Studies Of Some Aspects On Abstract Semilinear Evolution Equations And Functional Differential Equations

This dissertation is divided into two parts. The first part, chapters 1-4, is concerned with the fundamental theory of abstract semilinear evolution equations (ASEEs), while the second part, chapters 5-6, is devoted to the asymptotic behavior of solutions, the existence, uniqueness and attractivity of periodic solutions for two classes of functional differential equations (FDEs). Chapter 1 deals with the global existence of solutions for abstract semilinear RFDEs with infinite delay. By utilizing the properties of evolution systems, some sufficient conditions under which a mild solution exists globally in any given interval are established, via Leray-Schauder Alternative approach. The continuous dependence of solutions on initial functions is also ob- tained using Banach contraction mapping theorem. Chapter 2 discusses monotone iterative techniques for ASEEs. First, the abstract iterative sequences for paraxneterized BVPs are constructed by resorting to the theory of positive C0- semigroups and the method of upper and lower solutions and, the existence of maximal and minimal solutions is established under suitable conditions. As an application, the perodic solutions of ASEEs with a parameter is derived. Next, the mixed monotone iteration is studied for Cauchy problems and, the existence of maximal and minimal coupled quasisolutions is gained using fixed-point theorems. Our results improve and generalize some known ones. Chapter 3 is to study the existence of positive solutions for ASEEs. By combining the theory of positive Cc-semigroups and the cone compression fixed-point theorem, some sufficient conditions under which positive solutions exist are established for abstract semilinear ODEs and FDEs, respectively. Employing the theory of positive Cc-semigroups and Schaefer fixed-point theorem, the controllability of abstract ordinary differential systems with nonlocal conditions and functional differential systems with infinite delay, respectively, is set up in Chapter 4. The purpose of Chapter 5 is to present the asymptotic behavior of solutions for NFDES with finite delay and RFDEs with infinite delay. General results on positively invariant sets, mono- tone solutions and contracting rectangles are obtained using the theory of monotone semiflows generated by FDEs. In the final chapter, the existence, uniqueness and attractivity of periodic solutions for NFDEs with finite delay are obtained by combining the theory of monotone semifiows and fixed-point theorem, then by resorting to the theory of discrete semiflows, the existence and attractivity of periodic solutions are also presented for RFDEs with infinite delay.