fixed point approximation

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.

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