Homotopy invariants for tangent vector fields on closed sets

In the paper we construct a topological degree theory for (single and convex-valued) tangent vector fields defined on locally compact closed subsets of a Banach space. The obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds. The degree allows to study vector fields on sets which are neither smooth nor convex and may be applied to study continuation and bifurcation of equilibria in parameterized families of closed sets.