A Perron-Frobenius theorem for positive quasipolynomial matrices associated with homogeneous difference equations (Q2472055)

The authors present sufficient and necessary conditions for the stability of the homogeneous difference equation \[ y(t) - \sum_{k=1}^N A_k y(t - r_k) = 0, \] where \(r_k\) are positive constants and \(A_k\) are \(n \times n\)-matrices with nonnegative elements. For this purpose, a Perron-Frobenius theorem for the positive quasi-polynomial matrix \(H(s) = I - \sum_{k=1}^N e^{-sr_k}A_k, s \in \mathbb{C}\) is proved.