Study On The Existence Of Solutions For Impulsive Delay Differential Equations

Many evolution processes and phenomena in nature science and technology such as physics, ecology, economics, control theory, which adequate mathematical models are impulsive differential systems, are characterized by the fact that at certain moments of time experience an abrupt change of state. The theory of impulsive differential systems is studied from 60th 20century, and from 80th 20century, it has been investigated by a number of authors. Many mathematicians in America, Russia, Bulgaria et al. have done much significant work, and some monographs have been published. The most prominent feature of impulsive differential system is the ability to fully take into account the instantaneous impact of the state, more accurately reflect the changes of things. When we applied the theory of impulsive differential equations to some specific practical areas, we have to consider the impact of delay frequently. Such as continuous mechanics, population ecology, electronics, nuclear reactor dynamics, and the modern control theory etc..So it is significant to do some research on its theory and application.This study is mainly about the following three aspects:(1) Existence of Solu-tions for Semilinear Impulsive Delay Differential Equations with Nonlocal Conditions. (2)Existence of Solutions for Neutral Impulsive Differential Equations with Nonlocal Conditions. (3) Existence Result for Second-Order Impulsive Differential Equations with Nonlocal Conditions. Through our research, we get the existence theorem of the solutions. In our study, we mainly using the Arzela-Ascoli fixed point theorem, Shaefer theorem, Darbo-Sadovskii theorem and some other lemmas.