### 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

 M. Abbas, M. Berzig, T. Nazir and E. Karapınar, Iterative approximation of fixed points for Preˇsi´c type Fcontraction operators, U.P.B. Sci. Bull. Series A 78 (2016), no. 2, 147–160.
 S. Banach, Sur les ope´rations dans les ensembles abstraits et leur application aux e´quationsinte´grales, Fund. Math.
3 (1922), 133–181.
 V. Berinde and M. Pˇacurar, An iterative method for approximating fixed points of Preˇsi´c nonexpansive mappings,
Rev. Anal. Numer. Theor. Approx. 38 (2009), no. 2, 144–153.
 V. Berinde and M. Pˇacurar, Two elementary applications of some Preˇsi´c type fixed point theorems, Creat. Math.
Inf. 20 (2011), no. 1, 32–42.
 C.W. Boyd and J.S.W. Wong, On nonlinear contractions,Proc. Am. Math. Soc. 20 (1969), 458–464.
 Y.Z. Chen, A Preˇsi´c type contractive condition and its applications, Nonlinar Anal. 71 (2009), 2012–2017.
 L.B. Ciri´c and S.B. Preˇsi´c, On Preˇsi´c type generalization of the Banach contraction mapping principle, Acta
Math. Univ. Comenianae. 76 (2007), no. 2, 143-147.
 R. Devault, G. Dial, V.L. Kocic and G. Ladas, Global behavior of solutions of xn+1 = axn + f(xn, xn−1), J.
Difference Eq. Appl. 3 (1998), 311–330.
 D. Duki´c, Z. Kadelburg and S. Radenovi´c, Fixed points of Geraghty-type mappings in various generalized metric
spaces, Abstract Appl. Anal. 2011 (2011), Art. ID 561245, 13 pages. H. El-Metwally, E.A. Grove, G. Ladas, R. Levins and M. Radin, On the difference equation xn+1 = α+αxn−1e
−xn ,
Nonlinear Anal. 47 (2001), no. 7, 4623–4634.
 M. Geraghty, On contractive mappings, Proc. Am. Math. Soc. 40 (1973), 604–608.
 M. S. Khan, M. Berzig and B. Samet, Some convergence results for iterative sequences of Preˇsi´c type and applications, Adv. Difference Equ. 2012 (2012), 38.
 V.L. Kocic, A note on the non-autonomous Beverton-Holt model, J. Difference Equ. Appl. 11 (2005), no. 4-5,
415–422.
 V.L. Kocic and G. Ladas, Global asymptotic behavior of nonlinear difference equations of higher order with
applications, Kluwer Academic Publishers, Dordrecht, 1993.
 S.A. Kuruklis, The asymptotic stability of xn+1 − axn + bxn−k = 0, J. Math. Anal. Appl. 188 (1994), 719–731.
 R. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988), no.
391, 1–137.
 M. Pˇacurar, Approximating common fixed points of Preˇsi´c-Kannan type operators by a multi-step iterative method,
An. stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009), no. 1, 153–168.
 M. Pˇacurar, A multi-step iterative method for approximating fixed points of Preˇsi´c-kannan operators, Acta Math.
Univ. Comenianae. 79 (2010), no. 1, 77–88.
 S. B. Preˇsi´c, Sur une classe d’in´equations aux diff´e rences finies et sur la convergence de certaines suites, Publ.
Inst. Math. (Beograd) 5 (1965), no. 19, 75–78.
 V. Parvaneh, N. Hussain and Z. Kadelburg, Generalized Wardowski type fixed point theorems via α-admissible
F G-contractions in b-metric spaces, Acta Math. Scientia 36 (2016), no. 5, 1445–1456.
 I.A. Rus, An abstract point of view in the nonlinear difference equations, Conf. An. Funct. Equ. App. Convexity,
Cluj-Napoca, 1999, October 15-16, p. 272–276.
 S. Shukla, Preˇsi´c type results in 2-Banach spaces, Afr. Mat. 25 (2014), no. 4, 1043–1051.
 S. Shukla, R. Sen and S. Radenovi´c, Set-valued Preˇsi´c type contraction in metric spaces, An. S¸tiint¸. Univ. Al. I.
Cuza Ia¸si. Mat. LXI (2015), 391–399.
 S. Shukla and R. Sen, Set-valued Preˇsi´c-Reich type mappings in metric spaces, Rev. R. Acad. Cienc. Exactas F´ıs.
Nat. Serie A. Mat. 108 (2014), no. 2, 431-440.
 S. Shukla, S. Radojevi´c, Z.A. Veljkovi´c and S. Radenovi´c, Some coincidence and common fixed point theorems
for ordered Preˇsi´c-Reich type contractions, J. Inequal. Appl. 2013 (2013), no. 1, 1–14.
 S. Stevi´c, Asymptotic behavior of a class of nonlinear difference equations, Discrete Dyn. Nature Soc. 2006 (2006),
Article ID 47156, 10 pages.
 A. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Am. Math. Soc. 14
(1963), 438–443.
 D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed Point
Theory Appl. 2012 (2012), Article ID 94.
###### Volume 13, Special Issue for selected papers of ICDACT-2021The link to the conference website is https://vitbhopal.ac.in/event/icdact_dec_21/March 2022Pages 1-15
• Receive Date: 15 August 2021
• Revise Date: 31 December 2022
• Accept Date: 01 January 2022
• First Publish Date: 01 March 2022