Research On Almost Periodic Solutions To Several Kinds Of Differential Equations

Due to it manifest the system's regular change, the periodic solutions and almost periodicsolutions of differential equations have attracted much attention these years. Compares withthe periodic phenomena, the almost periodic phenomena are easier to see in real world. SinceH. Bohr proposed the theory of almost periodic function, it had been found with successful andinteresting applications in the differential equations, functional differential equations, impulsivedifferential equations, partial differential equations and other areas, which consumedly enlargesthe theory of almost periodic. The differential equations of almost periodic type are interestedin the theory of mathematics and have broad prospects in practical applications.In this dissertation, three kinds of differential equations such as delayed Lotka-Volterrasystems, a class of vectorial Liénard systems and a class of generalized Liénard systems areinvestigated. Some sufficient conditions to the existence and uniqueness of almost periodicsolutions for these differential equations are obtained by different methods. This dissertation isorganized by four parts.Chapter 1 simply introduce the development of almost periodic solutions and the background for three differential system and some preliminary notes.Since Liénard system have broad prospects in practical applications such as physics, nonlin-ear dynamics, engineering technology and other fields, and its inherent complicated nonlinearitydynamics characteristic, the research on solutions of Liénard system always is a subject of con-siderable interest. In Chapter 2, we investigates bounded solutions or the existence of almostperiodic solutions of the generalized Liénard system. By constructing suitable monotonous func-tions, we prove the uniqueness of bounded solutions. And the authors obtain some results thatthe ultimate boundedness with respect to part of the solution implies the existence of an almostperiodic solution by using Amerio's results. Our results are the natural generalization of the main result by Cieutat [Nonlinear Analysis, 58(2004), 885-898].The question of Liénard system, in presence of the scalar term, has been studied intensively.However, results about vectorial case are much fewer. In Chapter 3, we investigates the existenceof almost periodic solutions to a class of vectorial Liénard system. By introducing the productapace endowed with the inner product, the definition of minimal solutions is enlarged. Theauthors obtain some results that the existence of a bounded solution implies the existence ofalmost periodic solutions by the method of minimal solutions. Our results are the naturalgeneralization of the main result by Cieutat [J.Differential Equations, 209(2005), 302-328].Chapter 2 and Chapter 3 studies two classes of ordinary differential equation. However, inmany applications, the system under consideration is not only determined by the present butalso dependent of the past states, which means ordinary differential equation can not describeexactly. So, chapter 4 studies a general class of delayed almost periodic Lotka-Volterra system,including cooperative systems, competitive systems and their hybrids, and also including time-varying delays and distributed delays. Most of papers on almost periodic system are based onconstructed appropriate Lyapunov functions. In this chapter, the authors adopt a direct methodto address the problem and some results that the system has a unique global exponential stabilitypositive almost periodic solution are obtained. The conditions we obtained are weaker than thepreviously known ones, and can be easily reduced to special cases.