On the relative fixed point index for a class of noncompact multivalued maps

In the paper we define a topological characteristic, the fixed point index with respect to a convex closed subset of a Banach space for a class of completely fundamentally restrictible multivalued maps which can be represented as a composition of maps with aspheric values. This class includes, in particular, maps which are condensing with respect to a monotone nonsingular measure of noncompactness. Maps of this type naturally arise while the study of nonlinear systems with impulse effects. Applications of the index to some fixed point theorems are considered.