Linear equations over cones and Collatz-Wielandt numbers (Q1870059)

Let \(K\) be a proper cone in \(\mathbb{R}^n\), let \(A\) be an \(n\times n\) real matrix that satisfies \(AK\subseteq K\), let \(b\) be a given vector of \(K\), and let \(\lambda\) be a given positive real number. The following two linear equations are considered in the paper: \[ \text{(i)}\quad (\lambda I_n-A)x=b,\;x \in K,\qquad \text{(ii)} \quad (A-\lambda I_n)x=b, \;x\in K. \] The authors obtain several equivalent conditions for the solvability of the first equation. For the second equation an equivalent condition for its solvability in case when \(\lambda >\rho_b(A)\), and also necessary condition when \(\lambda= \rho_b(A)\) and when \(\lambda< \rho_b(A)\), sufficiently close to \(\rho_b(A)\), where \(\rho_b(A)\) denotes the local spectral radius of \(A\) at \(b\), are given. With \(\lambda\) fixed, the authors also consider the questions of when the set \((A-\lambda I_n) K \cap K\) equals \(\{0\}\) or \(K\), and what the face of \(K\) generated by the set is. Then they derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of \(M\)-matrices among \(Z\)-matrices in terms of alternating sequences.