## Global behavior of the max-type difference equation $$x_{n+1}=\max\{1/x_{n},A_{n}/x_{n - 1}\}$$.(English)Zbl 1167.39303

Summary: We study global behavior of the following max-type difference equation $$x_{n+1}=\max\{1/x_{n},A_{n}/x_{n - 1}\}, n=0,1,\dots,$$ where $$\{A_n\}_{n=0}^{\infty}$$ is a sequence of positive real numbers with $$0\leq \text{inf} A_{n}\leq \text{sup} A_{n}<1$$. The special case when $$\{A_n\}_{n=0}^{\infty}$$ is a periodic sequence with period $$k$$ and $$A_{n}\in (0,1)$$ for every $$n\geq 0$$ has been completely investigated by Y. Chen [J. Difference Equ. Appl. 11, No. 15, 1289–1294 (2005; Zbl 1086.39003)]. Here, we extend his results to the general case.

### MSC:

 39A11 Stability of difference equations (MSC2000)

Zbl 1086.39003
Full Text:

### References:

 [1] A. M. Amleh, J. Hoag, and G. Ladas, “A difference equation with eventually periodic solutions,” Computers & Mathematics with Applications, vol. 36, no. 10-12, pp. 401-404, 1998. · Zbl 0933.39030 [2] K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193-1199, 2006. · Zbl 1116.39001 [3] W. J. Briden, E. A. Grove, G. Ladas, and L. C. McGrath, “On the nonautonomous equation xn+1=max {An/xn,Bn/xn - 1},” in Proceedings of the 3rd International Conference on Difference Equations, pp. 49-73, Gordon and Breach, Taipei, Taiwan, September 1997. · Zbl 0938.39012 [4] W. J. Briden, E. A. Grove, G. Ladas, and C. M. Kent, “Eventually periodic solutions of xn+1=max {1/xn,An/xn - 1},” Communications on Applied Nonlinear Analysis, vol. 6, no. 4, pp. 31-43, 1999. · Zbl 1108.39300 [5] Y. Chen, “Eventual periodicity of xn+1=max {1/xn,An/xn - 1} with periodic coefficients,” Journal of Difference Equations and Applications, vol. 11, no. 15, pp. 1289-1294, 2005. · Zbl 1086.39003 [6] C. \cCinar, S. Stević, and I. Yal\ccinkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307-314, 2005. · Zbl 1074.39002 [7] J. Feuer, “On the eventual periodicity of xn+1=max {1/xn,An/xn - 1} with a period-four parameter,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467-486, 2006. · Zbl 1095.39016 [8] E. A. Grove, C. Kent, G. Ladas, and M. A. Radin, “On xn+1=max {1/xn,An/xn - 1} with a period 3 parameter,” Fields Institute Communication, vol. 29, pp. 161-180, 2001. · Zbl 0980.39012 [9] 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 [10] C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation xn+1=max {An/xn,Bn/xn - 1} with periodic parameters,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, supplement, pp. 11-15, 2003. [11] W. T. Patula and H. D. Voulov, “On a max type recurrence relation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 329-338, 2004. · Zbl 1050.39017 [12] S. Stević, “On the recursive sequence xn+1=A+(xnp/xn - 1r),” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007. · Zbl 1151.39011 [13] S. Stević, “On the recursive sequence xn+1=max {c,xnp/xn - 1p},” Applied Mathematics Letters, vol. 21, no. 8, pp. 791-796, 2008. · Zbl 1152.39012 [14] F. Sun, “On the asymptotic behavior of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008. · Zbl 1155.39008 [15] I. Szalkai, “On the periodicity of the sequence xn+1=max {A0/xn, ... ,Ak/xn - k},” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25-29, 1999. · Zbl 0930.39011 [16] H. D. Voulov, “Periodic solutions to a difference equation with maximum,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2155-2160, 2003. · Zbl 1019.39005 [17] H. D. Voulov, “On the periodic nature of the solutions of the reciprocal difference equation with maximum,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 32-43, 2004. · Zbl 1053.39023 [18] J. Bibby, “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63-65, 1974. · Zbl 0291.40003 [19] E. T. Copson, “On a generalisation of monotonic sequences,” Proceedings of the Edinburgh Mathematical Society. Series II, vol. 17, no. 2, pp. 159-164, 1971. · Zbl 0223.40001 [20] S. Stević, “A note on bounded sequences satisfying linear inequalities,” Indian Journal of Mathematics, vol. 43, no. 2, pp. 223-230, 2001. · Zbl 1035.40002 [21] S. Stević, “A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality,” Indian Journal of Mathematics, vol. 43, no. 3, pp. 277-282, 2001. · Zbl 1034.40002 [22] S. Stević, “A global convergence result,” Indian Journal of Mathematics, vol. 44, no. 3, pp. 361-368, 2002. · Zbl 1034.39002 [23] S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium Mathematicum, vol. 93, no. 2, pp. 267-276, 2002. · Zbl 1029.39006 [24] S. Stević, “Asymptotic behaviour of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681-1687, 2003. · Zbl 1049.39012
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.