×

zbMATH — the first resource for mathematics

The behavior of positive solutions of a nonlinear second-order difference equation. (English) Zbl 1146.39018
Summary: The authors study the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation \(x_{n}=f(x_{n - 2})/g(x_{n - 1})4, 4n\in \mathbb N_{0}\), where \(f,g\in C[(0,\infty ),(0,\infty )]\). It is shown that if \(f\) and \(g\) are nondecreasing, then for every solution of the equation the subsequences \(\{x_{2n}\}\) and \(\{x_{2n - 1}\}\) are eventually monotone. For the case when \(f(x)=\alpha +\beta x\) and \(g\) satisfies the conditions \(g(0)=1\), \(g\) is nondecreasing, and \(x/g(x)\) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then \(f(x) = c_{1}/x\) and \(g(x) = c_{2}x\), for some positive \(c_{1}\) and \(c_{2}\).

MSC:
39A11 Stability of difference equations (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence xn+1=\alpha +xn - 1/xn,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790-798, 1999. · Zbl 0962.39004 · doi:10.1006/jmaa.1999.6346
[2] F. Balibrea, A. Linero Bas, G. S. López, and S. Stević, “Global periodicity of xn+k+1=fk(xn+k)\cdots f1(xn+1),” Journal of Difference Equations and Applications, vol. 13, no. 10, pp. 901-910, 2007. · Zbl 1127.39004 · doi:10.1080/10236190701351144
[3] K. S. Berenhaut, K. M. Donadio, and J. D. Foley, “On the rational recursive sequence yn=A+yn - 1/yn - m for small A,” to appear in Applied Mathematics Letters. · Zbl 1152.39304 · doi:10.1016/j.aml.2007.07.033
[4] K. S. Berenhaut, J. E. Dice, J. D. Foley, B. D. Iri\vcanin, and S. Stević, “Periodic solutions of the rational difference equation yn=yn - 3+yn - 4/yn - 1,” Journal of Difference Equations and Applications, vol. 12, no. 2, pp. 183-189, 2006. · Zbl 1090.39003 · doi:10.1080/10236190500539295
[5] K. S. Berenhaut and S. Stević, “A note on the difference equation xn+1=1/xnxn - 1+1/xn - 3xn - 4,” Journal of Difference Equations and Applications, vol. 11, no. 14, pp. 1225-1228, 2005. · Zbl 1088.39017 · doi:10.1080/10236190500331370
[6] K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation xn=A+(xn - 2/xn - 1)p,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909-918, 2006. · Zbl 1111.39003 · doi:10.1080/10236190600836377
[7] L. Berg, “Nonlinear difference equations with periodic solutions,” Rostocker Mathematisches Kolloquium, no. 61, pp. 13-20, 2006. · Zbl 1145.39302 · ftp.math.uni-rostock.de
[8] L. Berg and S. Stević, “Periodicity of some classes of holomorphic difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 827-835, 2006. · Zbl 1103.39004 · doi:10.1080/10236190600761575
[9] M. Csörnyei and M. Laczkovich, “Some periodic and non-periodic recursions,” Monatshefte für Mathematik, vol. 132, no. 3, pp. 215-236, 2001. · Zbl 1036.11002 · doi:10.1007/s006050170042
[10] R. DeVault, C. Kent, and W. Kosmala, “On the recursive sequence xn+1=p+xn - k/xn,” Journal of Difference Equations and Applications, vol. 9, no. 8, pp. 721-730, 2003. · Zbl 1049.39026 · doi:10.1080/1023619021000042162
[11] H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equation xn+1=\alpha +xn - 1p/xnp,” Journal of Applied Mathematics & Computing, vol. 12, no. 1-2, pp. 31-37, 2003. · Zbl 1052.39005 · doi:10.1007/BF02936179
[12] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. · Zbl 1078.39009
[13] B. D. Iri\vcanin, “A global convergence result for a higher order difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 91292, 7 pages, 2007. · Zbl 1180.39003 · doi:10.1155/2007/91292
[14] B. D. Iri\vcanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 13, no. 3-4, pp. 499-507, 2006. · Zbl 1098.39003
[15] G. Karakostas, “Asymptotic 2-periodic difference equations with diagonally self-invertible responses,” Journal of Difference Equations and Applications, vol. 6, no. 3, pp. 329-335, 2000. · Zbl 0963.39020 · doi:10.1080/10236190008808232
[16] R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics, vol. 26, pp. 1356-1371, 1974. · Zbl 0313.26019 · doi:10.4153/CJM-1974-129-6
[17] R. C. Lyness, “1581. Cycles,” The Mathematical Gazette, vol. 26, no. 268, 62 pages, 1942. · doi:10.2307/3606036
[18] R. C. Lyness, “1847. Cycles,” The Mathematical Gazette, vol. 29, no. 287, pp. 231-233, 1945. · doi:10.2307/3609268
[19] R. C. Lyness, “2952. Cycles,” The Mathematical Gazette, vol. 45, no. 353, pp. 207-209, 1961. · doi:10.2307/3612778
[20] S. Stević, “On the recursive sequence xn+1=xn+1/g(xn),” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405-414, 2002. · Zbl 1019.39010
[21] S. Stević, “Asymptotic behavior of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681-1687, 2003. · Zbl 1049.39012
[22] S. Stević, “On the recursive sequence xn+1=\alpha +\beta xn - 1/1+g(xn),” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 12, pp. 1767-1774, 2002. · Zbl 1019.39011
[23] S. Stević, “On the recursive sequence xn+1=A/\Pi i=0kxn - i+1/\Pi j=k+22(k+1)xn - j,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249-259, 2003. · Zbl 1054.39008
[24] S. Stević, “On the recursive sequence xn+1=\alpha n+xn - 1/xn II,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 6, pp. 911-916, 2003. · Zbl 1051.39012
[25] S. Stević, “A note on periodic character of a difference equation,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 929-932, 2004. · Zbl 1057.39005 · doi:10.1080/10236190412331272616
[26] S. Stević, “Periodic character of a difference equation,” Rostocker Mathematisches Kolloquium, no. 59, pp. 3-10, 2005. · Zbl 1083.39011
[27] S. Stević, “On the recursive sequence xn+1=\alpha n+xn - 1p/xnp,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229-234, 2005. · Zbl 1078.39013 · doi:10.1007/BF02936567
[28] S. Stević, “On the recursive sequence xn+1=\alpha +\beta xn - k/f(xn,\cdots ,xn - k+1),” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 583-593, 2005. · Zbl 1100.39014
[29] S. Stević, “A note on periodic character of a higher order difference equation,” Rostocker Mathematisches Kolloquium, no. 61, pp. 21-30, 2006. · Zbl 1151.39012 · ftp.math.uni-rostock.de
[30] S. Stević, “On global periodicity of a class of difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 23503, 10 pages, 2007. · Zbl 1180.39005 · doi:10.1155/2007/23503 · eudml:116968
[31] S. Stević, “On the recursive sequence xn=\alpha +\sum i=1k\alpha ixn - pi/1+\sum j=1m\beta jxn - qj,” Journal of Difference Equations and Applications, vol. 13, no. 1, pp. 41-46, 2007. · Zbl 1113.39011 · doi:10.1080/10236190601069325
[32] S. Stević, “On the recursive sequence xn=1+\sum i=1k\alpha ixn - pi/\sum j=1m\beta jxn - qj,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 39404, 7 pages, 2007. · Zbl 1180.39006 · doi:10.1155/2007/39404
[33] S. Stević, “On the recursive sequence xn+1=A+xnp/xn - 1p,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 34517, 9 pages, 2007. · Zbl 1180.39007 · doi:10.1155/2007/34517
[34] T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation xn+1=xn - 1/(p+xn),” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 20652, 7 pages, 2006. · Zbl 1149.39301 · doi:10.1155/DDNS/2006/20652
[35] S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence f(xn - 1,xn),” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631-638, 2006. · Zbl 1100.39001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.