Pseudo Almost Automorphic Functions And Applications To Evolution Equations

Properties of pseudo almost automorphic functions and related ones are discussedin this thesis. Applications of these properties to evolution equations are also presented. This thesis consists of five chapters.Chapter 1 presents the research background and main results of this thesis.Chapter 2 is about preliminaries of this thesis. Definitions and basic properties of almost periodic, almost automorphic, asymptotically almost periodic, asymptoticallyalmost automorphic, pseudo almost periodic and pseudo almost automorphic functions are stated. Moreover, definitions of C₀ semigroups and cosine operator functions are revoiewed.Chapter 3 is concerned with basic properties of almost automorphic functions and pseudo almost automorphic functions, which is a basis for exploring further applications of these functions to evolution equations.In Section 3.1, some basic properties of almost automorphic functions and functions with 0 mean value are investigated.In Section 3.2, we study the composition of pseudo almost automorphic functions f(t,x) and x(t) under Lipschitz conditions.In Section 3.3, we show the uniqueness of decomposition of pseudo almost automorphicfunctions, and then prove that pseudo almost automorphic functions form a Banach space under the normSection 3.4 is mainly about generalized pseudo almost automorphic functions, whose corrective terms are not bounded and continuous.Chapter 4 studies the applications of pseudo almost automorphic functions to linear evolution equations u^'(t) = Au(t)+f(t), t∈R, and semilinear evolution equations u^'(t) = Au(t) + f(t,u(t)), t∈R. For the case of A generating an exponentially stable C₀ semigroup and the case of A generating a compact C₀ semigroup, respectively, we establish new existence and uniqueness theorems for mild pseudo almost automorphic solutions to these evolution equations.In Chapter 5, we investigate exponentially asymptotically almost periodic functions.Applications to first order and incomplete second order evolution equations are discussed.