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
[
mathcal{u}_{n+k} = mathcal{q} + frac{1}{k} sum_{j=0}^{k-1} mathcal{a}^* psi(mathcal{u}_{n+j}) mathcal{a}, n=1,2,3ldots,
]
where $mathcal{p}(n)$ is the set of $n times n$ hermitian positive definite matrices, $k$ is a positive integer,
$mathcal{q}$ 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 $mathcal{fg}$-prev{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
[
x_{n+k} = f(x_n, x_{n+1}, ldots, x_{n+k-1}), n = 1, 2, ldots
]
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.