A variant of the Krein-Rutman theorem for Poincaré difference equations

摘要

Let,x _n, n ∈ ?, be a non-vanishing solution of the Poincaré difference equationwhere A _n, n ∈ ?,are K × k real matrices such that the limit A=lim _(n rarr); ∞A _n exists (entrywise). According to a Perron type theorem, the limit limit p=lim _(n rarr;) ∞ ~n√{pipe}X _n{pipe} exists and is equal to the modulus of one of the eigenvalues of A. In this paper, we show that if the solution belongs to a given order cone K in ? ~k, then p is an eigenvalue of A with an eigenvector in K. In the case of constant coefficients, this result implies the finite-dimensional version of the Krein-Rutman theorem.