| Many of the problems in natural science and social science often can be described by the delay differential equations. According to the boundedness of delay, we can divided delay differential equation into unbounded delay differential system and the bounded one. Generally speaking, the results of bounded delay differential system does not apply to the unbounded system. In 1978, the basic theory of bounded delay system was only initially established. The theory system is not perfect and cumbersome. Therefore, it is necessary that unbounded delay differential system is distinguishable from bounded system.Functional differential equation with pantograph delay which is also called pantograph equation, is one kind of the unbounded delay differential equations which has a broad real application background. People have found that practical problems of biological, electrodynamics and so on can be described by pantograph delay equations and have built corresponding mathematical models. Analysis found that we still have a lot of problems of qualitative theory of pantograph delay differential system for further study, so it is of great theoretical and practical significance to carry out study. Impulsive differential system has a wide range of practical application in many field. Therefore, studying it is very significative. The paper which about stability of behavior of the subject with impulsive forces published by scholars of former Soviet Union Milman and Myshkis is the first result in the field of impulsive differential system. But viewed as a whole, the studying of qualitative theory of pantograph differential system was at avery early stage, many issues need to be settled.In recent years, the studying of impulsive differential system has attracted a great deal of attention and many results have been obtained. In this paper, by employing the analytic method, Lyapunov method, Razumikhin method, Lyapunov-Razumikhin method and piecewise idea, we investigate the stability, asymptotic behavior and oscillation of the solutions of some kinds of pantograph equations with impulses. This thesis is divided into four chapters, the summary of each chapter is as follows.In Chapter one, in the part of introduction summarize the research background and significance of natural sciences and social sciences, research status of impulsive pantograph equations. And the main work of this paper is given.In Chapter two, the oscillation of a kind of pantograph equation with impulsive perturbations is studied. By employing the analytical method and new technique, some oscillation criterion are established. We reveal out impulsive effect the oscillation of solutions of this kind of equations.In Chapter three, by using Lyapunov-Razumikhin method and piecewise idea, we study the asymptotic behavior of the solutions of linear pantograph equation with impulsive perturbations and establish some relevant asymptotic conditions.In Chapter four, by employing the Lyapunov functions and Razumikhin technique, some stability results are obtained for pantograph equations with impulses. Our results reveal the fact that certain impulses may make an unstable system stable and that the stability of pantograph equations may also be inherited by impulsive pantograph ones under appropriate impulsive perturbations. |
