Algorithms For Iterative Roots And Iterative Equations And Their Stability

Since the initial works by N.H. Abel, B. Babbage et al, increasing attentionshave been paid to the old problem of iterative roots. An n-th iterative root f ofa given self-mapping F:X→X, where X is a nonempty set, is a solution ofthe functional equation fⁿ(x)=F(x), (?)x∈X. (0.0.1)Generally, iterative equations are those equations which involve iteration as abasic operation. In recent years, promoting by the developing of nonlinear science and the theory of iteration, the problem of iteration, especially the methodsof computing the solutions have become important subject and has attractedinterests from the community of information sciencist such as L. Kinderman,P. Protzel and N. Iannella et al. For monotonic and portion of non-montoniccases, the general method of constructing the iterative roots on the interval hasobtained by M. Kuczma and Cy. Targonski et al. Owing to the results, whenF is an increaing polygonal function, J. Kobza consider the polygonal iterativeroots for n=2 and give the algorithms for computing the roots. Concerningthe polynomial-like iterative equations, S. Nabeya and J. Matkowski et al. givethe method of finding the characteristic solutions, which provide the ideas forsolving the problems. From the route of J. Kobza's work, the iterative rootsof polygonal functions should by considered firstly, then we can use polygonalfunctions to approach the general continuous solutions. So, the problem of stability of solutions was brougth up. B. Xu and W. Zhang discuss the Hyers-Ulamstability of polynomial-like iterative equation and iterative roots, which makethe work becoming more meaningful when we study the algorithms. We state some basic knowledge on iteration, dynamical system and tunctional equationtheories which will be used in the thesis in chapter 1. Many known results onpolynomial-like iterative equations, iterative roots and the idea of characteristicsolutions are summarized.Based on Kobza's work, we discuss iterative roots in the class of polygonal non-monotonic functions and give an algorithm to compute polygonal iterative roots for a given polygonal non-monotonic function f in chapter 2. UnlikeKobza's work, we discuss on a compact interval [a, b] instead of the whole real lineR, and decreasing polygonal f are also discussed for polygonal iterative rootswith general n. Moreover, we also discuss composition of polygonal functionsand exhibit a property of such composition, through Having different domainsand ranges possibly as well as different distribution of vertices, composition ofpolygonal functions usually makes more complex than iteration of a polygonalfunction, which actually reinforces the corresponding result in Kobza's work.For general form of polynomial-like iterative equations, how to compute thesolutions also depend on the existence and the methods of construct the generalsolutions. In chapter 3, owing to the work by D. Yang and W. Zhang, we studysome properties of continuous solutions which didn't be discussed by D. Yangand W. Zhang. We discuss properites of solutions of a third order polynomial-likeiterative equations in some characteristic conditions. The properties of continuous solutions are given for r_j≠0, j=1, 2, 3. Moreover, for |r₁|=1, we alsoconsider the properties of continuous solutions for 0＜r₂, r₃≠1, r₂＜r₃and-1≠r₂, r₃＜0, r₂＜r₃. All of those results play an important role inconctructing the general solutions.Recently, K. Nikodem and W. Zhang studied the second order multivaluediterative equation and obtained the existence of strictly increaing upper semicontinuous solutions of the equation. Continuing their work, we discuss theHyers-Ulam stability and the Hyers-Ulam Rassias stability of the equation inchapter 4. The existence of continuous approximate solutions of the equation the classical existence theorems in functional equation are not valid and the sta-bility of the solutions are involved if we consider the existence and uniqueness ofsolutions near its approximate solutions. We prove that there exists a solution ofthe equation near its an approximate multivalued solution by constructing theCauchy sequence which is convergent.