About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 969813, 11 pages
http://dx.doi.org/10.1155/2012/969813
Research Article

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

1School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia
2Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi Arabia

Received 1 September 2012; Accepted 11 October 2012

Academic Editor: Mustafa Kulenovic

Copyright © 2012 S. Atawna et al. 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 consider the second order rational difference equation   n = 0,1,2,…, where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).

1. Introduction

Difference equations proved to be effective in modelling and analysing discrete dynamical systems that arise in signal processing, populations dynamics, health sciences, economics, and so forth. They also arise naturally in studying iterative numerical schemes. Furthermore, they appear when solving differential equations using series solution methods or studying them qualitatively using, for example, Poincaré maps. For an introduction to the general theory of difference equations, we refer the readers to Agarwal [1], Elaydi [2], and Kelley and Peterson [3].

Rational difference equations; particularly bilinear ones, that is, attracted the attention of many researchers recently. For example, see the articles [424], monographs Kocić and Ladas [25], Kulenović and Ladas [26], and Camouzis and Ladas [27], and the references cited therein. We believe that behavior of solutions of rational difference equations provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one [26, page 1].

Our aim in this paper is to study the second order bilinear rational difference equation where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. Our concentration is on the periodic character of the positive solutions of (1.2). Indeed, our interest in (1.2) was stimulated by the following conjecture proposed by Camouzis and Ladas in [27, Conjecture  5.201.2].

Conjecture 1.1. Show that the period-two solution of (1.2) is locally asymptotically stable.

It is worth mentioning that the aforementioned conjecture appeared previously in Conjecture in the Kulenović and Ladas monograph [26].

To this end and using transformations similar to the ones used by [22, 27], let then (1.2) reduces to where are positive real numbers, and the initial conditions are nonnegative real numbers.

That being said, the remainder of this paper is organized as follows. In the next section, we present some definitions and results that are needed in the sections to follow. Next, using elementary mathematics and nontrivial combinations of ideas, we establish our main results in Section 3. Our main results provide positive confirmation of Conjecture 1.1. Finally, we conclude in Section 4 with suggestions for future research.

2. Preliminaries

For the sake of self-containment and convenience, we recall the following definitions and results from [26].

Let be a nondegenerate interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation has a unique solution .

A constant sequence, for all where , is called an equilibrium solution of (2.1) if

Definition 2.1. Let be an equilibrium solution of (2.1). (i) is called locally stable if for every , there exists such that for all , with , we have (ii) is called locally asymptotically stable if it is locally stable, and if there exists , such that for all , with , we have (iii) is called a global attractor if for every , we have (iv) is called globally asymptotically stable if it is locally stable and a global attractor. (v) is called unstable if it is not stable. (vi) is called a source, or a repeller, if there exists such that for all , with , there exists such that Clearly a source is an unstable equilibrium.

Definition 2.2. Let denote the partial derivatives of evaluated at the equilibrium of (2.1). Then the equation is called the linearized equation associated with (2.1) about the equilibrium solution .

Definition 2.3. A solution of (2.1) is said to be periodic with period if
A solution of (2.1) is said to be periodic with prime period , or a -cycle if it is periodic with period and is the least positive integer for which (2.9) holds.

Theorem 2.4 (Linearized stability). (a) If both roots of the quadratic equation lie in the open unit disk , then the equilibrium of (2.1) is locally asymptotically stable. (b) If at least one of the roots of (2.10) has absolute value greater than one, then the equilibrium of (2.1) is unstable. (c) A necessary and sufficient condition for both roots of (2.10) to lie in the open unit disk is In this case, the locally asymptotically stable equilibrium is also called a sink. (d) A necessary and sufficient condition for both roots of (2.10) to have absolute value greater than one is In this case, is a repeller. (e) A necessary and sufficient condition for one root of (2.10) to have absolute value greater than one and for the other to have absolute value less than one is In this case, the unstable equilibrium is called a saddle point. (f) A necessary and sufficient condition for a root of (2.10) to have absolute value equal to one is or In this case, the equilibrium is called a nonhyperbolic point.

3. Main Result

In this section, we give a necessary and sufficient condition for (1.4) to have a prime period-two solution. We show that the period-two solution of (1.4) is locally asymptotically stable.

Equation (1.4) has a unique positive equilibrium given by Furthermore, the linearized equation associated with (1.4) about the equilibrium solution is given by Therefore, its characteristic equation is

Lemma 3.1. (a) When Equation (1.4) has no nonnegative prime period-two solution. (b) When Equation (1.4) has prime period-two solution, if and only if where and are the positive and distinct solutions of the quadratic equation

Proof. (a) Suppose and assume, for the sake of contradiction, there exist distinct positive real numbers and such that is a prime period-two solution of (1.4). Then, and satisfy Furthermore, Subtracting (3.11) from (3.12), we have But, implies that which contradicts the hypothesis that and are distinct positive real numbers.(b) Now, suppose , and let and be two positive real numbers such that is a prime period-two solution of (1.4). Thus, and satisfy Moreover, Subtracting (3.17) from (3.18), we have Furthermore, adding (3.17) to (3.18), we have Hence, and satisfy the quadratic equation In other words, and are given by

Theorem 3.2. Suppose (1.4) has a prime period-two solution. Then, the period-two solution is locally asymptotically stable.

Proof. To start off, we first vectorize (1.4) by introducing the following change of variables: and write (1.4) in the following equivalent form: where
Now and generate a period-two solution of (1.4) if and only if is a fixed point of , the second iterate of . Furthermore, where
The prime period-two solution of (1.4) is asymptotically stable if the eigenvalues of the Jacobian matrix , evaluated at lie inside the unit disk. But, and, hence, its characteristic equation is where By Theorem 2.4(c), the eigenvalues of lie inside the unit disk if and only if
To this end, without loss of generality, assume that . Then, by (3.15), Hence, Similarly, we observe that Furthermore, since , (3.19) implies the sum of and is less than and, a fortiori, each is less than 1. Indeed, we have
With that in mind, it is clear that In addition, with the understanding that we have The second inequality follows immediately since and However, the proof of the first inequality is a bit tricky.
By (3.19), Therefore, if and only if which is true if and only if The latter inequality holds true because and This completes the proof.

4. Conclusion

Without dispute, the existence of a periodic solution or an equilibrium solution does not imply its local stability. As such, it is natural to ask about the class of difference equations which afford the ELAS property, that is, the existence of a periodic solution implies its local asymptotic stability. In particular, one may want to investigate the class of bilinear difference equations (1.1) and completely characterize those equations enjoying the ELAS property.

References

  1. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at Zentralblatt MATH
  2. S. N. Elaydi, An introduction to Difference Equations, Springer, New York, NY, USA, 2nd edition, 1999.
  3. W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Harcour Academic, New York, NY, USA, 2nd edition, 2001.
  4. A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence xn+1=α+xn-1/xn,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790–798, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Basu, “Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist,” Journal of Difference Equations and Applications. In press.
  6. S. Basu and O. Merino, “Global behavior of solutions to two classes of second-order rational difference equations,” Advances in Difference Equations, vol. 2009, Article ID 128602, 27 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Brett and M. R. S. Kulenović, “Global asymptotic behavior of yn+1=(pyn+yn-1)/(r+qyn+yn-1),” Advances in Difference Equations, Article ID 41541, 22 pages, 2007. View at Zentralblatt MATH
  8. K. Cunningham, M. R. S. Kulenović, G. Ladas, and S. V. Valicenti, “On the recursive sequence xn+1=(α+βxn1)/(Bxn+Cxn1),” Nonlinear Analysis: Theory, Methods & Applications A, vol. 47, no. 7, pp. 4603–4614, 2001. View at Publisher · View at Google Scholar
  9. M. Dehghan, M. J. Douraki, and M. J. Douraki, “Dynamics of a rational difference equation using both theoretical and computational approaches,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 756–775, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. C. H. Gibbons, M. R. S. Kulenovic, and G. Ladas, “On the recursive sequence xn+1=(α+βxn-1)/(γ+xn),” Mathematical Sciences Research Hot-Line, vol. 4, no. 2, pp. 1–11, 2000. View at Zentralblatt MATH
  11. C. H. Gibbons, M. R. S. Kulenović, G. Ladas, and H. D. Voulov, “On the trichotomy character of xn+1=(α+βxn+γxn-1)/(A+xn),” Journal of Difference Equations and Applications, vol. 8, no. 1, pp. 75–92, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. L.-X. Hu and W.-T. Li, “Global stability of a rational difference equation,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1322–1327, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. L.-X. Hu, W.-T. Li, and S. Stević, “Global asymptotic stability of a second order rational difference equation,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 779–797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. L.-X. Hu, W.-T. Li, and H.-W. Xu, “Global asymptotical stability of a second order rational difference equation,” Computers & Mathematics with Applications, vol. 54, no. 9-10, pp. 1260–1266, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. S. Huang and P. M. Knopf, “Boundedness of positive solutions of second-order rational difference equations,” Journal of Difference Equations and Applications, vol. 10, no. 11, pp. 935–940, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. Karatas and A. Gelişken, “Qualitative behavior of a rational difference equation,” Ars Combinatoria, vol. 100, pp. 321–326, 2011.
  17. W. A. Kosmala, M. R. S. Kulenović, G. Ladas, and C. T. Teixeira, “On the recursive sequence yn+1=(p+yn-1)/(qyn+yn-1),” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 571–586, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. R. S. Kulenovic and G. Ladas, “Open problems and conjectures: On period two solutions of xn+1=(α+βxn+γxn1)/(a+bxn+cxn1),” Journal of Difference Equations and Applications, vol. 6, no. 5, pp. 641–646, 2000.
  19. M. R. S. Kulenović, G. Ladas, and N. R. Prokup, “On the recursive sequence xn+1=(αxn+βxn-1)/(1+xn),” Journal of Difference Equations and Applications, vol. 6, no. 5, pp. 563–576, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. R. S. Kulenović, G. Ladas, and W. S. Sizer, “On the recursive sequence xn+1=(αxn+βxn-1)/(γxn+cxn-1),” Mathematical Sciences Research Hot-Line, vol. 2, no. 5, pp. 1–16, 1998. View at Zentralblatt MATH
  21. G. Ladas, “Open problems on the boundedness of some difference equations,” Journal of Difference Equations and Applications, vol. 1, no. 3, pp. 317–321, 1995.
  22. M. Saleh and S. Abu-Baha, “Dynamics of a higher order rational difference equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 84–102, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequence xn+1=axn-bxn/(cxn-dxn-k),” Communications on Applied Nonlinear Analysis, vol. 15, no. 2, pp. 47–57, 2008. View at Zentralblatt MATH
  24. E. M. E. Zayed, A. B. Shamardan, and T. A. Nofal, “On the rational recursive sequence xn+1=(α-βxn)/(γ-δxn-xn-k),” International Journal of Mathematics and Mathematical Sciences, Article ID 391265, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256, Kluwer Academic, Dordrecht, The Netherlands, 1993.
  26. M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problem and Conjectures, Chapman & Hall, Boca Raton, Fla, USA, 2002.
  27. E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, vol. 5, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008.