A Degree Theory for Multivalued Perturbations of Densely Defined (S+)-Operators and Application to a Feedback Control Problem

We study the topological degree for densely defined operators of class (S+) with compact multivalued perturbations. It is assumed that a perturbation can be represented as a sum of two terms: a multivalued mapping which is itself a composition of a continuous single-valued mapping and an upper semicontinuous multivalued mapping with aspherical images, and a single-valued compact operator. This makes it possible to extend the standard degree theory for convex-valued operators to multivalued mappings whose values may have a complicated geometry, as well as to operators defined not on the whole space but only on a dense subset. Theoretical aspects concerning the definition and main properties of the topological degree for such multivalued mappings are discussed. In particular, it is shown that the introduced degree is homotopy invariant and can serve as a convenient tool for establishing the solvability of the corresponding operator inclusions. As an application, a feedback control problem for a class of nonlinear elliptic equations is considered. Using the proposed topological degree, the solvability of this problem in the weak setting is proved.