| Neutral evolution equations are an important branch in the field of differential equations.The existence,asymptotic properties of solutions and control theory of this kind of equations are essential and generally interested topics having important theoretical and practical research value.In this dissertation,by means of semigroup theory,theory of fractional power operators,theory of resolvent operators and fixed point principle,the asymptotic properties and controllability of some semilinear neutral(integro-)differential evolution equations with nonlocal conditions are studied systematically.This full text is divided into five chapters.In Chapter 1,we mainly introduce the research background and significance of neutral(integro-)differential evolution equations,and focus on the relevant recent research situation on neutral evolution equations in recent years.We also sketch in this part out the main work of this dissertation.In Chapter 2,we study the existence,regularity and asymptotic properties of solutions for a class of neutral differential evolution equations with discrete nonlocal conditions.These problems are mainly discussed by using the theory of semigroups and the Sadvoskii fixed point theorem.In particular,since the nonlinear term of the equation contains the partial derivatives of space variables,we take advantage of theory of fractional power operators,α-norm to discuss the problems.Finally,an example is given to show the applications of the obtained results.Chapter 3 is devoted to the discussion of the existence and regularity of solutions for a class of neutral differential evolution equations with state-dependent nonlocal conditions.By utilizing the theory of fractional power operators and fixed point theorem,the existence and regularity of solutions are studied under certain conditions for nonlinear functions,including the existence of Holder continuous solutions.It is pointed out here that since the nonlocal Cauchy problem is defined on the infinite interval[0,+∞),the work in this chapter mainly uses the so-called generalized Arzela-Ascoli theorem with regards to the functional space Ce([0,+∞);Xα)to prove the compactness of operators.The results obtained obviously generalize the corresponding conclusions of neutral differential evolution equations on finite intervals.We then consider in chapter 4 a class of semilinear neutral integro-differential equations with nonlocal conditions.Firstly,by means of the theory of resolvent operators established recently for linear neutral integro-differential evolution systems and fixed point theorem,we study the existence,uniqueness and regularity of the solution of the equation on infinite interval.Furthermore,the asymptotic properties of the solution,including stability and asymptotic periodicity,are discussed by Gronwall’s inequality.In particular,since the nonlocal function H does not satisfy compactness or Lipschitz continuity,we use the approximate method to prove the existence results and thus weaken the conditions of the nonlocal function H in the existing literature.Chapter 5 is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equations with nonlocal condition.The basic tool of this study is the theory of resolvent operators of linear neutral integro-differential equation established recently in literature.Fundamental solutions of linear neutral integro-differential equations is also constructed to express mild solutions of the considered system to deal with the non-uniform boundedness of the extra linear term.Sufficient conditions of approximate controllability are then obtained through the resolvent condition.It is worth mentioning that we overcome the obstacle which requires the nonlinear function be uniformly bounded in the literature through the fundamental solution method. |
