Asymptotics of solution of a system of difference equations (Q1589213)

The paper deals with a system of difference equations \[ x_n = \sum_{k=0}^n A_{n-k} x_k + f_n, \qquad n \geq 0, \] where \(\{x_n\}\) and \(\{f_n\}\) are sequences of \(m\)-dimensional vectors, the series \(\sum_{k=0}^{\infty} A_k = A > 0\) of \(m \times m\)-matrices \(A_n\) with non-negative components is convergent, \(|\lambda_j (A)|\leq 1, \lambda_1(A) = 1,\) where \(\lambda_j (A)\) are eigenvalues of the matrix \(A.\) The asymptotic behavior of a solution \(x_n\) as \(n \to \infty\) is obtained.