Research On The Qualitative For Solutions Of Some Kinds Of Impulsive Delay Differential Equations

Impulse and delay extensively exist in natural sciences and social sciences.When we consider a differential system with impulses and delays, we can betterdescribe the change and development of things and accurately reveal theircharacteristic. In this thesis of Master, by using some analysis technique, we establishsome sufficient conditions for the oscillation of solutions of a class of impulsivedifferential equations with proportional delays and positive and negative coefficients.Our results reveal the fact that the oscillatory behavior of all solutions of differentialequations without impulses can be inherited by impulsive differential equations undercertain impulsive perturbations. It is also seen that the oscillatory properties of allsolutions of impulsive differential equations may be caused by the impulsiveperturbations, though the corresponding differential equations without impulses admita non-oscillatory solution. Some examples are given to illustrate the applicability ofthe results obtained. We also establish some sufficient conditions for asymptoticstability of solutions of a class of impulsive differential equations with proportionaldelays by use of some analysis techniques and Lyapunov functional method. Someexamples are also presented to illustrate the applicability of the results obtained.The full text divides into three chapters. The first chapter briefly introduces theresearch background, development status,some preliminary knowlege and the mainwork of this paper. In the second chapter, we consider a class of impulsive differentialequations with proportional delays and positive and negative coefficients. Somesufficient conditions are obtained for the oscillation of its solution. The third chapteris devoted to investigate a class of impulsive delay differential equations withproportional delays and some sufficient conditions are obtained for the asymptoticstability of its solution.