| Abstract
|
:
|
The discrete dynamical systems are usually described by maps as f: N usd\rightarrowusd Rn or f: Z usd\rightarrowusd Rn. In other words, we have to deal with sequences whose values are n-vectors. The sequence may be infinite in only one direction (in case N is the domain), or in both directions (in case Z is the domain). The space Rn is the phase space, while the "time" is assumed to be discrete: 1,2,3,usd\dotsusd,m,usd\dotsusd or usd\dots-usdm,m+1,usd\dotsusd,1,0,1,usd\dotsusd,m,usd\dotsusd. Both situations present interest in the applications. The first chapter is dedicated to a systematic investigation of the "sequence spaces", with all basic properties as Banach spaces or, more general, as Frechet spaces (linear metric spaces). Various categories of sequence space are defined and basic properties are illustrated. The most comprehensive sequence space is the space of all sequences with values in Rn, which can be organized as a Frechet space. Most common sequence spaces are closed subspaces of the above, or just algebraically imbedded in it. Two original results are included in this chapter: first a compactness criterion for families of convergent sequences, and second a compactness criterion for families of almost periodic sequences. The second chapter is concerned with operators acting in between sequence spaces as those discussed in the first chapter. Linear operators and nonlinear ones are considered. Continuity and compactness criteria are derived, in connection with the results of the first chapter. A very useful class of nonlinear operators can be defined as follow: if x = (xn) is an arbitrary sequence belonging to a given sequence space X, then y = (yn), with yn = f(n,xn), must belong to another sequence space Y. It is obvious that the properties of the function f(n,x) determines the properties of the operator x usd\rightarrowusd y. Such kind of operators are important in connection with the investigation of discrete equations such as xm+1 = f(m,xm), where m usd\inusd N or Z, or xm+1 = f(m,xm,xm-1). The third chapter contains various results in regard to the behavior of solutions to discrete equations of the form shown above, xm+1 = f(m,xm), or xm+1 = f(m,xm,xm-1), based on the results obtained in the first two chapters. There are two types of asymptotic behavior we are interested in: the convergent solutions, i.e., solutions in the space c(N,Rn), or almost periodic solutions, i.e., solutions in the space AP(Z,Rn). Both situations are discussed and criteria are given for these two types of behavior of solutions.
|