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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 891564, 6 pages
http://dx.doi.org/10.1155/2010/891564
Research Article

On a Higher-Order Difference Equation

1Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11120, Serbia
2College of Computer Science, Chongqing University, Chongqing 400044, China

Received 20 May 2010; Accepted 23 June 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Bratislav D. Iričanin and Wanping Liu. 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.

Abstract

We describe in an elegant and short way the behaviour of positive solutions of the higher-order difference equation , , where and , extending some recent results in the literature.

1. Introduction

Studying difference equations has attracted a considerable interest recently, see, for example, [139] and the references listed therein. The study of positive solutions of the following higher-order difference equations: and where are natural numbers such that , , and was proposed by Stević in several talks, see, for example, [21, 26]. For some results concerning equations related to (1.1) see, for example, [6, 7, 10, 29, 31, 32, 34, 38], while some results on equations related to (1.2) can be found, for example, in [3, 8, 9, 1114, 1820, 22, 25, 29, 32, 33, 35] (see also related references cited therein).

Case is of some less interest, since in this case positive solutions of (1.1) and (1.2), by using the change , become solutions of a linear difference equation with constant coefficients. However, some particular results for the case recently appeared in the literature, see [16, 17, 39].

Nevertheless, motivated by the above-mentioned papers, we will describe the behaviour of positive solutions of the higher-order difference equation where and in, let us say, an elegant and short way.

Let us introduce the following.

Definition 1.1. A solution of (1.3) is said to be eventually periodic with period if there is such that for all If , then we say that the sequence is periodic with period

For some results on equations all solutions of which are eventually periodic see, for example, [2, 4, 8, 15, 28, 37] and the references therein.

Definition 1.2. One says that a solution of a difference equation converges geometrically to if there exist and such that

Now we return to (1.3).

First, note that if , then (1.3) becomes from which easily follow the following results:(a)if , then all positive solutions of (1.5) are periodic with period (b)if , then each positive solution of (1.5) converges to zero. Moreover, its subsequences converges decreasingly to zero as (c)if , then each positive solution of (1.5) tends to infinity as . Moreover, its subsequences tend increasingly to infinity as .

We may assume that and are relatively prime integers, that is, (the greatest common divisor of numbers and ). Namely, if , then by using the changes , (1.3) is reduced to copies of the following equation: where , and .

Further, note that from (1.3), we have that which implies that the sequence satisfies the following simple difference equation:

2. Main Results

Here we formulate and prove our main results.

Theorem 2.1. Assume that , , and is odd, then all positive solutions of (1.3) are eventually periodic with period .

Proof. By using repeatedly relation (1.7) -times, we obtain Now, note that from (1.8), it follows that in this case is periodic with period . On the other hand, since for each , we have that Hence, the indices , and , belong to different subsequences. From this and the periodicity of , it follows that from which the theorem follows.

By taking the logarithm of (1.3) and using the change , we get The characteristic polynomial of the homogeneous part of (2.4) is from which it follows that all its roots are expressed by These roots are simple if and only if Clearly, if is odd, inequality (2.7) holds. If is even, that is, , for some , then, since must be odd. Then, for and , we will get that inequality (2.7) does not hold.

From the above consideration and Theorem 2.1, we get the next corollary.

Corollary 2.2. Assume that and . Then all positive solutions of (1.3) are eventually periodic if and only if is odd. Moreover, if is odd, then the period is .

Since the root of characteristic polynomial (2.5) is a simple one, a particular solution of nonhomogeneous (2.4) has the form from which, by a direct calculation, we easily get that .

Hence, if is odd, the general solution of (1.3) is Note that from (2.9), it follows that and that is a positive solution of (1.3) with .

From (2.9), (2.10), and Theorem 2.1 the following results directly follow.

Theorem 2.3. Assume that , , and is odd, then every positive solution of (1.3) converges geometrically to zero. Moreover, for each , the subsequence converges monotonically to zero as .

Theorem 2.4. Assume that , , and is odd, then every positive solution of (1.3) tends to infinity. Moreover, for each , the subsequence converges increasingly to infinity as .

Finally, there are two concluding interesting remarks.

Remark 2.5. Note that, since the functions and are periodic with period and the functions and are periodic with period , from the representation (2.9) we also obtain Theorem 2.1.

Remark 2.6. The results in papers [16, 17, 39], which are obtained in much complicated ways, are particular cases of our results. Namely, in [16] Özban studied a system which is transformed into (1.3) with and , in [17] he studied a system which is transformed into (1.3) with , and , while in [39] the authors considered a system which is transformed into (1.3) with , but they only considered the case when

Acknowledgments

The authors are indebted to the anonymous referees for their advice resulting in numerous improvements of the text. The research of the first author was partly supported by the Serbian Ministry of Science, through The Mathematical Institute of SASA, Belgrade, Project no. 144013.

References

  1. M. Aloqeili, “Global stability of a rational symmetric difference equation,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 950–953, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. Balibrea, A. Linero, S. López, and S. Stević, “Global periodicity of xn+k+1=fk(xn+k)f1(xn+1),” Journal of Difference Equations and Applications, vol. 13, no. 10, pp. 901–910, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Berg, “On the asymptotics of nonlinear difference equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 21, no. 4, pp. 1061–1074, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. L. Berg and S. Stević, “Linear difference equations mod 2 with applications to nonlinear difference equations,” Journal of Difference Equations and Applications, vol. 14, no. 7, pp. 693–704, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. M. Elsayed, B. Iričanin, and S. Stević, “On the max-type equation xn+1=max{An/xn,xn1},” Ars Combinatoria, vol. 95, no. 2, pp. 187–192, 2010. View at Google Scholar
  7. E. M. Elsayed and S. Stević, “On the max-type equation xn+1=max{A/xn,xn2},” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 910–922, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2005. View at MathSciNet
  9. L. Gutnik and S. Stević, “On the behaviour of the solutions of a second-order difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 27562, 14 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. D. Iričanin and E. M. Elsayed, “On the max-type difference equation xn+1=max{A/xn,xn3},” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 675413, 13 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Iričanin and S. Stević, “On a class of third-order nonlinear difference equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479–483, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. L. Karakostas and S. Stević, “On a difference equation with min-max response,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 53–56, pp. 2915–2926, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. L. Karakostas and S. Stević, “On the recursive sequence xn+1=B+xnk/α0xn++αk1xnk+1+γ,” Journal of Difference Equations and Applications, vol. 10, no. 9, pp. 809–815, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. L. Karakostas and S. Stević, “On the recursive sequence xn+1=A+f(xn,,xnk+1)/xnk,” Communications on Applied Nonlinear Analysis, vol. 11, no. 3, pp. 87–99, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics, vol. 26, pp. 1356–1371, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Y. Özban, “On the positive solutions of the system of rational difference equations xn+1=1/ynk, yn+1=yn/xnmynmk,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 26–32, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Y. Özban, “On the system of rational difference equations xn=a/yn3, yn=byn3/xnqynq,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 833–837, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Stević, “On the recursive sequence xn+1=A/i=0kxni+1/j=k+22(k+1)xnj,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249–259, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Stević, “On the recursive sequence xn+1=αn+(xn1/xn). II,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 6, pp. 911–917, 2003. View at Google Scholar · View at MathSciNet
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Stević, “Some open problems and conjectures on difference equations,” http://www.mi.sanu.ac.rs/colloquiums/mathcoll_programs/mathcoll.apr2004.htm.
  22. S. Stević, “On the recursive sequence xn+1=α+(xn1p/xnp),” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229–234, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. Stević, “On positive solutions of a (k+1)-th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Stević, “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 56813, 20 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Stević, “Boundedness character of a max-type difference equation,” in Conference in Honour of Allan Peterson, p. 28, Novacella, Italy, July-August 2007.
  27. S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. S. Stević, “On the recursive sequence xn+1=A+(xnp/xn1r),” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. S. Stević, “Nontrivial solutions of higher-order rational difference equations,” Matematicheskie Zametki, vol. 84, no. 5, pp. 772–780, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  31. S. Stević, “On the recursive sequence xn+1=max{c,xnp/xn1p},” 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
  32. S. Stević, “Boundedness character of a class of difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 839–848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Stević, “Boundedness character of a fourth order nonlinear difference equation,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2364–2369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  34. S. Stević, “Global stability of a difference equation with maximum,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 525–529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. Stević, “On a class of higher-order difference equations,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 138–145, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  36. S. Stević, “Global stability of some symmetric difference equations,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 179–186, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  37. S. Stević and K. S. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation,” Abstract and Applied Analysis, vol. 2008, Article ID 653243, 8 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. X. Yang, Y. Liu, and S. Bai, “On the system of high order rational difference equations xn=a/ynp, yn=bynp/(xnqynq),” Applied Mathematics and Computation, vol. 171, no. 2, pp. 853–856, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet