### 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

[1] M. Abbas, M. Berzig, T. Nazir and E. Karapınar, Iterative approximation of fixed points for Presic type Fcontraction operators, U.P.B. Sci. Bull. Series A 78 (2016), no. 2, 147–160.
[2] S. Banach, Sur les ope´rations dans les ensembles abstraits et leur application aux equationsintegrales, Fund. Math. 3 (1922), 133–181.
[3] V. Berinde and M. Pˇacurar, An iterative method for approximating fixed points of Presic nonexpansive mappings, Rev. Anal. Numer. Theor. Approx. 38 (2009), no. 2, 144–153.
[4] V. Berinde and M. Pacurar, Two elementary applications of some Presic type fixed point theorems, Creat. Math. Inf. 20 (2011), no. 1, 32–42.
[5] C.W. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969), 458–464.
[6] Y.Z. Chen, A Presic type contractive condition and its applications, Nonlinear Anal. 71 (2009), 2012–2017.
[7] L.B. Ciric and S.B. Presic, On Presic type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenianae. 76 (2007), no. 2, 143-147.
[8] 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.
[9] D. Dukic, Z. Kadelburg and S. Radenovic, Fixed points of Geraghty-type mappings in various generalized metric spaces, Abstract Appl. Anal. 2011 (2011), Art. ID 561245, 13 pages.
[10] 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.
[11] M. Geraghty, On contractive mappings, Proc. Am. Math. Soc. 40 (1973), 604–608.
[12] M.S. Khan, M. Berzig and B. Samet, Some convergence results for iterative sequences of Presic type and applications, Adv. Difference Equ. 2012 (2012), 38.
[13] V.L. Kocic, A note on the non-autonomous Beverton-Holt model, J. Difference Equ. Appl. 11 (2005), no. 4-5,
415–422.
[14] V.L. Kocic and G. Ladas, Global asymptotic behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, 1993.
[15] S.A. Kuruklis, The asymptotic stability of xn+1 − axn + bxn−k = 0, J. Math. Anal. Appl. 188 (1994), 719–731.
[16] R. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988), no. 391, 1–137.
[17] M. Pacurar, Approximating common fixed points of Presic-Kannan type operators by a multi-step iterative method, An. stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009), no. 1, 153–168.
[18] M. Pacurar, A multi-step iterative method for approximating fixed points of Preˇsi´c-kannan operators, Acta Math. Univ. Comenianae. 79 (2010), no. 1, 77–88.
[19] S. B. Presic, Sur une classe dinequations aux diff´e rences finies et sur la convergence de certaines suites, Publ. Inst. Math. (Beograd) 5 (1965), no. 19, 75–78.
[20] 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.
[21] 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, pp. 272–276.
[22] S. Shukla, Presic type results in 2-Banach spaces, Afr. Mat. 25 (2014), no. 4, 1043–1051.
[23] S. Shukla, R. Sen and S. Radenovic, Set-valued Presic type contraction in metric spaces, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. LXI (2015), 391–399.
[24] S. Shukla and R. Sen, Set-valued Presic-Reich type mappings in metric spaces, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Serie A. Mat. 108 (2014), no. 2, 431-440.
[25] S. Shukla, S. Radojevic, Z.A. Veljkovic and S. Radenovic, Some coincidence and common fixed point theorems for ordered Presic-Reich type contractions, J. Inequal. Appl. 2013 (2013), no. 1, 1–14.
[26] S. Stevic, Asymptotic behavior of a class of nonlinear difference equations, Discrete Dyn. Nature Soc. 2006 (2006), Article ID 47156, 10 pages.
[27] A. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Am. Math. Soc. 14 (1963), 438–443.
[28] 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