Stability For Saveral Classes Of Impulsive Delay Differential Equations And Their Numerical Solutions

In real life, impulsive delay differential equations have important significance andwide applications. The exact solutions of impulsive delay differential equations havebeen widely studied. Since the exact solutions are very difficult to obtain, it is necessaryto study the numerical solutions. But there is a lack of the research on impulsive delaydifferential equations.In this paper, we not only study the stability, asymptotical stability and exponentialstability of impulsive delay differential equations, but also study the stability, asymptoticalstability and exponential stability of numerical solutions of impulsive delay differentialequations.Sufficient conditions for the exponential stability of impulsive delay differential e-quations are obtained by the Lyapunov-Razumikhin method. These sufficient conditionsare applied to a class of linear impulsive delay differential equations. Under these con-ditions, the exponential stability of Euler method for the linear impulsive delay differen-tial equations is considered. Furthermore, the exponential stability of the Runge-Kuttamethod for the linear impulsive delay differential equations is studied.The results obtained by Wang Q and Liu X’s in2005are applied to a class of lin-ear impulsive delay differential equations, then sufficient conditions for the exponentialstability of this kind of equations are obtained. Under these conditions, the exponen-tial stability of Runge-Kutta methods for this kind of equations is investigated, and theexponential stability of θ-method for this kind of equations is derived.The asymptotic stability and exponential stability of a class of linear impulsive delaydifferential equations(IDDEs) are studied. We prove that the solution of the linear IDDEcan be obtained by multiplying a periodic function and the solution of a linear delay differ-ential equation(DDE) without impulsive perturbations. And the asymptotic stability andexponential stability of exact solutions of IDDEs are studied by stability theory of DDEs.Given different periodic functions, different sufficient conditions for asymptotic stabilityand exponential stability of exact solutions of IDDEs are obtained. The asymptotic stabil-ity and exponential stability of the numerical solutions of IDDEs are also studied. Someexperiments are given to illustrate our results. The stability, asymptotic stability and exponential stability of a class of nonlinearIDDEs are considered. We also prove that the solution of the IDDE can be obtained bymultiplying a periodic function and the solution of a DDE without impulsive perturba-tion. And the stability, asymptotic stability and exponential stability of exact solutionsof IDDEs are studied by stability theory of DDEs without impulsive perturbation. Givendifferent periodic functions, different sufficient conditions for the stability, asymptoticstability and exponential stability of exact solutions of IDDEs are obtained. The stability,asymptotic stability and exponential stability of the numerical solutions of IDDEs are alsostudied. Our results are illustrated by some experiments.