Existence Of Solutions For Impulsive Differential Systems

This dissertation mainly deals with the existence of solutions for impulsive di?erential systems from three parts.The first part mainly discusses the existence of solutions for one type of fourth-order impulsive di?erential systems,the second part discusses the existence of solutions for two types of fractional-order impulsive di?erential systems,the third part discusses the existence and exponential stability of mild solutions for three types of stochastic impulsive di?erential systems.It is consists of four chapters.The first chapter introduces the research background and research status of impulsive di?erential systems related to this paper and state our main works in this paper.The second chapter discusses the existence of solutions for one type of fourth-order impulsive di?erential systems.By employing the critical point theory,un-der the case that the impulsive functions require to satisfy the superlinear growth conditions,the su cient conditions of the existence of at least two nontrivial solu-tions and infinitely many solutions for a class of fourth-order impulsive di?erential equation have been obtained and we also generalize the range of M₁which in cor-responding lemmas of literature[14,15,16,17,18,19,20],the obtained results are e?ectively improved some corresponding results.The third chapter discusses the existence of solutions for two types of fractional-order impulsive di?erential systems.Firstly,by weakening the existing literature[24]conditions and using the critical point theory,a new principle to ensure the ex-istence of infinitely many nontrivial solutions for this kind of fractional-order im-pulsive di?erential equation is obtained.Then,we give more accurate conditions than the corresponding condition in existing literature[25]and take the impulsive disturbance into the original fractional order di?erential system,by applying the variational method,the su cient conditions for the existence of at least three nontrivial solutions of the fractional order impulsive di?erential equation are ob-tained.Without impulsive e?ects,our results can also correct the results of[25].The fourth chapter discusses the existence and exponential stability of mild solutions for three types of impulsive stochastic di?erential systems.Firstly,by using the Hausdor?measure of noncompactness,the M¨onch fixed point theorem and inequality technique,new principle to ensure the existence of mild solution for a class of impulsive stochastic di?erential equation with noncompact semi-group is obtained;Then,by establishing a new impulsive-integral inequality,we obtained that the mild solution of the impulsive stochastic di?erential equation with noncompact semigroup is mean-square exponential stability,the inequality can improve the corresponding results in previous literatures[32,62].Next,by the new impulsive-integral inequality,we also obtain the results that the mild so-lutions of the impulsive stochastic partial di?erential equation with Poisson jumps are exponential stability in pth moment,the model we investigated is simple which can include the models which have been investigated in previous literatures.Fi-nally,by applying the Schaefer’s and Banach fixed point theorems,the su cient conditions of the existence and uniqueness of mild solution for impulsive stochas-tic pantograph equations with Caputo fractional derivative have been obtained,then,we prove the mild solutions for considered equations is Hyers-Ulam stable in pth moment by the well-know gronwall inequality,the obtained results are new and e?ectively generalized some previous results.