Abstract. In this paper we consider composite multifunctions, i. e. multifunctions which are of the form $F=S\circ G$ with $S$ and $G$ both multifunctions which are u.s.c. and have compact and convex values in a Banach space. The resulting composition need not be convex-valued. Nevertheless, we show that a degree function can be defined for such compositions so as to satisfy the three basic properties (normalization, additivity and homotopy invariance). Also we show that under certain conditions this degree is independent of the way the composition is defined and we examine the relation between the boundary conditions and the degree. We also derive a product formula for condensing multifunctions and define a degree function for semicondensing multivalued vector fields. Finally we examine the fixed point index for weakly inward multifunctions.

AMS Subject Classification (1991): 47H11, 55M25, 47H04

Received December 30, 1996. (Registered under 2688/2009.)