×

Forbidden set and solutions of a higher order difference equation. (English) Zbl 1390.39006

Summary: In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation \[ x_{n+1}\frac{ax_nx_{n-k}}{bx_n+cx_{n-k-1}},\quad n=0,1,\dots \] where \(a,b,c\) are positive real numbers, the initial conditions \(x_{-k-1}\), \(x_{-k},\dots,x_{-1},x_0\) are real numbers and \(k\) is a nonnegative integer.

MSC:

39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: Link

References:

[1] R. Abo-Zeid, Global behavior of a third order rational difference equation,Math. Bohem., 139 (1), (2014) 25-37. · Zbl 1340.39014
[2] R. Abo-Zeid, Global behavior of a rational difference equation with quadratic term, Math. Morav., 18 (1), (2014) 81− 88. · Zbl 1349.39014
[3] R. Abo-Zeid and C. Cinar, global behavior of the difference equation xn+1= Axn−1 · Zbl 1413.39027
[4] R. Abo-Zeid and M. A. Al-Shabi, Global behavior of a third order difference equation, Tamkang J. Math., 43 (3), (2012) 375-383. · Zbl 1268.39007
[5] R. Abo-Zeid, Global asymptotic stability of a second order rational difference equation, J. Appl. Math. & Inform., 28 (3), (2010) 797 - 804. · Zbl 1294.39010
[6] R.P. Agarwal, Difference Equations and Inequalities, First Edition, Marcel Dekker, 1992. · Zbl 0925.39001
[7] M.A. Al-Shabi, R. Abo-Zeid, Global asymptotic stability of a higher order difference equation, Appl. Math. Sci., 4 (17), (2010) 839 - 847. · Zbl 1198.39025
[8] K. S. Berenhaut, J. D. Foley, S. Stevic, The global attractivity of the rational difference yn−k+yn−m · Zbl 1131.39006
[9] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/HRC Boca Raton, 2008. · Zbl 1129.39002
[10] E. Camouzis, G. Ladas, I. W. Rodrigues and S. Northshield, On the rational recursive βx2n · Zbl 0806.39002
[11] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz, H. Sedaghat, Dynamics of rational difference equations containing quadratic terms, J. Difference Equ. Appl., 14 (2), (2008) 191-208. · Zbl 1196.39006
[12] E.A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, 2005. · Zbl 1078.39009
[13] E. A. Grove, E. J. Janowski, C. M. Kent, G. Ladas, On the rational recursive sequence xn+1=(γxαxn+β, Comm. Appl. Nonlinear Anal., 1 (3), (1994) 61-72. · Zbl 0856.39011
[14] G. Karakostas, Convergence of a difference equation via the full limiting sequences method, Diff. Equ. Dyn. Sys., 1 (4), (1993) 289-294. · Zbl 0868.39002
[15] V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993. · Zbl 0787.39001
[16] N. Kruse and T. Nesemann, Global asymptotic stability in some discrete dynamical systems, J. Math. Anal. Appl., 253 (1), (1999) 151-158. · Zbl 0933.37016
[17] M.R.S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures, Chapman and Hall/HRC Boca Raton, 2002.
[18] H. Levy and F. Lessman, Finite Difference Equations, Dover, New York, NY, USA, 1992. · Zbl 0092.07702
[19] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Eq. Appl., 15 (3), (2009) 215-224. · Zbl 1169.39006
[20] S. Stevic, On positive solutions of a (k +1)th order difference equation, Appl. Math. Lett., 19 (5), (2006) 427-431. · Zbl 1095.39010
[21] S. Stevic, Global stability and asymptotics of some classes of rational diffference equations, J. Math. Anal. Appl., 316 (1), (2006), 60-68. · Zbl 1090.39009
[22] S. Stevic, More on a rational recurrence relation, Appl. Math. E-Notes, 4, (2004) 80-84. · Zbl 1069.39024
[23] X. Yang, D. J. Evans and G. M. Megson, On two rational difference equations, Appl. Math. Comput., 176 (2), (2006) 422-430. · Zbl 1096.39022
[24] X. Yang, On the global asymptotic stability of the difference equation xn+1= xn−1xn−1xn−2+xn−3+α
[25] X.Yang, W.Su, B.Chen, G.M.Megson, D.J.Evans, On the recursive sequence xn+1
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.