CONTINUOUS DEPENDENCE OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS ON THE INITIAL CONDITIONS AND BARRIER CURVES

The basic objects of investigation in this article are nonlinear impulsive dif- ferential equations. The impulsive moments coincide with the moments of meeting of the integral curve and some of the so-called barrier curves. For such type of equations, suf- ficient conditions are found under which the solutions are continuously dependent on the perturbations with respect to the initial conditions and barrier curves. The results are applied to a mathematical model of population dynamics.

A NOVEL METHOD FOR NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS IN BROKEN REPRODUCING KERNEL SPACE

In this article,a new algorithm is presented to solve the nonlinear impulsive differential equations.In the first time,this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations.Then,the uniform convergence of the numerical solution is proved,and the time consuming Schmidt orthogonalization process is avoided.The algorithm is employed successfully on some numerical examples.

Oscillation Criteria for Second-order Half-linear Impulsive Differential Equations with Damping

By employing the inequality of [8] and a positive continuous function g（t）, t[t0, ＋∞）, oscillation criteria for the second-order half-linear impulsive differential equ-ations with damping are established. Our results generalize and improve some known ones.

Controllability of Second Order Stochastic Impulsive Differential Equations in Hilbert Spaces

The exact controllability of second order stochastic impulsive differential equations in Hilbert spaces is studied.By using the Holder's inequality,stochastic analysis and fixed point strategy,some sufficient conditions are given,but no compactness condition is imposed on the cosine family of operators.This work improves some previous results without impulses or stochastic factors.

Multiplicity Results for Second Order Impulsive Differential Equations via Variational Methods

In this paper we investigate a class of impulsive differential equations with Dirichlet boundary conditions. Firstly, we define new inner product of <img src="Edit_890fce38-e82b-4f36-be40-9d05e8119b88.png" width="40" height="17" alt="" /> and prove that the norm which is deduced by the inner product is equivalent to the usual norm. Secondly, we construct the lower and upper solutions of (1.1). Thirdly, we obtain the existence of a positive solution, a negative solution and a sign-changing solution by using critical point theory and variational methods. Finally, an example is presented to illustrate the application of our main result.

Positive Solutions for Second Order Impulsive Differential Equations with Dependence on First Order Derivative

We study positive solutions for second order three-point boundary value problem：{″（t）＋f（t,x（t）,x＇（t））=0,t≠ti△x（ti）=Ii（x（ti）,x＇（ti））,i=1,2,3,…,k△x＇（ti）=Ji（x（ti）,x＇（t））,x（0）=0=x（1）-αx（η）,where 0〈η〈1,0〈α〈1,and f：[0,1]×（0,∞）×R→[0,∞）,Ii：[0,∞）×R→R,Ji：[0,∞）×R→R,（i=1,2,…,k）are continuous. Based on a new extension of Krasnoselskii fixed-point theorem （which was established by Guo Yan-ping and GE Wei-gao, the existence of positive solutions for the boundary value problems is obtained. In particular, we obtain the Green function of the problem, which makes the problem simpler.

The Global Solutions of Impulsive Retarded Functional Differential Equations

In this paper, we present some results on the existence of the golbal solutions of nonlinear impulsive retarded functional differential equations.

Multiplicity of Positive Solutions to M-point Boundary Value Problem of Second Order Impulsive Differential Equations

In this paper, by using Avery-Peterson theorem on a convex cone, we consider the m-point boundary value problems for second order impulsive differential equations with the nonlinear term depending on the first order derivative, the multiplicity result of three positive solutions are obtained.

THE PERIODIC BOUNDARY PROBLE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH VARIABLE TIME

In this paper, we establish the general comparison principles for IVP of impulsive differential equations with variable time. Our results extend and improve the previous comparison results obtained by V.Lak. et al and S.K.Kaul [3-7]. Using this comparison result, we construct two monotonic iterative sequences of solutions for (IVP) which converge to the minimal and maximal solutions of PBVP for impulsive differential equations with variable time. The results improve the corresponding results in [5].

On the Existence of Positive Solutions for Impulsive Neutral Differential Equations

Aim To investigate the existence of positive solutions for impulsive neutral differential equations. Methods The Banach contraction principle was used to establish our results. Results and Conclusion The results of the existence of positive solutions for impulsive neutral differential equations are obtained.

Strict Stability of Impulsive Differential Equations

In this paper, we will extend the strict stability to impulsive differential equations. By using Lyapunov functions, we will get some criteria for the strict stability of impulsive differential equations, and we can see that impulses do contribute to the system＇s strict stability behavior. An example is also given in this paper to illustrate the efficiency of the obtained results.

Asymptotic stability for impulsive functional differential equations

In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen＇s inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.

Stability Criteria of Nonlinear Impulsive Differential Equations with Infinite Delays

In this paper, we consider the uniform stability and uniformly asymptotical stability of nonlinear impulsive infinite delay differential equations. Instead of putting all components of the state variable x in one Lyapunov function, several Lyapunov-Razumikhin functions of partial components of the state variable x are used so that the conditions ensuring that stability are simpler and less restrictive; moreover, examples are discussed to illustrate the advantage of the results obtained.

Numerical Analysis of Balanced Methods for the Impulsive Stochastic Differential Equations

Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.

THE EXISTENCE OF PERIODIC SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS OF MIXED TYPE

The existence of periodic solutions for a class of impulsive differential equations of mixed type is studied by constructing periodic sequence solutions of difference equations.

Existence of Solutions for Semilinear Impulsive Delay Differential Equations with Nonlocal Conditions

In this paper,by using Schaefer fixed-point theorem,the existence of mild solutions of semilinear impulsive delay differential equations with nonlocal conditions is studied.The results obtained are a generalization and continuation of the recent results on this issue.In the end,an example is given to show the application of the results.

OSCILLATION OF FIRST ORDER NONLINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH MIXED ARGUMENT

This paper devotes to study the oscillatory behavior of solutions of a first order nonlinear impulsive differential equation with mixed argument. First, without assuming the deviating argument to be retarded or advanced, a sufficient condition is established for all solutions of the differential equation to be oscillatory. Next, a sufficient condition for the differential equation to have nonoscillatoty solution is given. Finally, a sufficient and necessary condition for all solutions of the differential equation to be oscillatory is obtained.

EXISTENCE OF EXTREME SOLUTION TO FIRST-ORDER IMPULSIVE DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper is concerned with the existence of extreme solutions to three-point boundary value problems with nonlinear boundary conditions for a class of first order impulsive differential equations. We obtain suficient conditions for the existence of extreme solutions by the upper and lower solutions method coupled with a monotone iterative technique.

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

Abstract In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm Liouville boundary value problems for nonlinear second order impulsive differential equations in Banach spaces.

The Existence of Solutions of Three-point Boundary Value Problems for Second Order Impulsive Differential Equation

We study the existence of solutions to the second order three-point boundary value problem：{x″（t）＋f（t,x（t）,x′（t））=0,t≠ti,△x（ti）=Ii（x（ti）,x′（ti））,i=1,2,…,k,△x′（ti）=Ji（x（ti）,x′（t））,i=1,2,…,k,x（0）=0=x（1）-αx（η）,where 0〈η〈1,α∈R,and f：[0,1]×R×R→R,Ii：R×R→R,Ji：R×R→R（i=1,2,…,k）are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.