A version of the infinite-dimensional Borsuk-Ulam theorem for multivalued maps (Q2821900)

The author proves a generalization of the Borsuk-Ulam theorem for multivalued maps. Let \(E_1\) and \(E_2\) be Banach spaces and \(A:D(A) \subseteq E_1\rightarrow E_2\) be a surjective closed linear operator. Let \(\alpha:D(A) \rightarrow \mathbb{R}_+\) be a continuous even function which is also positively homogeneous and nonnegative, such that \(d \| y \| \leq \alpha(y) \leq D \| y \|\) for each \(y \in D(A)\), for some positive constants \(d,D\). Let \(\mathbb{S}_R\) denote the level set of \(\alpha\) and let \(F:\mathbb{S}_R \rightarrow Kv(E_2)\) (meaning that the image of each point \(x \in \mathbb{S}_R\) is a convex compact set) be an odd upper semicontinuous \(A\)-compact multivalued map. Let \(N(A,F)\) denote the set of all solutions of the inclusion \(A(x) \in F(x)\). It is shown that, if the dimension of the kernel of \(A\) is at least \(n\), then the dimension of \(N(A,F)\) is at least \(n-1\). Applications to Fredholm operators and differential equations are discussed.