Degree theory for noncompact multivalued vector fields

Abstract

Introduction. In this note we indicate the development and state the properties of a degree theory for a rather general class of multivalued mappings, the so-called ultimately compact vector fields, and then use this degree to obtain fixed point theorems. As will be seen, these results unite and extend the degree theory for single-valued ultimately compact vector fields in [13] and the degree theory for multivalued compact vector fields in ([5], [8]) and also serve to extend to multivalued mappings the fixed point theorems for single-valued mappings obtained in [1], [2], [3], [9], [10], [13], and others (see [13]) and to more general multivalued mappings the fixed point theorems in [4], [6], [8]. The detailed proofs of the results presented in this note will be published elsewhere.