Asymptotic behavior of a nonhomogeneous linear recurrence system
Consider the linear recurrence system \[ x_{n+1}=(A+B_n)c_n+g_n,\tag{1} \] where \(A,B_n\) are \(k\times k\) complex matrices, and \(g_n\) are \(k\)-dimensional complex vectors, \(n=0,1,\dots\) The main result of the paper establishes an asymptotic representation of the solutions \((x_n)_{n=n_0}^{\infty}\) of (1) in the form \(\lim_{n\to\infty}(\lambda_{0}^{-n}x_n)= \alpha\xi\), where \(\lambda_0\) is assumed to be a simple dominant eigenvalue of the matrix \(A\), \(\xi\) is an eigenvector of \(A\) corresponding to \(\lambda_0\), and \(\alpha\) is determined in terms of a solution of the adjoint equation \(y_n=(A^*+B_n^*)y_{n+1}\). It is also assumed that the series \(\sum_{n=0}^{\infty}\|B_n\|\) and \(\sum_{n=0}^{\infty} |\lambda_0|^{-n}\|g_n\|\) are convergent. From the results obtained for system (1), the author also derives interesting results of the same type for the higher order difference equations \[ u_{n+1}=\sum_{j=0}^{k}(c_j+d_j(n))u(n-j)+h(n), \] where \(c_j\) are complex constants, and \(\{d_j(n)\}_{n=0}^{\infty}\), \(\{h(n)\}_{n=0}^{\infty}\) are complex sequences.
scientific article; zbMATH DE number 1749674
scientific article