Krasnosel’skii Numbers and Non Simply Connected Orthogonal Polygons

Let \(S = T \sim (\cup\{A : A \in \mathcal{A}\})\), where \(T\) is a simply connected orthogonal polygon and \(\mathcal{A}\) is a collection of \(n\) pairwise disjoint open rectangular regions contained in \(T\). Point \(x\) belongs to the staircase kernel of \(S\), Ker \(S\), if and only if \(x\) belongs to Ker \(T\) and neither the horizontal nor the vertical line through \(x\) meets any \(A\) in \(\mathcal{A}\). This produces a Krasnosel’skii-type theorem for \(S\) in terms of \(n\). However, an example shows that, independent of \(n\), no general Krasnosel’skii number exists for \(S\).