Linear equations over cones and Collatz–Wielandt numbers

AbstractLet K be a proper cone in Rn, let A be an n×n real matrix that satisfies AK⊆K, let b be a given vector of K, and let λ be a given positive real number. The following two linear equations are considered in this paper: (i) (λIn−A)x=b,x∈K, and (ii) (A−λIn)x=b, x∈K. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when λ>ρb(A), and we also find a necessary condition when λ=ρb(A) and also when λ