Existence Of Mild Solution For Several Classes Of Evolution Equations With Impulses

In recent years,impulse differential equation has been studied in many fields,such as population dynamics and microbial simulation systems,and its own research significance and application value has attracted the interest of many domestic and foreign scholars.Based on operator semigroup theory,evolution family theory,fractional power operator theory and the theory of measure of noncompactness,the existence of mild solutions of several kinds of evolution equations with impulses are obtained in this paper,which are divided into the following four chapters:The chapter 1 is the introduction,which introduces the domestic and foreign research background and status of impulse evolution equations,summarizes the main work of this paper,and gives the preparatory knowledge needed for the follow-up work.In chapter 2,we study the existence of mild solutions for a class of first-order integro-differential equations with non-instantaneous impulses in Banach spaces.Previous studies on the evolution equations of non-instantaneous impulses are different,in this chapter,we consider the case that the operator family ()dependent on time generates the two-parameter resolvent operator (,).By using the resolvent operator theory,the measure of noncompactness and the Darbo’s fixed point theorem,we get the existence conclusion of the mild solution of this equation,which enriches and improves the research results of the first-order integro-differential nonautonomous equation to a certain extent.In chapter 3,The existence of mild solutions for a class of second-order nonautonomous stochastic evolution equations with instantaneous impulses is studied in Hilbert space.Stochastic term and non-local condition are added to the equation on the basis of the existing research.by using evolution systems theory,the theory of measure of noncompactness and Sadovskii’s fixed point theorem,the existence of the mild solution of the equation is obtained without requiring the corresponding evolution family is noncompact.In addition,an example is given to illustrate the results obtained in this chapter.In chapter 4,In this work,we consider a class of non-instantaneous impulsive neutral stochastic integro-differential equations driven by Brownina motion and fractional Brownian motion in separable Hilbert spaces.Based on the equations in Chapter 3,delay term,neutral term and fractional Brownian motion are added.Based on the theory of analytic semigroup,fractional power of operators,stochastic analysis and Banach fixed point theorem,an existence and uniqueness result of mild solution for the equation are obtained.Further,by using an integral equation and some sufficient conditions,we get the exponential stability of mild solution in the mean square moment.Finally,an example is given to illustrate the feasibility of the results obtained.