An affine eigenvalue problem on the nonnegative orthant

In this paper, we consider the conditional affine eigenvalue problemλx=Ax+b,λ∈R,x⩾0,‖x‖=1,where A is an n × n nonnegative matrix, b a nonnegative vector, and ∥·∥ a monotone vector norm. Under suitable hypotheses, we prove the existence and uniqueness of the solution (λ∗, x∗) and give its expression as the Perron root and vector of a matrix A+bc∗T, where c∗ has a maximizing property depending on the considered norm. The equation x = (Ax + b)/∥Ax + b∥ has then a unique nonnegative solution, given by the unique Perron vector of A+bc∗T.