Iterative Equations On High Dimensional Space And Wiener Type Banach Algebra

Nonlinear science is one of the most important topics in today's science. 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 mapping. Many mathematical models in physics, mechanics, biology and astronomy are given in such forms. Hence, it is the important topics of modern dynamical systems to study the continuous motion described by differential equations and to study the discrete dynamical systems described by iteration of mappings. Iterative equations are the equivalent form that includes unknown iteration of mappings. It has many important applications in the nature. For example, the Feigen-baum phenomena as investigating universality of period-doubling bifurcation cascade, invariant curves and manifolds of a differential equation, invariant tori and curves of Hamiltonian systems and the normal form problem can be reduced to the research of iterative equations. Iterative equations have become an important mathematics equation form which closely link with differential equations, difference equations, interal equations and dynamical systems, and have been given attention by many scholars. In recent several ten years, many results have been achieved in this field. In preface of this paper, concepts ofiteration, iterative equations, the relationship between iteration and dynamical systems and the iterative equations in some mathematics fields are intruduced. Many known results on iterative roots, linear type iterative equations and nonlinear type iterative equations are summarized. The problems that have not been resolved in the research fields of iterative equations are given.At present, many known results are obtained only on one dimensional functional space, and the research on high dimensional space is little, only the continuous solutions are discussed on RN, neither the differentiate solutions on RN nor the solutions on abstract space and manifold. In the condition of one dimensional space, functions often have better properties such as monotonicity, the derivates of reverse mappings are Lipschitzian (often need to be considered in studying the C1 solutions), but in high dimensional space, there have not these good properties. Hence it need to make some improvement and break through in discussing high dimensional condition.In Chapter 2, first, existence, uniqueness and stability of continuously dif-ferentiable solutions of a kind of polynomial-like iterative equations with infanty items on RN are studied. By constructing a new structure operator different from before in order to improve the ways used previous, it avoids the conditions that the derivates of reverse mappings of the unknown mappings are bi-Lipschitzian, and simplifies the the demonstration process. Then, by using the way of nonlinear functional analysis and topological degree theory as well as Schauder fixed points theory, the continuous solutions of a class of mapping iterative equations on Banach space are discussed.In Chapter 3, by using the way of lifting and continuous extention, a kind of equation of nonlinear mapping iterates is discussed on the one dimensional compact manifold S1. By defining the partial relationship and orientation-preserving properties, using the techniques of lifting and continuous extensionas well as the results on R1, continuous solutions and differentiable solutions are obtained on S1.On the other hand, as the combination of abstract algebra and Banach space theory, Banach algebra plays an important role in coping with many typical analysis problems (such as Wiener theorem). In Chapter 4, first, the conce...