Abstract and Applied Analysis

Volume 2010 (2010), Article ID 237129, 6 pages

http://dx.doi.org/10.1155/2010/237129

## Global Behavior of the Difference Equation

^{1}College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China^{2}Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003, China

Received 31 March 2010; Revised 17 April 2010; Accepted 30 April 2010

Academic Editor: Stevo Stević

Copyright © 2010 Taixiang Sun 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 study the following difference equation , where and the initial conditions . We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.

#### 1. Introduction

Kulenović and Ladas in [1] studied the following difference equation: where and the initial conditions , and they obtained the following theorems.

Theorem A (see [1, Theorem ]). *Equation (1.1) has a prime period-two solution
**
if and only if . Furthermore, when , the prime period-two solution is unique and the values of and are the positive roots of the quadratic equation
*

Theorem B (see [1, Theorem ]). *Let be a solution of (1.1). Let be the closed interval with end points 1 and and let and be the intervals which are disjoint from and such that
**
Then either all the even terms of the solution lie in and all odd terms lie in , or vice-versa, or for some ,
**
when (E1) holds, except for the length of the first semicycle of the solution, if , the length is one; if , the length is at most two.*

Theorem C (see [1, Theorem ]). *(a) Assume . Then the equilibrium of (1.1) is global attractor.**(b) Assume . Then every solution of (1.1) eventually enters and remains in the interval .*

In [1], they proposed the following conjecture.

Conjecture 1 (see [1, Conjecture ]). *Assume that . Show that every positive solution of (1.1) either converges to a finite limit or to a two cycle.**Gibbons et al. in [2] trigged off the investigation of the second-order difference equations such that the function is increasing in and decreasing in . Motivated by [2], Berg [3] and Stević [4] obtained some important results on the existence of monotone solutions of such equations which was later considerably developed in a series of papers [5–14] (for related papers see also [15–19]). The monotonous character of solutions of the equations was explained by Stević in [20]. For some other papers in the area, see also [1, 17–19, 21–26] and the references cited therein. In this paper, we shall confirm that the Conjecture 1 is true. The main idea used in this paper can be found in papers [24, 26].*

#### 2. Global behavior of (1.1)

Theorem 2.1. *Let be a nonoscillatory solution of (1.1); then converges to the unique positive equilibrium of (1.1).*

*Proof. *Since is a nonoscillatory solution of (1.1), we may assume without loss of generality that there exists such that for any . We claim for any . Indeed, if for some , then
which implies ; this is a contradiction. Let ; then and . The proof is complete.

In the sequel, let and the unique prime period-two solution of (1.1) with . Define by for any and by for any . Then

Lemma 2.2. *Let , then the following statements are true.*

(i)* if and only if .*(ii)* if and only if .*(iii)*If , then and . If , then and .*

*Proof. *(i) Since is decreasing in and , if and only if .

(ii) Since is a decreasing function for , if and only if .

(iii) Since
it follows that
By (i), we obtain if and if . The proof is complete.

Lemma 2.3. *Let and is a positive solution of (1.1); then and do exactly one of the following.*

(i)*Eventually, they are both monotonically increasing.*(ii)*Eventually, they are both monotonically decreasing.*(iii)*Eventually, one of them is monotonically increasing and the other is monotonically decreasing.*

*Proof. *See [20] (also see [27]).

*Remark 2.4. *Stević in [20] noticed the relationship between the monotonicity of the subsequences and of solution of a second-order difference equation and the monotonicity of the function in variables and . A simple observation shows that Stević's proof works in the general case if the function is replaced by . The result was later used for many times by Stević and his collaborators (see, e.g., [21, 23–26]).

Lemma 2.5. *Let . Assume that there exists some such that ; then .*

*Proof. *Since , it follows that . By Lemma 2.2(ii), we get , which with Lemma 2.2(iii) implies . Since is increasing in () and , it follows that
By Lemma 2.2(iii), we have as . Thus . The proof is complete.

Theorem 2.6. *Let and be an oscillatory solution of (1.1); then converges to the unique prime period-two solution of (1.1).*

*Proof. *It follows from Theorem C(b) that there exists such that for any ,
and and . We assume without loss of generality that
and and . Since
is decreasing in and increasing in , it follows that and for any .

If is eventually increasing or is eventually decreasing, then it follows from Theorem A that and .

If is eventually decreasing and is eventually increasing, we may assume without loss of generality that for any . It follows from Lemma 2.5 that for any . By Theorem A, we obtain and . The proof is complete.

We confirm from Theorems 2.1, 2.6, and C(a) that the Conjecture 1 is true.

#### Acknowledgment

The project is supported by NNSF of China(10861002) and NSF of Guangxi (2010GXNSFA013106) and SF of Education Department of Guangxi (200911MS212).

#### References

- M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. View at MathSciNet - C. H. Gibbons, M. R. S. Kulenovic, and G. Ladas, “On the recursive sequence ${x}_{n+1}=(\alpha +\beta {x}_{n-1})/(\gamma +{x}_{n})$,”
*Mathematical Sciences Research Hot-Line*, vol. 4, no. 2, pp. 1–11, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - S. Stević, “On the recursive sequence ${x}_{n+1}={x}_{n-1}/g({x}_{n})$,”
*Taiwanese Journal of Mathematics*, vol. 6, no. 3, pp. 405–414, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Berg, “Inclusion theorems for non-linear difference equations with applications,”
*Journal of Difference Equations and Applications*, vol. 10, no. 4, pp. 399–408, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - L. Berg, “On the asymptotics of the difference equation ${x}_{n-3}={x}_{n}(1+{x}_{n}{{}_{-}}_{1}{x}_{n-2})$,”
*Journal of Difference Equations and Applications*, vol. 14, no. 1, pp. 105–108, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic behavior of a class of nonlinear difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 47156, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - S. Stević, “On monotone solutions of some classes of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 53890, 9 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic periodicity of a higher-order difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 13737, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - 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 - B. Iričanin and S. Stević, “Eventually constant solutions of a rational difference equation,”
*Applied Mathematics and Computation*, vol. 215, no. 2, pp. 854–856, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. M. Kent, “Convergence of solutions in a nonhyperbolic case,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 7, pp. 4651–4665, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, “On non-oscillatory solutions of the recursive sequence ${x}_{n+1}=p+{x}_{n-k}/{x}_{n}$,”
*Journal of Difference Equations and Applications*, vol. 11, no. 6, pp. 483–485, 2005. View at Google Scholar - T. Sun and H. Xi, “On the solutions of a class of difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 766–770, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun and H. Xi, “Existence of monotone solutions of a difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 917560, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+{x}_{n-1}/{x}_{n}$ II,”
*Dynamics of Continuous, Discrete & Impulsive Systems A*, vol. 10, no. 6, pp. 911–916, 2003. View at Google Scholar · View at MathSciNet - K. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation ${x}_{n}=A+{({x}_{n-2}/{x}_{n-1})}^{p}$,”
*Journal of Difference Equations and Applications*, vol. 12, no. 9, pp. 909–918, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+{x}_{n-1}/{x}_{n}$,”
*International Journal of Mathematical Sciences*, vol. 2, no. 2, pp. 237–243, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}^{p}/{x}_{n}^{p}$,”
*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 - S. Stević, “On the difference equation ${x}_{n+1}={\alpha}_{n}+{x}_{n-1}/{x}_{n}$,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1159–1171, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - S. Stević and K. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation ${x}_{n}=f({x}_{n-2})/g({x}_{n-1})$,”
*Journal of Difference Equations and Applications*, vol. 12, no. 9, pp. 909–918, 2006. View at Publisher · View at Google Scholar - E. Camouzis and G. Ladas, “When does local asymptotic stability imply global attractivity in rational equations?”
*Journal of Difference Equations and Applications*, vol. 12, no. 8, pp. 863–885, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet