Research Article | Open Access
Global Asymptotic Stability for a Fourth-Order Rational Difference Equation
Our aim is to investigate the global behavior of the following fourth-order rational difference equation: , where and the initial values . To verify that the positive equilibrium point of the equation is globally asymptotically stable, we used the rule of the successive lengths of positive and negative semicycles of nontrivial solutions of the aforementioned equation.
There has been a great interest in studying global behaviors of rational difference equations. One can easily see that it is hard to understand thoroughly the global behaviors of solutions of rational difference equations although they have simple forms. And there has not been any general method to identify the global behaviors of rational difference equations of order greater than one and so far [1–3].
Let us consider the following fourth-order difference equation: where and the initial values in this paper. By determining the rule for the positive and negative semicycles, we assigned the global behavior of the positive equilibrium point. The unique positive equilibrium point of (1.1) is obtained by solving
Definition 1.3. A positive semicycle of a solution of (1.1) consists of a “string” of terms , all greater than or equal to the equilibrium point , with and such that A negative semicycle of a solution of (1.1) consists of a “string” of terms , all greater than or equal to the equilibrium point , with and such that And also the lengths of a semicycle is , the number of the terms contained in it. And we denote that the lengths of a positive semicycle are by and the lengths of a negative semicycle are by [1, 2, 4–6].
Definition 1.4. A solution of (1.1) is called nonoscillatory about , or simply nonoscillatory, if there exists such that either or Otherwise, the solution is called oscillatory about , or simply oscillatory [1, 2].
Lemma 1.5. A positive solution of (1.1) is eventually trivial if and only if
Proof. To prove the lemma, first we assume that and then we must show that for any .
Assume that for some , and that for . So and we obtain ; hence when we solve the equation above. It is easy to see that , , or contradicts with for .
If (1.7) holds, it is clear that the following conclusions hold:(i)if , for ,(ii)if , for ,(iii)if , for ,(iv)if , for .It is obvious that if the initial conditions do not satisfy (1.7), then the positive solution of (1.1) is eventually nontrivial.
Lemma 1.6. If is a nontrivial positive solution of (1.1), then the following conclusions are satisfied:(i)(ii)(iii)(iv)
Proof. (i) The proof of the inequality (i) is obtained by subtracting 1 from (1.1)
The dominator of this fraction is positive so also.
(ii) If we subtract from (1.1), we obtain The expression is positive, and so we get The proofs for inequalities (iii) and (iv) are similar to the one for (ii).
2. Main Results and Their Proofs
The trajectory of (1.1) and global asymptotic stability of the positive solution are considered in this part of the paper.
Proof. Assume that is a strictly oscillatory solution of (1.1), then the initial values must satisfy one of the following four cases: (i), , , (ii), , , (iii), , , (iv), , , If (i) occurs, it follows from Lemma 1.6(i) that
It means that the rule for the lengths of positive and negative semicycles of the solution of (1.1) occurs successively as
If (ii) happens, the positive and negative semicycles are The regulation for the lengths of positive and negative semicycles which occur successively is
The other cases can be shown similarly.
Theorem 2.2. The positive equilibrium point of (1.1) is globally asymptotically stable.
Proof. Let us show that the positive equilibrium point of (1.1) is locally asymptotically stable and also globally attractive, then it is globally asymptotically stable. The linearized equation of (1.1) about the positive equilibrium is
where and .
And we obtain thereby is locally asymptotically stable.
Now we must show that . The proof is as follows.(1)If initial values of (1.1) satisfy (1.7), then according to Lemma 1.5, so . (2)If the initial values of (1.1) do not satisfy (1.7), then for any solution of (1.1), for . (i)If the solution is nonoscillatory about the positive equilibrium point of (1.1), then is monotonic and bounded because of Lemma 1.6. So, the limit exists and is finite. If we take limits on both sides of (1.1), then we obtain and thereby . So .(ii)If the solution is strictly oscillatory, then trajectory structure of nontrivial solutions of (1.1) is or or or First, we investigate the case where the rule of the trajectory structure is in a period. We denote positive semicycles by and negative semicycles by . The rule for the positive and negative semicycles can be “periodically’’ expressed as follows: By using Lemma 1.6 we obtain(i); (ii); This relations give rise to It is easy to see that is decreasing with its lower bound 1 because of the inequality . Hence the limit exists and is finite. The limit is then Similarly is increasing and its upper bound 1 because of the inequality . So, the limit exists and is finite. And we derive .
Now, we must show that . To do this, let us take and taking the limit on both sides of the above equality, we obtain By solving this equation we have and then . Similarly to obtain by using we are taking And taking the limit on both sides of the above equality we get By solving this equation we have and . Hence, we derive , so It can be shown that for the other rules of the trajectory structures with the same manner. Therefore, the positive equilibrium point is globally asymptotically stable.
- E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, vol. 5 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008.
- 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.
- C. Çınar, S. Stević, and İ. Yalçınkaya, “A note on global asymptotic stability of a family of rational equations,” Rostocker Mathematisches Kolloquium, no. 59, pp. 41–49, 2005.
- X. Li, “Qualitative properties for a fourth-order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 1, pp. 103–111, 2005.
- X. Li, “Global behavior for a fourth-order rational difference equation,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 555–563, 2005.
- D. Li, P. Li, and X. Li, “Dynamical properties for a class of fourth-order nonlinear difference equations,” Advances in Difference Equations, vol. 2008, Article ID 678402, 13 pages, 2008.
Copyright © 2009 Meseret Tuba Gülpinar and Mustafa Bayram. 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.