Almost Periodicity,Almost Automorphy And Asymptotic Behavior Of Nonlinear Equations

In this thesis, we discuss mainly the existence of (pseudo) almost periodic solutions and (asymptotically) almost automorphic solutions for some nonlinear equations. Also, we study the existence of global attractors for a class of semilinear hyperbolic equations. This thesis is divided into six chapters.In Chapter 1, we introduce the research backgrounds and the main results of this thesis.Chapter 2 is preliminaries, mainly including some definitions and basic properties about almost periodic functions, pseudo almost periodic functions and almost automorphic functions. Moreover, we introduce briefly the definitions and related notations of C₀ semigroups and evolution systems.In Chapter 3, of concerned is pseudo almost periodicity of the solutions to some nonautonomous evolution equations. In§3.2, we consider the following nonau-tonomous semilinear evolution equations with delayu'(t) = A(t)u(t) + f(t, u(t - h))in a Banach space and present some sufficient conditions which ensure the existence and uniqueness of pseudo almost periodic mild solutions. In§3.3, we also study a class of nonautonomous semilinear evolution equations, i.e.,u'(t) = A(t)u(t) + f(t,u(t)).But we obtain an existence theorem of pseudo almost periodic mild solutions with no Lipschitz conditions on the nonlinear term f.In Chapter 4, we investigate the following abstract semilinear integrodifferential equations with a nonlocal initial conditionsUnder some suitable hypotheses, we establish some new theorems about the existence of asymptotically almost automorphic solutions to the above nonlocal problems. Chapter 5 is concerned with some nonlinear delay integral equations arising in epidemic problems. In§5.1, we consider the following equationswhich is a model for the spread of some infectious disease. A new fixed point theorem for mixed monotone operator in a cone is presented, and with its help we establish existence theorems of positive almost automorphic solutions. Even in the case of almost periodicity, our theorems extend some earlier results. In§5.2, we study some neutral nonlinear delay integral equations as followsAn existence theorem for positive almost automorphic solutions to the above equations is obtained. As a corollary, we present some existence theorems of positive almost periodic solutions, which generalize some existing results.In the last Chapter, we handle a class of semilinear hyperbolic equations:By using the generalized semiflow theory and semigroup theory, we obtained some results including the existence of global attractors with no Lipschitz condition on the nonlinear term.