Analytic Solutions And Smooth Solutions Of Iterative 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 iterative differential equations. They are differential equations with deviating argument of the unknown function, and the delay function depends not only on the 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.It has been used in many fields,the two-body problem in a classic electrodynamics,some population models,some modles of commodity price fluetuations and models of blood cell productions are given in the form of iterative differeutial equations.In this paper we study the analytic solutions and smooth solutions of two classes iterative differential equations.In Chapter 1,concepts and developpings of iteration,iteration and dynamical system,iterative differential equation are introduced.It provid the basic theory for the next two chapter.Iterative differential equations are quite different from ordinary differential equations for the appearance of iterates of the unknown function,which has affect the solution greatly,so the classic existence theorem for the ordinary differential equations is not applicable.For iterative differential equations whether exist existence,uniquness and stability like ordinary differential equations is a question which need to be answered.In Chapter 2(section 1)and Chapter 3,we study the existence of analytic solutions and the structure of such solutions for two kinds of iterative differential equations,we use the Schr(?)der transformation to change the iterative differential equation to another without iterates of the unknown function. Further,we obtain the existence of analytic solutions of such an equation by means of majorant series.We also use the Schr(?)der transformation,power series theory to disscns the existence of analytic solution for an extensive class of nonlinear iterative equations.In menthod requires the eigenvalues of the solutions at their fixed point is off the unit circle or lies on the unit circle with the Diophantine condition.In this paper,the existence of analytic solutions is closedly related to the distribution of eigenvalues 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 converence is equivalent to the well-known "small divisor problems").We break through the restriction of Diophantine condition and obtain results of analytic solutions in the case of unit rootαand Brjuno condition which weaker than Diophantine condition.Although there are many results on existence,uniquness and stability of continuous solutions and differentiable solutions for iterative differential equations,because of complicated computation for higher order derivatives of high order iteration,it remains difficult,to study higher order smoothness.In the second part of the chapter 2,we use the fixed point theory to give conditions of the existence,the tmiquness and the stability for higher order smooth solutions of the iterative differential equation, then we get the similar colusions with ordinary differential equations'.