Analytic Theory Of Iterative 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, Feigenbaum equation g(x) = -g(g(-x/)) comes from the study of Feigenbaum phenomena as investigating universality of period-doubling bifurcation cascade, invariant curves or mainfolds of a diferential equation can be found by solving a functional equation, and invariant tori of Hamiltonian systems are also related to functional equations. Besides, 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. In this paper we study several forms of iterative functional equations and iterative functional differential equations. The existence, uniquness and the stability of the smooth solutions and analytic ones are discussed.In Chapter 1, concepts of iteration, dynamical system, iterative functional equation and iterative functional differential equation are introduced. Many known results on iterative roots, linear iterative functional equation, nonlinear iterative functional equation, analytic invariant curves of plane mappings and iterative functional differential equations are summarized.Although there are many results on existence, uniquness and stability of continuous solutions and differentiable solutions for polynomial-like iterative functional equations, because of complicated computation for higher order derivatives of high order iteration, it remains difficult to study higher order somoothness. In Chapter 2, we use the fixed point theory to give conditions of the existence, the uniquness and the stability for smooth solutions and analytic ones of polynomial-like iterative equations with variable coefficients, which answers an open problem by Jingzhong Zhang, Lu Yang, Weinian Zhang and John A. Baker in [32] and [54]. Moreover, we investigate the existence of analytic solutions for polynomial-like iterative equations with variable coefficients and improve some known results. Invariant curves of the area preserving maps play an important role in the theory of periodic stability of discrete dynamical systems. In this chapter, we also discuss the existence of analytic invariant curves for two kinds of planar mappings. We reduce the existence of invariant vurves to the existence of an iterative functional equation. Then we use the Schroder transformation to change the iterative functional 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 Schroder transformation, Abel transformation, power series and Dirichlet series theory to discuss the existence and uniquness of analytic solutions for an extensive class of nonlinear iterative equations. Previous works require a, the eigenvalue of the linearization of the unknown function at its fixed point, is not on the unit circle or lies on the circle with the Diophantine condition. We breakthrough the restriction of Diophantine condition and obtain results of analytic solutions in the case of unit root a and the case that given functions have a regular singular point.Iterative fun...