Analytic Solutions Of One-kind Iterative Functional Equations

Nonlinear science is one of the most important topics in today's science.The theory of iterative dynamical systems involves iterative functional differential equaptions.They are differential equaptions with deviating argument of the unkown function,and the delay function depends not only on the argument of the unkown function,but also state or state derivative,even higher order derivatives.Such equaptions are kinds of new functions quite different from the usual differential equaptions(Retared FDE,Neutral FDE,Advanced FDE) which formed a systemic theory.The purpose of dynamical system theory is to study rules of change in state which depens on time.Usually there are two basic forms of dynamical system:continuous dynamical systems described by differential equaptions 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 equaption or an iterative functional differential equaption.For example,the two-body problem in a classic electrodynamic some population models,some models of commodity price fluctuations and models of blood cell productions are given in the form of iterative functional differential equaptions,In this paper we study several forms of iterative functional differ- entail equaptions.The existence,the stability of the analytic ones are discussed.In Chapter 1,concepts of uteration,dynamical system,iterative functional differential equaptions are introduced.As well as for second,three chapter of proofs provides the essential theory knowledge.Iterative functional differential equaptions are quite different from ordinary differential equaptions for the appearance of iterates of the unknown function,so the classic existence theorem for the ordinary differential equaptions is not applicable In Chapter 2,we use the.Schr(O|¨)der transformation to change the iterative functional differential equaption to another without iterates of the unknown function.Further,we obtain the existence of analytic solutionas of such an equaption by means of majorant series.We also use the Schr(O|¨)der transformation,power series theory to discuss the discuss of the existence analytic solutions for an existensive class of nonlinear iterative equaptions.In this chapter,the existence analytic solutions is closely related to the distribution of eigenvalues of linearized solutions at the fixed point.The convergence of formal solutions is very complicated when the eigenvalues lie on the unit circle. We not only prove the convergence of the formal solution under the Diophantine condition(i.e,eigenvalues is "far from" unit roots),but also make progresses without the Diophantine condition (i.e,the convergence is equivalent to the well-known"small divisor problems").