Pseudo Almost Periodic Solutions For Differential Equations

This thesis which consists of two parts mainly investigates the existence and uniqueness of pseudo almost periodic solutions for differential equations, and it also considers the relationship between almost periodic solutions and pseudo almost periodic solutions.Chapter 1 considers the pseudo almost periodic solutions of first order differential equations. Section 1 introduces some relating definitions, symbols and theorems of almost periodic type functions. Using fixed point theorem, Section 2 investigates the existence and uniqueness of the pseudo almost periodic solutions for x '(t) = a(t)x{t) + f(t). Sections 3 are based on sections 2 discuss the pseudo almost periodic solutions of nonliear equations x'(t) = a(t,x(t))x(t) + f(t,x(t)). Using Schauder fixed point theorem, Sections 4 studies the periodic solution of the equations x '(t) = a(t)x(t) + g(t, x(t), x(t - Ï„)) + f(t).Chapter 2 investigates existence and unqueness of pseudo almost periodic solutions for a class of high dimensional differential equations. Section 1 mainly introduces the concept of exponential dichotomy and some relating definitions. Using exponential dichotomy, matrix decomposition and fixed point theorem, section 2 investigates the pseudo almost periodic solution of x'(t) = A(t)x(t) + f(t). Section 3 consideres therelationship between bounded solutions of the almost periodic system and pseudo almost periodic system. Section 4 uses fixed point theorem to discuss the solutions of nonliear system x'(t) = A(t)x(t) + f(t,x(t)). The last section presents some sufficient conditions of satisfying exponential dichotomy.