Global attractivity results for a class of matrix difference equations

Document Type : Research Paper

Authors

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India

Abstract

In this chapter, we investigate the global attractivity of the recursive sequence $\{\mathcal{U}_n\} \subset \mathcal{P}(N)$ defined by
\[
\mathcal{U}_{n+k} = \mathcal{Q} + \frac{1}{k} \sum_{j=0}^{k-1} \mathcal{A}^* \psi(\mathcal{U}_{n+j}) \mathcal{A}, n=1,2,3\ldots,
\]
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 $\mathcal{FG}$-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
\[
x_{n+k} = f(x_n, x_{n+1}, \ldots, x_{n+k-1}), n = 1, 2, \ldots
\]
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.

Keywords

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Volume 13, Special Issue for selected papers of ICDACT-2021
The link to the conference website is https://vitbhopal.ac.in/event/icdact_dec_21/
March 2022
Pages 1-15
  • Receive Date: 15 August 2021
  • Revise Date: 31 December 2022
  • Accept Date: 01 January 2022
  • First Publish Date: 01 March 2022