Stability for impulsive functional differential equations with infinite delays

In this paper, some theorems of uniform stability and uniform asymptotic stability for impulsive functional differential equations with infinite delay are proved by using Lyapunov functionals and Razumikhin techniques. An example is also proved at the end to illustrate the application of the obtained results.     

On the stability of impulsive functional differential equations with infinite delays

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays

This paper studies the global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays by using Lyapunov functions and improved Razumikhin technique. The results obtained in this paper improve and complement ones from some recent works. Moreover, the Razumikhin condition obtained is very simple and effective to implement in real problems and it is helpful for investigating the stability of control systems and synchronization control of chaotic systems. Finally, two examples and their simulations are given to show the effectiveness and advantages of our results.     

Existence results for impulsive neutral functional differential equations with infinite delay

In this work, we prove the existence of mild solutions for impulsive partial neutral functional differential equations with infinite delay in a Banach space. The results are obtained by using the Krasnoselski–Schaefer type fixed point theorem.     

Existence results for nondensely defined impulsive neutral functional differential equations with infinite delay

In this paper, we study a class of impulsive neutral functional differential equations with infinite delay. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille–Yosida theorem. We give some sufficient conditions ensuring the existence of integral solutions and strict solutions. To illustrate our abstract results, we conclude this work with an example.     

On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays

This paper considers the impulsive functional differential equations with infinite delays or finite delays. Some new sufficient conditions are obtained to guarantee the global exponential stability by employing the improved Razumikhin technique and Lyapunov functions. The result extends and improves some recent works. Moreover, the obtained Razumikhin condition is very simple and effective to implement in real problems and it is helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation. Finally, a numerical example and its simulation is given to show the effectiveness of the obtained result in this paper.     

STABILITY OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

In this paper,the stability of a class of impulsive functional differential equations with infinite delays is investigated. A uniform stability theorem and a uniform asymptotic stability theorem are established.     

Asymptotic stability of impulsive stochastic partial differential equations with infinite delays

In this paper, we study the existence and asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with infinite delay. By employing a fixed point approach, sufficient conditions are derived for achieving the required result. These conditions do not require the monotone decreasing behaviour of the delays.     

Impulsive stabilization of functional differential equations with infinite delays

In this paper, we study the stability of a class of impulsive functional differential equations with infinite delays. We establish a uniform stability theorem and a uniform asymptotic stability theorem, which shows that certain impulsive perturbations may make unstable systems uniformly stable, even uniformly asymptotically stable.     

New comparison results for impulsive functional differential equations

Comparison principles play an important role in the qualitative and quantitative study of differential equations. In this paper, we establish new maximum principles for impulsive functional differential equations.     

Global existence results for impulsive functional differential equations

In this paper, we investigate the existence of solutions of impulsive delay differential equation

     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Boundedness and periodicity in impulsive ordinary and functional differential equations

In this paper, we study the boundedness and periodicity of solutions for impulsive ordinary differential equations (ODEs) and functional differential equations (FDEs). By using Horn’s fixed point theorem, Hale–Yoshizawa type criteria for the existence of T$T$-periodic solutions are established. To use these criteria, we also give two new boundedness theorems, and establish new existence results for T$T$-periodic solutions which show that the impulsive perturbations do contribute to yielding periodic solutions even when the underlying systems do not enjoy periodic solutions.     

Existence results of second order impulsive functional differential equations with infinite delay and fractional damping

In this paper we prove the existence, uniqueness, regularity and continuous dependence of mild solutions for second order impulsive functional differential equations with infinite delay and fractional damping in Banach spaces. We generalize the existence theorem of integer order differential equations to the fractional order case. The results obtained here improve and generalize some known results.     

Existence results for fractional order functional differential equations with infinite delay

The Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.     

Boundedness of solutions for impulsive differential equations with integral jump conditions

The boundedness of solutions for certain nonlinear impulsive differential equations are obtained, the jumping conditions at discontinuous points are related to the integral of the past states, rather than a left hand limit at the discontinuous points. These results are obtained by new built impulsive integral inequalities with integral jumping conditions using the method of successive iteration.     

Stability of impulsive infinite delay differential equations

In this work, we consider the stability of impulsive infinite delay differential equations. By using Lyapunov functions and the Razumikhin technique, we get some results that are more general than ones given before. And in using the Razumikhin technique, we use a new technique that has been given by Shunian Zhang; we extend this technique to study impulsive systems. An example is also discussed in this work to illustrate the advantage of the results obtained.     

Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays

ABSTRACT

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.