Studies On Analytic Solutions For A Kind Of Iterative Functional Differential Equations

Nonlinear science is one of the most important topics in today's sciences. The theory of iterative dynamical systems plays an important role in nonlinear science. The study of iterative dynamical systems involves self-mappings on intervals,iterative roots of functions,iterative functional equations, iterative functional differential equations and embedding flows.The purpose of dynamical system theory is to study rules of change in state which depends on time. Usually there are two basic forms of dynamical systems: continuous dynamical systems described by differential equations and discrete dynamical systems described by iteration of mappings. Many mathematical models in physics, mechanics, biology and astronomy are given in such forms. Many problems of dynamical systems can be reduced to an iterative functional equation or an iterative functional differential equation. For example,the two-body problem in a classic electrodynamics,some population models, some models of commodity price fluctuations and models of blood cell productions are given in the form of iterative functional differential equations. So, the study of iterative dynamical systems involves iterative functional differential equations. In this paper we study two forms of iterative functional differential equations. The existence of the analytic solutions are discussed.Iterative functional differential equations are quite different from ordinary differential equations for the appearance of iterates of the unknown function,so the classic existence theorem for the ordinary differential equations is not applicable. The problem whether the existence and continuity theorem of the iterative functional differential equations is similar to that of the ordinary differential equations or not is eagerly to be answered.In Chapter 1,concepts of iteration,dynamical system,iterative functio- nal differential equation are introduced. We study the existence of analytic solutions and structure of solutions about two iterative functional differential equations in chapter 2 and chapter 3. We use the Schr(o|¨)der transformation to change the iterative functional differential equation to another without iterates of the unknown function. We obtain the existence of analytic solutions of the auxiliary equation by means of majorant series and power series theory, then we get the analytic solutions of the original equation. The result is perfect.