On a local degree for a class of multi-valued vector fields in infinite dimensional Banach spaces (Q1809751)

A version of the topological degree theory for set-valued maps of the form \(f- F\), where \(f:\overline U\to E_1\) is a nonlinear proper \(C^1\)-smooth Fredholm operator of index \(0\) and \(F:\overline U\to E_1\) is an upper semicontinuous admissible compact map with compact values in a Banach space \(E_1\) defined on the closure of an open subset \(U\) of a Banach space \(E\), is studied. Admissibility (in the sense of Górniewicz) means that there is a Vietoris map \(p: X\to\overline U\) and \(q: X\to E_1\), where \(X\) is a topological space, such that \(q(p^{-1}(x))\subset F(x)\) for \(x\in\overline U\). The constructed degree is a positive integer and, when applied to the classical (i.e. single-valued finite-dimensional) situation, equals the absolute value of the ordinary (i.e. Brouwer) degree. This is done in order to avoid a possible orientation incompatibility. The usual degree properties and existence issues are established and used to investigate the solvability of a certain class of operator inclusions.