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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 327437, 9 pages
http://dx.doi.org/10.1155/2012/327437
Research Article

On the Max-Type Equation 𝑥 𝑛 + 1 = m a x { 1 / 𝑥 𝑛 , 𝐴 𝑛 𝑥 𝑛 1 } with a Period-Two Parameter

Department of Mathematics, Ahmet Kelesoglu Education Faculty, Konya University, 42090 Meram Campus, Meram Yeni Yol, Konya, Turkey

Received 30 September 2011; Accepted 12 November 2011

Academic Editor: Cengiz Çinar

Copyright © 2012 İbrahim Yalçınkaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. D. Myškis, “Some problems in the theory of differential equations with deviating argument,” Uspekhi Matematicheskikh Nauk, vol. 194, no. 32-2, pp. 173–202, 1977. View at Google Scholar
  2. A. Gelisken, C. Cinar, and R. Karatas, “A note on the periodicity of the Lyness max equation,” Advances in Difference Equations, vol. 2008, Article ID 651747, 5 pages, 2008. View at Google Scholar · View at Zentralblatt MATH
  3. A. Gelisken, C. Cinar, and I. Yalcinkaya, “On the periodicity of a difference equation with maximum,” Discrete Dynamics in Nature and Society, Article ID 820629, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Gelişken and C. Çinar, “On the global attractivity of a max-type difference equation,” Discrete Dynamics in Nature and Society, Article ID 812674, 5 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation xn+1=maxAn/xn,Bn/xn-1 with periodic parameters,” Dynamics of Continuous, Discrete & Impulsive Systems B, supplement, pp. 11–15, 2003. View at Google Scholar
  7. D. P. Mishev, W. T. Patula, and H. D. Voulov, “A reciprocal difference equation with maximum,” Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1021–1026, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. P. Mishev, W. T. Patula, and H. D. Voulov, “Periodic coefficients in a reciprocal difference equation with maximum,” Panamerican Mathematical Journal, vol. 13, no. 3, pp. 43–57, 2003. View at Google Scholar · View at Zentralblatt MATH
  9. D. Simsek, C. Cinar, and I. Yalcinkaya, “On the solutions of the difference equation xn+1=maxxn-1,1/xn-1,” International Journal of Contemporary Mathematical Sciences, vol. 10, no. 1, pp. 481–487, 2006. View at Google Scholar · View at Zentralblatt MATH
  10. D. Simsek, “On the solutions of the difference equation xn+1=maxxn-2,1/xn-2,” Selcuk University Journal of Education Faculty, vol. 23, pp. 367–377, 2007. View at Google Scholar
  11. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4, Chapman & Hall, Boca Raton, Fla, USA, 2005.
  12. E. C. Pielou, An Introduction to Mathematical Ecology, Wiley-Interscience, New York, NY, USA, 1969.
  13. E. C. Pielou, Population and Community Ecology, Gordon and Breach, 1974.
  14. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the periodic nature of some max-type difference equations,” International Journal of Mathematics and Mathematical Sciences, no. 14, pp. 2227–2239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Global attractivity and periodic character of a fractional difference equation of order three,” Yokohama Mathematical Journal, vol. 53, no. 2, pp. 89–100, 2007. View at Google Scholar · View at Zentralblatt MATH
  16. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of higher order difference equation,” Soochow Journal of Mathematics, vol. 33, no. 4, pp. 861–873, 2007. View at Google Scholar · View at Zentralblatt MATH
  17. E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of some max-type difference equations,” Vietnam Journal of Mathematics, vol. 36, no. 1, pp. 47–61, 2008. View at Google Scholar · View at Zentralblatt MATH
  18. E. M. Elabbasy and E. M. Elsayed, “On the solution of the recursive sequence xn+1=maxxn-2,1/xn-2,” Fasciculi Mathematici, no. 41, pp. 55–63, 2009. View at Google Scholar · View at Zentralblatt MATH
  19. E. M. Elsayed, “On the solutions of higher order rational system of recursive sequences,” Mathematica Balkanica, vol. 22, no. 3-4, pp. 287–296, 2008. View at Google Scholar · View at Zentralblatt MATH
  20. E. M. Elsayed and S. Stević, “On the max-type equation xn+1=maxA/xn,xn-2,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 910–922, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. E. M. Elsayed and B. D. Iričanin, “On a max-type and a min-type difference equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 608–614, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. E. M. Elsayed, “Dynamics of a recursive sequence of higher order,” Communications on Applied Nonlinear Analysis, vol. 16, no. 2, pp. 37–50, 2009. View at Google Scholar · View at Zentralblatt MATH
  23. E. J. Janowski, V. L. Kocic, G. Ladas, and S. W. Schultz, “Global behavior of solutions of xn+1=maxxn,A/xn-1,” in Proceedings of the 1st International Conference on Difference Equations, pp. 273–282, Gordon and Breach Science, San Antonio, Tex, USA. View at Google Scholar · View at Zentralblatt MATH
  24. E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.
  25. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. I. Szalkai, “On the periodicity of the sequence xn+1=maxA0/xn,A1/xn-1,,Ak/xn-k,” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25–29, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. I. Yalçinkaya, B. D. Iričanin, and C. Çinar, “On a max-type difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 47264, 10 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. I. Yalcinkaya, C. Cinar, and A. Gelisken, “On the recursive sequence xn+1=max{xn,A}/xn2xn1,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 583230, 13 pages, 2010. View at Publisher · View at Google Scholar
  31. J. Feuer, “On the eventual periodicity of xn+1=max1/xn,An/xn-1 with a period-four parameter,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467–486, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. J. Feuer and K. T. McDonnell, “On the eventual periodicity of xn+1=max1/xn,An/xn-1 with a period-five parameter,” Computers and Mathematics with Applications, vol. 56, no. 4, pp. 883–890, 2008. View at Publisher · View at Google Scholar
  33. J. Feuer, “Periodic solutions of the Lyness max equation,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 147–160, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. R. Abu-Saris and F. Allan, “Periodic and nonperiodic solutions of the difference equation xn+1=maxxn2,A/xnxn-1,” in Advances in Difference Equations, pp. 9–17, Gordon and Breach, Amsterdam, The Netherlands, 1997. View at Google Scholar · View at Zentralblatt MATH
  36. R. M. Abu-Sarris and F. M. Allan, “Rational recursive sequences involving the maximum function,” Far East Journal of Mathematical Sciences, vol. 1, no. 3, pp. 335–342, 1999. View at Google Scholar · View at Zentralblatt MATH
  37. S. Stević, “On the recursive sequence xn+1=maxc,xnp/xn-1p,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation xn+1=max1/xn,An/xn-1,” Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  39. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet