Linear Impulsive Differential Equations Research Articles

Asymptotic integration of second-order impulsive differential equations

We initiate a study of the asymptotic integration problem for second-order nonlinear impulsive differential equations. It is shown that there exist solutions asymptotic to solutions of an associated linear homogeneous impulsive differential equation as in the case for equations without impulse effects. We introduce a new constructive method that can easily be applied to similar problems. An illustrative example is also given.

Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations

For a new linear non-instantaneous impulsive differential equations, we introduce the notation of non-instantaneous impulsive Cauchy matrix and analyze its exponential structure in terms of eigenvalues of matrix via the distance between impulsive points and junction points. Many useful criteria for asymptotic stability of linear non-instantaneous impulsive problems are derived, which allow us to establish a uniform framework to deal with asymptotic stability of linear non-instantaneous impulsive differential equations with perturbation under mild sufficient conditions. In particular, two examples are given to demonstrate the application of theoretical results for linear part. In addition, we study nonlinear non-instantaneous impulsive equations and investigate existence, uniqueness of their solutions and Ulam–Hyers–Rassias stability under the restriction of exponential growth or stable conditions for non-instantaneous impulsive Cauchy matrix, respectively, which provide an approach to find approximate solution to nonlinear non-instantaneous impulsive equations in the sense of Ulam–Hyers–Rassias stability. The obtained results cover the standard results for instantaneous impulsive differential equations, which are essentially extend the theory in the previous literatures.

Oscillation of Runge-Kutta methods for advanced impulsive differential equations with piecewise constant arguments

The purpose of this paper is to study oscillation of Runge-Kutta methods for linear advanced impulsive differential equations with piecewise constant arguments. We obtain conditions of oscillation and nonoscillation for Runge-Kutta methods. Moreover, we prove that the oscillation of the exact solution is preserved by the θ-methods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution. We give some numerical examples to confirm the theoretical results.

Oscillation of a First Order Linear Impulsive Delay Differential Equation with Continuous and Piecewise Constant Arguments

A class of first order linear impulsive delay differential equation with continuous and piecewise constant arguments is studied. Using a connection between impulsive delay differential equations and non- impulsive delay differential equations sufficient conditions for the oscillation of the solutions are obtained.

Exponential dichotomies for impulsive equations via quadratic functions

We give a characterization of the existence of a nonuniform exponential dichotomy for a linear impulsive differential equation. The characterization uses quadratic functions and the symmetric matrices defining them. As an application, we give a simple proof of the robustness property of a nonuniform exponential dichotomy under sufficiently small linear perturbations.

Stability of impulsive delay differential equations

In this paper, we introduce impulsive delayed matrix function and give its norm estimation. With the help of impulsive delayed Cauchy matrix and the variation of constants method, we obtain representation of solutions to linear impulsive delay differential equations. Moreover, we derive some sufficient conditions to guarantee the trivial solution is locally asymptotically stable. Finally, two examples are given to illustrate our theoretical results.

Double Perturbations for Impulsive Differential Equations in Banach Spaces

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

ON EXACT SOLUTIONS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH NON-INTEGER ORDERS

This paper deals with linear impulsive differential equations with non-integer orders. We provide the explicit representation of solutions of linear impulsive fractional differential equations with constant coefficient by mean of the Mittag-Leffler functions.

Matrix measure approach to Lyapunov-type inequalities for linear Hamiltonian systems with impulse effect

We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2n-first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems.

Asymptotical stability of Runge-Kutta for a class of impulsive differential equations

The aim of this paper is to study asymptotical stability of Runge-Kutta methods for a class of linear impulsive differential equations with piecewise continuous arguments. New results about the asymptotical stability region of Runge-Kutta methods for these equations are obtained by the theory of the Pade approximation. Finally, some numerical examples are given to illustrate the theoretical results.

Robustness for Stable Impulsive Equations via Quadratic Lyapunov Functions

For a linear impulsive differential equation, we give a complete characterization of the existence of a nonuniform exponential contraction in terms of quadratic Lyapunov functions and of the operators defining them. This corresponds to consider a nonuniform exponential stability of the dynamics, which is typical for example in the context of ergodic theory. As an application, we use this characterization to establish in a very simple manner the robustness property of a nonuniform exponential contraction under sufficiently small linear perturbations. In addition, we obtain versions of the robustness property for perturbations of the jumping times and of a strong nonuniform exponential contraction. The latter corresponds to consider not only an upper bound for the dynamics but also a lower bound.

Stability of Runge–Kutta methods for linear impulsive delay differential equations with piecewise constant arguments

This paper is concerned with a class of linear impulsive delay differential equations with piecewise continuous argument. The conditions for stability of the exact solution are obtained. Under these conditions, it is proven that the θ-method preserves stability of the equations. Furthermore, stability of Runge–Kutta methods for this kind of equations is studied. Some numerical examples are given to confirm the theoretical results.

Galerkin method for constrained variational equations and a collage-based approach to related inverse problems

For a general variational equation with constraints we present both the stability of the corresponding Galerkin method and a collage theorem for a related inverse problem. In addition we demonstrate the applicability of the results by considering forward and inverse boundary value problems of some linear impulsive differential equations.

Asymptotical stability of Runge–Kutta methods for advanced linear impulsive differential equations with piecewise constant arguments

This paper is concerned with a class of advanced linear impulsive differential equations with piecewise continuous argument. The sufficient and necessary condition for asymptotical stability of the exact solution is obtained. Under this condition, asymptotical stability of Runge–Kutta methods for this kind of equations is studied. Some numerical examples are given to confirm the theoretical results.

Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations

This paper is concerned with exponential stability of a class of linear impulsive delay differential equations (IDDEs). Exponential stability of this kind of equations is studied by the properties of delay differential equations (DDEs) without impulsive perturbations. When different delay differential equations (DDEs) without impulsive perturbations are chosen, different sufficient conditions for exponential stability of the linear impulsive delay differential equations (IDDEs) are provided. Numerical methods for this kind of equations are constructed. The convergence and exponential stability of the numerical solutions are studied and some experiments are given.

Oscillation properties of higher order linear impulsive delay differential equations

Oscillation properties of higher order linear impulsive delay differential equations

Mean-square stability of analytic solution and Euler–Maruyama method for impulsive stochastic differential equations

From the view of algebra, the mean-square stability of analytic solutions and numerical solutions for impulsive stochastic differential equations are considered. By the logarithmic norm, the conditions under which the analytic and numerical solutions for a linear impulsive stochastic differential equation are mean-square stable (MS-stable) respectively are obtained. The conditions are simple and easy to use. Some numerical experiments are given to illustrate the results.

Quasilinearization of the Initial Value Problem for Impulsive Differential Equations with Multi-Point Delay Jump Conditions

An initial value problem for nonlinear impulsive differential equations with jump conditions at several delay time points is studied. An algorithm for constructing successive approximations of the solution of the considered problem is given. This algorithm is based on the method of quasilinearization. Every successive approximation is the unique solution of an appropriately chosen initial value problem for linear impulsive differential equation with multi-point delay jump conditions. Also, every approximation is a lower/upper solution of the given problem. The rapid convergence is proved.

Linearized Oscillations in Autonomous Delay Impulsive Differential Equations

In recent years, there have been intensive efforts to establish linearised oscillation results for onedimensional delay, neutral delay and advanced impulsive differential equations. An impressive number of these efforts have yielded fruitful results in many analytical and applied areas. This is particularly obvious in the areas of applied disciplines such as the linear delay impulsive differential equations. However, there still remains a lot more to be explored in this direction, especially, in the area of non-linear autonomous differential equations. In this paper, we are proposing the development of linearised oscillation techniques for some general non-linear autonomous impulsive differential equations with several delays.

Convergence and stability of Euler method for impulsive stochastic delay differential equations

This paper deals with the mean square convergence and mean square exponential stability of an Euler scheme for a linear impulsive stochastic delay differential equation (ISDDE). First, a method is presented to take the grid points of the numerical scheme. Based on this method, a fixed stepsize numerical scheme is provided. Based on the method of fixed stepsize grid points, an Euler method is given. The convergence of the Euler method is considered and it is shown the Euler scheme is of mean square convergence with order 1/2. Then the mean square exponential stability is studied. Using Lyapunov-like techniques, the sufficient conditions to guarantee the mean square exponential stability are obtained. The result shows that the mean square exponential stability may be reproduced by the Euler scheme for linear ISDDEs, under the restriction on the stepsize. At last, examples are given to illustrate our results.