Computing the topological degrees via semi-concave functionals (Q446216)

In this paper, the authors provide two theorems stating that the Leray-Schauder degree of \(I-A\), where \(A\) is a completely continuous map defined on the closure of a bounded open subset \(\Omega\) containing 0 in an infinite-dimensional Banach \(E\) space space is 0 (resp. 1) depending on whether \(A\) is expanding (resp. contracting) with respect to a bounded, even and uniformly continuous semi-concave functional \(\alpha\) on \(E\) satisfying some other technical assumptions, i.e., \(\alpha(Ax)\geq \alpha(x)\) (resp. \(\alpha(Ax)\leq\alpha(x)\)) for all \(x\) in the boundary of \(\Omega\).