Iteration And Iterative Roots For Polygonal Functions And Multifunctions

Iteration is an important phenomenon in nature. The penetration of X-ray, infiltration of liquid, growth of organism and application of computer are examples of iteration. In scientific computation, iteration is as an effective tool to deal with reckon problems. All recurrence relations from equal difference, equal ratios sequences to approximation by Picard are the course of iterating in mathematics. Iteration emerges dynamical system, which describe the main segments and tendency of development. By iterating, we can forecast future, that is long-time behavior and final statements which we were concerned. On the other hand, we also pay attention to its process, especially the relationship between all the segments, which refers to the reversal operation of iteration called iterative root, so that the discrete parts can be connected reasonably.Iteration is a complicated nonlinear operation. We usually use fixed point and conjugacy method to compute iteration. For fixed point method, we should know its iteration's algebraic form first, and for conjugacy method, a bridge function must be found. Therefore, it is difficult to compute its general iteration in many cases. In chapter 2, we investigate the iteration of polygonal functions on intervals. Although polygonal function is a nonlinear mapping with elementary form, computing its iterates is not easy because the concerned different lines on distinct sub-intervals may interact each other in iteration. We will make use of the dynamics of vertices' orbits to discuss the conditions under which the number of vertices either does not increase or has a bound under iteration. In some cases the explicit expressions of their general iteration are given.As we know, for a kind of functions without continuity, the property is usually no good enough, but upper semi-continuous multifunctions with finite set-valued points reflect the properties of first set functions which have finite disjoint points, so it is meaningful to consider multifunctions. Multivalued analysis as an effective method in building up and solving nonlinear mathematical models has become an important part in nonlinear analysis. It also has comprehensive applications in control theory, differential decision, economics, bio-mathematics, physics, differential inclusion and so on. The number of set-valued points increasing is the main reason for the complexity of iteration of multifunctions, the problem becomes easier as the reduction of set-valued points. As we discussed for polygonal functions, in chapter 3, we investigate the iteration of upper semi-continuous multifunctions with one set-valued point, giving conditions for the number of set-valued points not to increase under iteration, and the explicit expressions of their general iteration are given in the cases.Based on chapter 3, we continue to discuss the problem of its iterative roots. In 2004, W. Jarczyk and Weinian Zhang investigate the 2-th order iterative roots of a kind multifunctions and give some conditions of nonexistence. In chapter 4, we continue to find more possibilities of nonexistence. In addition, we also present the condition of existence for a kind of multifunctions.