundary conditions

In this paper, we obtain some new existence results of solutions of impulsive differential equations with periodic boundary conditions. The main tool that we use is critical point theory. Our results generalize some existing results on periodic solutions for second order ordinary differential equations even when the impulses are absent.     

Strict practical stability of nonlinear impulsive systems by employing two Lyapunov-like functions

This paper develops the concepts of strict practical stability of ordinary differential equations to impulsive differential system. Strict practical stability, known as stability in tube-like domain, can be made to estimate upper bound and lower bound of the solutions of impulsive differential equations. This note provides several stability criteria for strict practical stability of nonlinear dynamical systems with impulse effects by employing two Lyapunov-like functions under general restrictions. It may provide a greater prospect to solve problems which exhibit impulsive effects.     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

Multiple Solutions of Nonlinear Impulsive Differential Equations with Dirichlet Boundary Conditions via Variational Method

In this paper, we consider the existence of multiple solutions for second-order nonlinear impulsive differential equations with Dirichlet boundary condition. We obtain some existence theorems of solutions for the nonlinear problem when the impulsive functions satisfies the superlinear growth conditions by critical point theory. We extend and improve some recent results.     

Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions

In this paper, we study the stability of nonlinear impulsive stochastic differential equations in terms of two measures. The concept of perturbing Lyapunov functions is introduced to discuss stability properties of solutions of nonlinear impulsive stochastic differential equations in terms of two measures. By using perturbing Lyapunov functions and comparison method, some sufficient conditions for the above stability are given.     

Strict stability of impulsive functional differential equations

Strict stability is the kind of stability that can give us some information about the rate of decay of the solutions. There are some results about strict stability of differential equations. In the present paper, we shall extend the strict stability to impulsive functional differential equations. By using Lyapunov functions and Razumikhin technique, we shall get some criteria for the strict stability of impulsive functional differential equations, and we can see that impulses do contribute to the system's strict stability behavior.     

脉冲泛函微分方程的整体解

本文给出了非线性脉冲泛函数微分方程整体解的存在性的一些结果。     

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper,oscillation of solutions to a class of impulsive delay para- bolic partial differential equations system with higher order Laplace operator is studied.Under two different boundary value conditions,we establish some sufficient criteria with respect to the oscillations of such systems,employing first-order impulsive delay differential inequalities.The results fully reflect the influence action of impulsive and delay in oscillation.     

Global attractability and permanence for a new stage-structured delay impulsive ecosystem

In this paper, a new stage-structured delay ecosystem with impulsive effect is formulated and some dynamical properties of this system is investigated.By using comparison theorem and the stroboscopic technique, we prove the existence of the predator-extinction periodic solution of this system and obtain some sufficient conditions to guarantee the global attractivity of the prey-extinction periodic solution. In the final, we also obtain the permanence of this system. It should be pointed out that the new mathematical method used in this paper can also be applied to investigate such other ecosystems corresponding to both impulsive and delay differential equations.     

Existence of solutions for impulsive Dirichlet problems with the parameter inequality reverse

In this paper, we consider the existence and multiplicity for second‐order nonlinear impulsive differential equations with Dirichlet boundary condition and a parameter. By using critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution or infinitely many solutions, assuming that the impulsive functions satisfy the superlinear growth condition and the parameter inequality is reverse. Our results extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

二阶时滞微分方程非振动性质在脉冲扰动下的不变性

本文建立了一类二阶脉冲时滞微分方程解的一个整体存在唯一性定理,并讨论了脉冲扰动对脉冲线性时滞微分程非振动性质的影响,获得了脉冲扰动对时滞微分方程非振动性质没有影响的一般性脉冲条件,推广了最近某些文献中的结论.     

高阶脉冲中立型偏泛函微分方程的强迫振动性

研究一类具高阶Laplace算子的高阶脉冲非线性中立型偏泛函微分方程的强迫振动性,利用Green公式和微分不等式方法将所讨论的脉冲中立型偏微分方程转化为脉冲中立型微分不等式的问题,获得了这类方程在三类不同边值条件下所有解强迫振动的若干充分条件.     

Variational approach to some damped Dirichlet problems with impulses

In this paper, we study the existence of solutions for damped nonlinear impulsive differential equations with Dirichlet boundary conditions. By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution. Some recent results are extended and significantly improved. Finally, some examples are presented to illustrate our main results. Copyright © 2013 John Wiley & Sons, Ltd.