Infinite-dimensional Krasnosel'skii-type criteria for cones
A point \(y\) in \(\text{cl }S\), the closure of a nonempty subset \(S\) of a real topological linear space \(L\) is said to be \(R\)-visible from a point \(x\) via \(S\) iff there exists a closed ray in \(S\) emanating from \(x\) and containing \(y\). Similarly, \(y\) is clearly \(R\)-visible from \(x\) via \(S\) iff there exists a neighbourhood \(N\) of \(y\) in \(L\) such that all points of \(S\cap N\) are \(R\)-visible from \(x\) via \(S\), and let \(A_ y^ R=\{x\in S\): \(y\) is clearly \(R\)-visible from \(x\) via \(S\}\). The set of all points \(x\) of \(S\) from which all points of \(S\) are \(R\)-visible is denoted by \(\ker_ R S\) and called the \(R\)-kernel of \(S\). It is shown that \(\ker_ RS\) is the intersection of affine hulls of sets \(A_ y^ R\) for selected boundary points. Two such intersection formulae are established for a nonempty proper and for a closed connected nonconvex subset \(S\) of \(L\). This extends a recent result by the author [Geom. Dedicata 39, No.3, 363-371 (1991; Zbl 0732.52002)] proved in \(\mathbb{R}^ d\) and, combined with the infinite-dimensional Helly-type theorem due to \textit{V. Klee} [Proc. Symp. Pure Math. 7, 349-360 (1963; Zbl 0136.188)] leads to infinite-dimensional Krasnosel'skii-type theorems for cones.
infinite-dimensional Krasnosel'skii-type theorems for cones
scientific article; zbMATH DE number 493372