In this paper we present a point of view which allows one to interpret the solutions of a non-autonomous differential equation as a classical dynamical system, without assuming uniqueness of solutions of the initial value problem. In addition we are able to construct a global flow without assuming the global existence of solutions of the given differential equations. This point of view seems appropriate in the sense that many applications of the results of classical topological dynamics to the study of the solutions of differential equations can now be performed, in a straight-forward manner, without the uniqueness assumption. We shall illustrate this claim with one important application concerning the existence of periodic or almost periodic solutions.
Bibliographical noteFunding Information:
* This work was done while the author was visiting the Istituto Matematico dell’ Universitil di Firenze under the auspices of the Italian Research Council (C.N.R.) Partial support for this research was also given by X’SF Grant No. GP-27275.