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

Let , , be a non-vanishing solution of the Poincaré difference equation where A n , , are real matrices such that the limit exists (entrywise). According to a Perron type theorem, the limit 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 , then 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.