Properties Of Solutions Of The Third Order Polynomial-like Iterative Equation And Generalization Of Functional Equations For The Square Root Spiral

Iteration is an extensive phenomenon in nature and human life, and one of the most important topics in nonlinear science. Iteration of a continuous map defines a discrete dynamical system. The study of iterative dynamical systems involves self-mappings on intervals, iterative roots of functions, iterative functional equations and embedding flows. Iterative equations are those equations which involve iteration as a basic operation. In recent years, iterative equations play an important role in experimental sciences and engineerings and closely link with differential equations, difference equations, integral equations and dynamical systems. In the preface of this paper, some basic concepts of iteration, itcrative equations, Hyers-Ulam stability, the relationship between iteration and dynamical systems and the iterative equation in some mathematics fields are in-troduced. Many known results on iterative roots, iterative equations and Hyers-Ulam stability of functional equations are summarized. The problems that have not been resolved in the research fields of iterative equations are given.Chapter 2 is devoted to properties of solutions of the third order polynomial-like iterative equation. The linear difference form of the n-order polynomial-like iterative equation has been studied and applied to describe the n-order polynomial-like iterative equation, to decrease the order of iteration and reduce number of involved characteristic roots so as to give a formula for the general it-crate of solutions of the n-order polynomial-like iterative equation where multiple...