Number Of Bounded Solutions In Critical Case

Nonlinear ordinary differential equations arise in a variety of areas such as engineering technology, diffusion and reaction equations, biology, fuzzy control ect. The linearization of nonlinear ordinary differential equations becomes an important topic in ordinary equations fields, plays an important role in different application fields. This thesis is composed of two chapters. In the first chapter, we introduce the historical background of problems, which will be investigated, and the main results of this paper. In Chapter 2, we use the method of linearization to discuss the number of bounded solutions in the following critical case.where .Hartman-Grobman's linearization theory tells us that if A is an matrix none of whose eigenvalues has zero real part, and f(x)is a continuous bounded function with a small Lipschitzian, then nonlinear system is homoeomorphous to linear system ,that is, there exists a self homoeomorphism ofsending the solutions ofonto the solutions of the linear system [1].In 1973,Palmer K.J. generalized Hartman's linearization theorem tononautonomous system, that is if the linear part of the system has an exponential dichotomy and if its nonlinear term is bounded and has a small Lipschitzian constant, then we attain the linearization of it[2].It is difficult to use the linearization theory directly ,when has no exponential dichotomy or the nonlinear term is unbounded. Of course it is more difficult to determine the number of the bounded solutions. However, using the method of linearization of papers [3,4],we can determine the number of bounded solutions in the above critical case with suitable structure. In a word,the purpose of this paper is to determine the number of bounded solutions in critical case with suitable structure by using the method of linearization of papers[3,4].