Non-instantaneous Impulsive Equations Research Articles

Robustness for Non-instantaneous Impulsive Equations via Quadratic Lyapunov Functions

Robustness for Non-instantaneous Impulsive Equations via Quadratic Lyapunov Functions

Smooth stable manifolds for the non-instantaneous impulsive equations with applications to Duffing oscillators.

In this paper, we present a theory of smooth stable manifold for the non-instantaneous impulsive differential equations on the Banach space or Hilbert space. Assume that the non-instantaneous linear impulsive evolution differential equation admits a uniform exponential dichotomy, we give the conditions of the existence of the global and local stable manifolds. Furthermore, -smoothness of the stable manifold is obtained, and the periodicity of the stable manifold is given. Finally, an application to nonlinear Duffing oscillators with non-instantaneous impulsive effects is given, to demonstrate the existence of stable manifold.

Center Manifolds for Non-instantaneous Impulsive Equations Under Nonuniform Hyperbolicity

Center Manifolds for Non-instantaneous Impulsive Equations Under Nonuniform Hyperbolicity

Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses

AbstractWe consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functionsgiare compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.

Mixed-order impulsive ordinary and fractional differential equations with initial conditions

In this paper, using the idea of separated intervals in non-instantaneous impulsive equations, we initiate the study of initial value problems for mixed-order ordinary and fractional differential equations with instantaneous impulsive effects. Existence and uniqueness results are established via standard fixed point theorems. Examples illustrating the main results are also presented.

Continuous Dependence of Solutions of Integer and Fractional Order Non-Instantaneous Impulsive Equations with Random Impulsive and Junction Points

This paper gives continuous dependence results for solutions of integer and fractional order, non-instantaneous impulsive differential equations with random impulse and junction points. The notion of the continuous dependence of solutions of these equations on the initial point is introduced. We prove some sufficient conditions that ensure the solutions to perturbed problems have a continuous dependence. Finally, we use numerical examples to demonstrate the obtained theoretical results.

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.