The Study About Analytic Solutions For Two Classes Of Iterative Functional Equations

Nonlinear science is one of the most important topics in today's sciences.The theory of iterative dynamical systems is an important part in nonlinear science. Many problems of dynamical systems can be reduced to 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.The study of iterative dynamical systems involves iterative functional differential equations.They are differential equations with deviating argument of the unknown function,and the delay function depends not only on argument of the unknown function,but also state or state derivative,even higher order derivatives. Such equations are kinds of new functions quite different from the usual differential equations(Retarded FDE,Neutral FDE,Advanced FDE)which formed a systemic theory[1].In Chapter 1,concepts of iteration,dynamical system and iterative functional differential equation are introduced.We study the existences of analytic solutions and structure of solutions about two iterative functional differential equations in chapter 2 and chapter 3.Using the Schroder transformation to change the iterative functional differential equations to another without iterates of the unknown function. Further,we obtain the existences of analytic solutions of such an equation by means of majoring series,then the analytic solutions of iterative functional differential equations were gotten.To study the existences of analytic solutions for an extensive class of nonlinear iterative equations,the method is related to the eigenvalues of the solutions at their fixed point is off the unit circle or lies on the unit circle.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"),which is weaker then the Diophantine condition,and the result is perfcctly.