Abstract and Applied Analysis

Global Behavior of the Max-Type Difference Equation x n + 1 = max { 1 / x n , A n / x n 1 }

Taixiang Sun, Bin Qin, Hongjian Xi, and Caihong Han

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Abstract

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 , , where { A n } n = 0 is a sequence of positive real numbers with 0 inf A n sup A n < 1 . The special case when { A n } n = 0 is a periodic sequence with period k and A n ( 0 , 1 ) for every n 0 has been completely investigated by Y. Chen. Here we extend his results to the general case.

Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 152964, 10 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745553

Digital Object Identifier
doi:10.1155/2009/152964

Mathematical Reviews number (MathSciNet)
MR2506993

Zentralblatt MATH identifier
1167.39303

Citation

Sun, Taixiang; Qin, Bin; Xi, Hongjian; Han, Caihong. Global Behavior of the Max-Type Difference Equation ${x}_{n+1}=\max \{1/{x}_{n},{A}_{n}/{x}_{n-1}\}$. Abstr. Appl. Anal. 2009 (2009), Article ID 152964, 10 pages. doi:10.1155/2009/152964. https://projecteuclid.org/euclid.aaa/1268745553


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References

  • 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.
  • 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.
  • W. J. Briden, E. A. Grove, G. Ladas, and L. C. McGrath, ``On the nonautonomous equation $x_n+1=\max \A_n/x_n,B_n/x_n-1\$,'' in Proceedings of the 3rd International Conference on Difference Equations, pp. 49--73, Gordon and Breach, Taipei, Taiwan, September 1997.
  • W. J. Briden, E. A. Grove, G. Ladas, and C. M. Kent, ``Eventually periodic solutions of $x_n+1=\max \1/x_n,A_n/x_n-1\$,'' Communications on Applied Nonlinear Analysis, vol. 6, no. 4, pp. 31--43, 1999.
  • Y. Chen, ``Eventual periodicity of $x_n+1=\max \1/x_n,A_n/x_n-1\$ with periodic coefficients,'' Journal of Difference Equations and Applications, vol. 11, no. 15, pp. 1289--1294, 2005.
  • C. Çinar, S. Stević, and I. Yalçinkaya, ``On positive solutions of a reciprocal difference equation with minimum,'' Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307--314, 2005.
  • J. Feuer, ``On the eventual periodicity of $x_n+1=\max \1/x_n,A_n/x_n-1\$ with a period-four parameter,'' Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467--486, 2006.
  • E. A. Grove, C. Kent, G. Ladas, and M. A. Radin, ``On $x_n+1=\max \1/x_n,A_n/x_n-1\$ with a period 3 parameter,'' Fields Institute Communication, vol. 29, pp. 161--180, 2001.
  • 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.
  • C. M. Kent and M. A. Radin, ``On the boundedness nature of positive solutions of the difference equation $x_n+1=\max \A_n/x_n,B_n/x_n-1\$ with periodic parameters,'' Dynamics of Continuous, Discrete & Impulsive Systems. Series B, supplement, pp. 11--15, 2003.
  • 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.
  • S. Stević, ``On the recursive sequence $x_n+1=A+(x_n^p/x_n-1^r)$,'' Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007.
  • S. Stević, ``On the recursive sequence $x_n+1=\max \c,x_n^p/x_n-1^p\$,'' Applied Mathematics Letters, vol. 21, no. 8, pp. 791--796, 2008.
  • 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.
  • I. Szalkai, ``On the periodicity of the sequence $x_n+1=\max \A_0/x_n,\cdots ,A_k/x_n-k\$,'' Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25--29, 1999.
  • 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.
  • 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.
  • J. Bibby, ``Axiomatisations of the average and a further generalisation of monotonic sequences,'' Glasgow Mathematical Journal, vol. 15, pp. 63--65, 1974.
  • 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.
  • S. Stević, ``A note on bounded sequences satisfying linear inequalities,'' Indian Journal of Mathematics, vol. 43, no. 2, pp. 223--230, 2001.
  • 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.
  • S. Stević, ``A global convergence result,'' Indian Journal of Mathematics, vol. 44, no. 3, pp. 361--368, 2002.
  • S. Stević, ``Asymptotic behavior of a sequence defined by iteration with applications,'' Colloquium Mathematicum, vol. 93, no. 2, pp. 267--276, 2002.
  • S. Stević, ``Asymptotic behaviour of a nonlinear difference equation,'' Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681--1687, 2003.