ABSTRACT
Differential inclusions can provide a convenient abstract framework for the study of certain control problems. In most of the literature, the results and the techniques available in connection with these two types of multifunctions have remained quite distinct. For differential inclusions with upper semicontinuous, convex valued right-hand side, the fundamental theory is entirely similar to the theory of ordinary differential equations. A quite different approach is thus needed. The present approach entirely avoids these technicalities, allowing us to deduce statements which are valid for the lower semicontinuous case, directly from the corresponding results known in the upper semicontinuous, convex-valued case. The basic link between upper and lower semicontinuous differential inclusions will be provided by the directionally continuous selections, recently considered in.
