Let K be a proper cone in R n , 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) (λI n−A)x=b, x∈K , and (ii) ( A− λI n ) 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> ρ b ( A), and we also find a necessary condition when λ= ρ b ( A) and also when λ< ρ b ( A), sufficiently close to ρ b ( A), where ρ b ( A) denotes the local spectral radius of A at b. With λ fixed, we also consider the questions of when the set ( A− λI n ) K∩ K equals {0} or K, and what the face of K generated by the set is. Then we 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.