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 plays an important role in nonlinear science. The study of iterative dynamical systems involves iterative functional differential equations . They are differetial 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 dcrivative,evcn 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 theoryThe 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/λ)) 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 arc given in the form of iterative functional differential equations. In this paper we study the existence of the analytic of two classes iterative functional differential equations.In Chapter 1, concepts of iteration, dynamical system, iterative functional differential equation are introduced. 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.In Chapter 2 and 3, we use the Schroder transformation to change the iterative functional 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 Schroder transformation, power series theory to discuss the existence analytic solutions for an extensive class of nonlinear iterative equations.The existence of 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. In this Chaper,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").