Topological dynamics of ordinary differential equations and kurzweil equations

Abstract

is to consider limit points (when 1 t 1 + co) of the translatesf, , wheref,(x, s) = f(~, t + s). A limit point defines a limiting equation and there is a tight relationship between the behavior of solutions of the original equation and of the limiting equations (see [l&13]). I n order to carry out this construction one has to specify the meaning of the convergence of ft , and this convergence has to have certain properties which we shall discuss later. In this paper we shall encounter a phenomenon which seems to be new in the context of applications of topological dynamics. The limiting equations will not be ordinary differential equations. We shall work under assumptions that do not guarantee that the space of ordinary equations is complete and we shall have to consider a completion of the space. Almost two decades ago Kurzweil [3] introduced what he called generalized ordinary differential equations and what we call Kurzweil equations. We shall see that if we embed our ordinary equations in a space of Kurzweil equations we get a complete and compact space, and the techniques of topological dynamics can be applied. Kurzweil has developed his theory to a great extent (see [3-71) and much more than we need here. Our assumptions (although not strictly included in [3-71) yield relatively simple proofs of the fundamental theory. Trying to present a self-contained paper, we shall give the basic definitions and properties of Kurzweil equations and in the Appendix present proofs of the basic existence uniqueness and continuous dependence results.