Regularity And Periodicity Of Solutions For Several Kinds Of Neutral Evolution Equations In Banach Spaces

The problems of regularity and periodicity of neutral evolution equations are an important research topics in the qualitative theory of infinite dimensional evolution systems.There are very meaningful and have great applications.In this dissertation,by using theory of resolvent operators,theory of evolution operators,fixed point principle and fractional power operators theory,we mainly discuss the regularity and periodicity of solutions of local and nonlocal Cauchy problems for several kinds of delay neutral evolution equations in Banach spaces.The whole thesis contains five chapters.In Chapter 1 we introduce some research backgrounds on neutral evolution equations and integro-differential evolution equations,and also present some recent relevant works on existence,regularity and periodicity of solutions for neutral evolution equations.Finally,we state briefly the main work of this dissertation.In Chapter 2 we study the existence and regularity of solutions for neutral integrodifferential equations with non-local condition by utilizing theory of resolvent operators.Since the nonlinear terms of the systems involve spacial derivatives,we make full use of theory of fractional power operators,-norm and Schauder's fixed point theorem to discuss the problems.An example is given to illustrate the applications of the obtained results.Chapter 3 discusses the existence and regularity of solutions of nonlocal Cauchy problem for a class of semilinear nondensely defined neutral integro-differential evolution equations.By using the theory of integrated resolvent operators and Banach fixed point theorem,the existence,continuous dependence and differentiability of solutions to this equations are obtained.It is assumed that the linear part of the considered equation is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida condition,thus it generates an integrated resolvent operator.The obtained results generalize the corresponding conclusions for the densely defined evolution equations,An example is also provided to illustrate the application of the obtained results.Chapter 4 is concerned with the existence of solutions and periodic solutions for a class of semilinear non-autonomous neutral functional differential equations with statedependent delay.We first establish the existence and regularity of bounded solutions for the considered equation,and then we show by using theory of evolution operators and Banach fixed point theorem that these solutions have periodicity property or asymptotic periodicity property respectively under some conditions.Finally,an example to illustrate the obtained results is given.In Chapter 5 we mainly study the asymptotic periodicity of solutions for neutral integro-differential evolution equations with infinite delay.By making use of theory of resolvent operators and Banach fixed point theorem,we first discuss the existence and regularity of mild solutions for neutral integro-differential evolution equations with infinite delay.Then we investigate the asymptotic periodicity of mild solutions under asymptotic periodic assumption on the nonlinear function.The obtained results extend somewhat the related conclusions in literature.