Poincaré-Hopf type formulas on convex sets of Banach spaces
Let \(X\) be a Banach space and \(C \subset X\) a closed convex subset. It is supposed that \(C\) has empty interior as, for instance, is the case for order intervals in Sobolev spaces. Locally Lipschitz and completely continuous maps \(A: C\to C\) are considered for which the vector field \(-\text{id}+A\) induces a semiflow \(\varphi: D(\varphi)\subset [0,\infty)\times C\to C\) on \(C\). The authors establish Poincaré-Hopf type formulae relating a fixed point index information about \(A\) with homology Conley index information about the semiflow \(\varphi\) on \(C\). When \(A\) is a gradient some results on the critical groups of isolated fixed points of \(A\) in \(C\) are obtained.
scientific article; zbMATH DE number 5734952