The Stability Of Certain First Order Impulsive Delay Differential System

In this paper, we mainly study the stability properties of the following impulsivedelay diferential system:There are so many great and widespread realistic backgrounds in natural scienceto research impulsive delay diferential equation.Many problems in mathematic model,such as electric analysis in physical aspect,ecology science,mental disorder in biologicalscience,population science and economics can be summarized to study impulsive delaydiferential equation, so There are much important practical value to research them. In20century60decade, the former Soviet Union scholar V.D.Milman and A.D.Myshkis[16]studied the phenomenon of the movement of objects under external force and publishedthesis, this was the earliest influential achievement in impulsive delay diferential equa-tion field. V.Lakshmikant ham[6]did some theory literary work in this field. D.D.Bainov,P.S.Simeonov[17], A. M. Samoilenko and N.A.Perestyuk[19]got perfect achievements andachieved significant success in impulsive delay diferential equation field. In recent years,more and more attention were paid on impulsive delay diferential equation and somepretty results were obtained. However, overall, the research about stability, asymptoticstability and the global stability of impulsive delay diferential systems are still in theprimary period, more problems need to be solved. So, we have a lot of work to do in thisfield.In nature, variation of objects are not only relied on the prevailing condition, butalso on the passed condition, which is to say there exists a time lag between reactionand the reason which caused it, and it often accompanied by instantaneous mutant phe-nomenon. The mathematic model of these phenomenon can be concluded to impulsivedelay diferential equation,it is more abundant than impulsive diferential equation, in the same time it accorded with actuality much more. There were many achievementsfor its study in the recent several ten years.K.Gopalsamy and Binggen Zhang[20]pub-lished several thesis about impulsive delay diferential equation in later of20century80decade. When we research impulsive delay diferential equations,3/2stability method isvery useful.In recent years, what attracted the interest of experts is that by the means ofstability method, we can research the impulsive diferential system, and so far we havemade some progress.However, the research of impulsive delay diferential system is quitescarce, so there are multiply works to do.In chapter one,first I get a sufcient condition of the solution for the delay difer-ential equation without impulse,then about the diferent initial values,by the means ofconstructing the solution of impulsive delay diferential equations with diferent initialvalues,I research a sufcient condition of the solution when impulse exists.In chapter two,I mainly research the stability of certain first order impulsive delaydiferential equation. In this chapter, I first introduce relational prior knowledge aboutimpulsive delay diferential equation,then by the specific Yorke condition, put forwardreasonable conditions, conduct a rational linear change,change the system into anotherequivalent system, use3/2method, I get a sufcient condition of the solution. Combinedwith specific example I verify the theorem.In chapter three,use3/2method, I research a new impulsive delay diferential equa-tion, restrict the relation between impulse and time delay. For the existence of impulse,the distributed of time delay has many diferent cases.It is more complexed.In this chap-ter, I prove every case separately and get a sufcient condition of the stability of thesolution.Last combined with specific example I verify the theorem.In chapter four,I first introduce relational prior knowledge of oscillatory and nonoscil-latory about impulsive delay diferential equation,then I define the oscillatory solutionand nonoscillatory of the system,and eventually positive and negative solution, I givesufcient conditions of boundedness of the oscillatory solution and nonoscillatory solu-tion,overcome afect with impulse, prove boundedness of the oscillatory solution impor-tantly.Then I get the asymptotically stability of the system.Using3/2method, I proveglobal attractivity in every case for impulsive delay diferential equation.