×

On a solvable nonlinear difference equation of higher order. (English) Zbl 1424.39023

Summary: In this paper we consider the following higher-order nonlinear difference equation \[ x_{n}=\alpha x_{n-k}+\frac{\delta x_{n-k}x_{n-\left( k+l\right)}}{\beta x_{n-\left( k+l\right)}+\gamma x_{n-l}},\ n\in \mathbb{N}_{0}, \] where \(k\) and \(l\) are fixed natural numbers, and the parameters \(\alpha \), \( \beta \), \(\gamma \), \(\delta \) and the initial values \(x_{-i}\), \(i=\overline{1,k+l}\), are real numbers such that \(\beta^{2}+\gamma^{2}\neq 0\). We solve the above-mentioned equation in closed form and considerably extend some results in the literature. We also determine the asymptotic behavior of solutions and the forbidden set of the initial values using the obtained formulae for the case \(l=1\).

MSC:

39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bajo I, Liz E. Global behaviour of a second-order nonlinear difference equation. J Differ Equ Appl 2011; 17: 14711486. · Zbl 1232.39014
[2] Berg L, Stević S. On some systems of difference equations. Appl Math Comput 2011; 218: 1713-1718. · Zbl 1243.39009
[3] Brand L. A sequence defined by a difference equation. Am Math Mon 1955; 62: 489-492.
[4] Camouzis E, DeVault R. The forbidden set of xn+1= p + xn−1/xn. J Differ Equ Appl 2003; 9: 739-750. · Zbl 1049.39024
[5] Dehghan M, Mazrooei-Sebdani R, Sedaghat H. Global behaviour of the Riccati difference equation of order two. J Differ Equ Appl 2011; 17: 467-477. · Zbl 1215.39012
[6] Dekkar I, Touafek N, Yazlik Y. Global stability of a third-order nonlinear system of difference equations with period-two coefficients. Racsam Rev R Acad A 2017; 111: 325-347. · Zbl 1370.39007
[7] Elsayed EM. Qualitative behavior of a rational recursive sequence. Indag Math 2008; 19: 189-201. · Zbl 1158.39005
[8] Elsayed EM. Qualitative properties for a fourth order rational difference equation. Acta Appl Math 2010; 110: 589-604. · Zbl 1201.39005
[9] Erdogan ME, Cinar C, Yalcinkaya I. On the dynamics of the recursive sequence xn+1=αxn−1∏. k=1n−2ktk=1xn−2k
[10] Haddad N, Touafek N, Rabago JFT. Well-defined solutions of a system of difference equations. J Appl Math Comput Doi:10.1007/s12190-017-1081-8. · Zbl 1390.39011
[11] Halim Y, Touafek N, Yazlik Y. Dynamic behavior of a second-order nonlinear rational difference equation. Turk J Math 2015; 39: 1004-1018. · Zbl 1339.39001
[12] Kulenovic MRS, Merino O. Discrete Dynamical Systems and Difference Equations with Mathematica. New York, NY, USA: CRC Press, 2002. · Zbl 1001.37001
[13] Levy H, Lessman F. Finite Difference Equations. New York, NY, USA: Macmillan, 1961. · Zbl 0092.07702
[14] McGrath LC, Teixeira C. Existence and behavior of solutions of the rational equation xn+1=axn−1+bxnxn. Rocky cxn−1+dxn
[15] Raouf A. Global behaviour of the rational Riccati difference equation of order two: the general case. J Differ Equ Appl 2012; 18: 947-961. · Zbl 1253.39010
[16] Raouf A. Global behavior of the higher order rational Riccati difference equation. Appl Math Comput 2014; 230: 1-8. · Zbl 1410.39021
[17] Rubió-Massegú J. On the existence of solutions for difference equations. J Differ Equ Appl 2007; 13: 655-664. · Zbl 1124.39001
[18] Sedaghat H. Existence of solutions for certain singular difference equations. J Differ Equ Appl 2000; 6: 535-561. · Zbl 0966.39002
[19] Sedaghat H. Global behaviours of rational difference equations of orders two and three with quadratic terms. J Differ Equ Appl 2009; 15: 215-224. · Zbl 1169.39006
[20] Stević S. More on a rational recurrence relation. Appl Math E-Notes 2004; 4: 80-85. · Zbl 1069.39024
[21] Stević S. On some solvable systems of difference equations. Appl Math Comput 2012; 218: 5010-5018. · Zbl 1253.39011
[22] Stević S. On the difference equation xn= xn−k/ (b + cxn−1· · · xn−k) . Appl Math Comput 2012; 218: 6291-6296. · Zbl 1246.39010
[23] Stević S. Domains of undefinable solutions of some equations and systems of difference equations. Appl Math Comput 2013; 219: 11206-11213. · Zbl 1304.39007
[24] Stević S. Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron J Qual Theo 2014; 67: 1-15. · Zbl 1324.39004
[25] Stević S, Alghamdi MA, Shahzad N, Maturi DA. On a class of solvable difference equations. Abstr Appl Anal 2013; 2013: Article ID 157943, 7 pages. · Zbl 1297.39005
[26] Stević S, Diblík J, Iričanin B, Šmarda Z. On the difference equation xn= anxn−k/ (bn+ cnxn−1· · · xn−k) . Abstr Appl Anal 2012; 2012: Article ID 409237, 20 pages. · Zbl 1246.39011
[27] Stević S, Diblík J, Iričanin B, Šmarda Z. Solvability of nonlinear difference equations of fourth order. Electron. J Differential Equations 2014; 2014: 1-14. · Zbl 1314.39013
[28] Stević S, Diblík J, Iričanin B, Šmarda Z. On a solvable system of rational difference equations. J Differ Equ Appl 2014; 20: 811-825. · Zbl 1298.39013
[29] Taskara N, Uslu K, Tollu DT. The periodicity and solutions of the rational difference equation with periodic coefficients. Comput Math Appl 2011; 62: 1807-1813. · Zbl 1231.39009
[30] Tollu DT, Yazlik Y, Taskara N. On the solutions of two special types of Riccati difference equation via Fibonacci numbers. Adv Differ Equ-Ny 2013; 2013: 174. · Zbl 1390.39020
[31] Tollu DT, Yazlik Y, Taskara N. On fourteen solvable systems of difference equations. Appl Math Comput 2014; 233: 310-319. · Zbl 1334.39002
[32] Touafek N. On a second order rational difference equation. Hacet J Math Stat 2012; 41: 867-874. · Zbl 1277.39021
[33] Yalcinkaya I. On the difference equation xn+1= α +xn−m. Discrete Dyn Nat Soc 2008; 2008: Article ID 805460, xkn
[34] Yazlik Y. On the solutions and behavior of rational difference equations. J Comput Appl Math 2014; 17: 584-594. · Zbl 1294.39005
[35] Yazlik Y, Elsayed EM, Taskara N. On the behaviour of the solutions of difference equation systems. J Comput Appl Math 2014; 16: 932-941. · Zbl 1293.39008
[36] Yazlik Y, Tollu DT, Taskara N. On the behaviour of solutions for some systems of difference equations. J Comput Appl Math 2015; 18: 166-178. · Zbl 1322.39003
[37] Yazlik Y, Tollu DT, Taskara N. On the solutions of a max-type difference equation system. Math Method Appl Sci 2015; 38: 4388-4410. · Zbl 1335.39019
[38] Yazlik Y, Tollu DT, Taskara N. On the solutions of a three-dimensional system of difference equations. Kuwait J Sci 2016; 4: 95-111. · Zbl 1463.39004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.