In this paper, we consider the conditional affine eigenvalue problem λx = Ax + b, λ ∈ R, x ≥ 0, ‖x‖ = 1, where A is an n × n nonnegative matrix, b a nonnegative vector, and ‖·‖ a 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∗ , where c∗ has a maximizing property depe...

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint ran...

We present a new approach of proving certain Carath\'{e}odory-type theorems using the Perron-Frobenius Theorem, classical result in matrix theory describing largest eigenvalue with positive entries. One problems left open this note is whether our may be extended to prove similar results area, particular Colourful Carath\'{e}odory Theorem.

On page 23 of his famous monograph [2], D. V. Anosov writes Every five years or so, if not more often, someone 'discovers' the theorem of Hadamard and Perron proving it either by Hadamard's method or Perron's. I myself have been guilty of this. If (X, d X) and (Y, d Y) are metric spaces and T : X → Y is a map then the Lipschitz constant of T is the quantity lip(T) = sup d Y (T (x 1), T (x 2))

Let V be a vector space over a fully ordered field F. In Sec. 2 we characterize cones K with ascending chain condition (ACC) on faces of Ii. In Sec. 3 we show that if K has ACC on faces, then an operator A is strongly irreducible if and only if A is irreducible. In Sec. 4 we prove theorems of Perron-Frobenius type for a strongly irreducible operator A in the case that FR, the real field, and K ...

We discuss variations of functions that provide conceptually similar descriptive definitions of the Lebesgue and Denjoy-Perron integrals.