Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses

In this paper, we investigate periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses. Several new existence results are obtained under different conditions via fixed point methods. Finally, two examples are given to illustrate our main results.     

Variational methods to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses

In this paper, the existence of solutions for the fourth-order linear and nonlinear differential equations with non-instantaneous impulses is studied by applying variational methods. The interesting point lies in that the variational structures corresponding to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses are established for the first time.     

Variational method to differential equations with instantaneous and non-instantaneous impulses

The aim of this paper is to study the existence of solutions for second-order differential equations with instantaneous and non-instantaneous impulses. Applying variational method, the existence result is obtained.     

On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses

In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.     

Stability of functional differential equations with impulses

In this paper, the stability of functional differential equations (FDE) with impulses is investigated. Some comparison theorems are given. Several Lyapunov-Razumikhin functions of partial components of the state variable x, which can be much easier constructed, are used so that the conditions ensuring that stability are simpler and less restrictive. The results improve and generalize the ones in the literature. An example is also given to illustrate the importance of our results.     

Stability caused by impulses for delay differential equations

In the present paper sufficient conditions are obtained for the stability of the zero solution of a nonlinear delay differential equation caused by impulses. This work is partially supported by NNSF of China and EYTF of National Educational Committee     

Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations

This paper studies the practical stability of the solutions of nonlinear impulsive functional differential equations. The obtained results are based on the method of vector Lyapunov functions and on differential inequalities for piecewise continuous functions. Examples are given to illustrate our results.     

Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations

In this paper, we study integral boundary value problems for two classes of nonlinear non-instantaneous impulsive ordinary differential equations. More precisely, one is integer order impulsive model, the other is fractional order impulsive model. By using standard fixed point approach, a series of existence results are presented under different conditions.     

A class of nonlinear differential equations with fractional integrable impulses

In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki’s normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki–Ulam’s type stability are introduced and generalized Ulam–Hyers–Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.     

Stability of delay equations via Lyapunov functions

The importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v^′=L(t)vt, we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations.     

Stability analysis of stochastic differential equations with the use of Lyapunov functions of constant sign

We prove a theorem on the asymptotic stability of stochastic differential equations using Lyapunov functions of constant sign.     

Stability of nonlinear variable-time impulsive differential systems with delayed impulses

This paper investigates the stability of nonlinear variable-time impulsive differential systems with delayed impulses. By using the comparison principle, the Lyapunov method and inequality techniques, some sufficient conditions for quasi-uniformly asymptotic stability and quasi-exponential stability of a given solution of variable-time impulsive differential systems with delayed impulses are established.     

Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses

This paper is concerned with a nonlinear neutral differential equations with impulses of the form

(*)     

Asymptotic behavior of solutions of second-order nonlinear delay differential equations with impulses

This paper is concerned with second-order nonlinear delay differential equations with impulses of the form

     

Method of Lyapunov functions for differential equations with piecewise constant delay

We address differential equations with piecewise constant argument of generalized type [5], [6], [7] and [8] and investigate their stability with the second Lyapunov method. Despite the fact that these equations include delay, stability conditions are merely given in terms of Lyapunov functions; that is, no functionals are used. Several examples, one of which considers the logistic equation, are discussed to illustrate the development of the theory. Some of the results were announced at the 14th International Congress on Computational and Applied Mathematics (ICCAM2009), Antalya, Turkey, in 2009.     

Asymptotically linear solutions of differential equations via Lyapunov functions

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.     

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.