Global attractivity results for a class of matrix difference equations

In this chapter, we investigate the global attractivity of the recursive sequence $\{\mathcal{U}_n\} \subset \mathcal{P}(N)$ defined by br / \[ br / \mathcal{U}_{n+k} = \mathcal{Q} + \frac{1}{k} \sum_{j=0}^{k-1} \mathcal{A}^* \psi(\mathcal{U}_{n+j}) \mathcal{A}, n=1,2,3\ldots, br / \] br / where $\mathcal{P}(N)$ is the set of $N \times N$ Hermitian positive definite matrices, $k$ is a positive integer, br / $\mathcal{Q}$ is an $N \times N$ Hermitian positive semidefinite matrix, $\mathcal{A}$ is an $N \times N$ nonsingular matrix, $\mathcal{A}^*$ is the conjugate transpose of $\mathcal{A}$ and $\psi : \mathcal{P}(N) \to \mathcal{P}(N)$ is a continuous. For this, we first introduce $\mathcal{FG}$-Pre\v{s}i\ c contraction condition for $f: \mathcal{X}^k \to \mathcal{X}$ in metric spaces and study the convergence of the sequence $\{x_n\}$ defined by br / \[ br / x_{n+k} = f(x_n, x_{n+1}, \ldots, x_{n+k-1}), n = 1, 2, \ldots br / \] br / with the initial values $x_1,\ldots, x_k \in \mathcal{X}$. We furnish our results with some examples throughout the chapter. Finally, we apply these results to obtain matrix difference equations followed by numerical experiments.