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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 930410, 11 pages
http://dx.doi.org/10.1155/2012/930410
Research Article

Global Attractivity of a Higher-Order Difference Equation

Department of Mathematics, College of Science & Arts-Rabigh, King Abdulaziz University, Rabigh 21911, Saudi Arabia

Received 7 May 2012; Accepted 8 July 2012

Academic Editor: M. De la Sen

Copyright © 2012 R. Abo-Zeid. 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

The aim of this work is to investigate the global stability, periodic nature, oscillation, and the boundedness of all admissible solutions of the difference equation where are positive real numbers and are nonnegative integers, such that .

1. Introduction and Preliminaries

Although some difference equations look very simple, it is extremely difficult to understand thoroughly the global behaviors of their solutions. One can refer to [1, 2]. The study of nonlinear rational difference equations of higher order is of paramount importance, since we still know so little about such equations. It is worthwhile to point out that although several approaches have been developed for finding the global character of difference equations [24], relatively a large number of difference equations have not been thoroughly understood yet [58].

Aloqeili in [9] discussed the stability properties and semicycle behavior of the solutions of the difference equation: with real initial conditions and positive real number .

In [10], the authors investigated the global asymptotic stability of the difference equation: where , , are nonnegative real numbers and are nonnegative integers, such that .

Also in [11], they discussed the existence of unbounded solutions under certain conditions of the difference equation: where , , are nonnegative real numbers and are nonnegative integers,

In [12], the global asymptotic stability of the difference equation: was discussed, where , , are nonnegative real numbers and , , are nonnegative integers such that and .

In [13], the global stability and periodic nature of the solutions of the difference equations: were discussed, where the initial conditions are real numbers.

In [14], we discussed the oscillation, boundedness, and the global behavior of all admissible solutions of the difference equation: where are positive real numbers.

In this paper, we study the global asymptotic stability of the difference equation where are nonnegative real numbers and are nonnegative integers, such that .

Consider the difference equation: where . An equilibrium point for (1.8) is a point such that .(1)An equilibrium point for (1.8) is called locally stable if for every such that every solution with initial conditions is such that , . Otherwise is said to be unstable. (2)The equilibrium point of (1.8) is called locally asymptotically stable if it is locally stable and there exists such that for any initial conditions , the corresponding solution tends to . (3)An equilibrium point for (1.8) is called global attractor if every solution converges to as . (4)The equilibrium point for (1.8) is called globally asymptotically stable if it is locally asymptotically stable and global attractor. The linearized equation associated with (1.8) is The characteristic equation associated with (1.9) is

Theorem 1.1 (see [2]). Assume that is a function and let be an equilibrium point of (1.8). Then the following statements are true.(1)If all roots of (1.10) lie in the open disk , then is locally asymptotically stable.(2)If at least one root of (1.10) has absolute value greater than one, then is unstable.

2. Linearized Stability Analysis

Consider the difference equation: where are nonnegative real numbers and are nonnegative integers, such that .

The change of variables reduces (1.7) to the difference equation: where .

Now we determine the equilibrium points of (2.2) and discuss their local asymptotic behavior. It is clear that the values of the equilibrium points depend on whether is even or odd.

When is odd, we have the equilibrium points and if and only if .

When is even, we have the equilibrium points and .

Now assume that .

The linearized equation associated with (2.2) about is The characteristic equation associated with this equation is We summarize the results of this section in the following two theorems.

Theorem 2.1. Assume that . Then the following statements are true. (1)The zero equilibrium point is locally asymptotically stable if and unstable (saddle point) if . (2)When is even, the equilibrium point is unstable if and unstable (saddle point) if . (3)When is odd, the equilibrium points are unstable.

Proof. The linearized equation (2.3) about is
The characteristic equation associated with this equation is So , . Therefore the result follows.
Suppose that is even. The linearized equation (2.3) about is
The associated characteristic equation (2.4) becomes
Let
We can see that has a real root in if and when , has a root in and some roots with . Therefore the result follows.
When is odd, has a root in and some roots with , if . Therefore are unstable.

Theorem 2.2. Assume that . Then the following statements are true. (1)The zero equilibrium point is locally asymptotically stable if and a source if . (2)If is even, then the equilibrium point is unstable (saddle point). (3)If is odd, then the equilibrium points are unstable (saddle points).

Proof. It is sufficient to consider the linearized equation about : and its associated characteristic equation:

3. Oscillation

Let be the largest nonnegative integer such that and let be the largest nonnegative integer such that .

Theorem 3.1. Assume that . Then the interval is an invariant interval for (2.2).

Proof. The proof is by induction. Suppose that , . Hence , .
This implies that . Then If for a certain we have , then This completes the proof.

Corollary 3.2. Assume that be a solution of (2.2) such that either . Then is positive (or negative). Moreover, converges to the zero equilibrium point.

Theorem 3.3. Let be a nontrivial solution of (2.2) such that either, , or, , is satisfied. Then oscillates about with semicycles of length one. Moreover and , and .

Proof. Assume that condition is satisfied. Then we have , and .
By induction we get , and .
If condition is satisfied, the result is similar and will be omitted.

4. Global Behavior of (2.2)

Theorem 4.1. The following statements are true.(1)If , then the zero equilibrium point is a global attractor with basin .(2)If , then (2.2) has prime period two solutions of the form , where . (3)If , then there exist solutions which are neither bounded nor persist.

Proof. Suppose that . Then using Theorem 3.1, we have that , .
Moreover, we have , .
That is, the subsequences , are decreasing.
From (2.2) we have
Now suppose that as , . Then the last inequality implies that If for a certain we have , then . This implies that
This is a contradiction as the the subsequences , are decreasing. Therefore, , , and converges to zero.
Clear!
Let be a solution of (2.2) with initial conditions, , , and , .
We consider only the case , , and , .
It follows that .
That is,
This implies that and so . Hence we have
Also implies that and so .
Hence we have
By induction we get and . Now suppose that as , .
But as then We claim that for each , .
For the sake of contradiction suppose that there exists with .
Then (4.10) gives This implies that As , , we have a contradiction.
Thus it is true that for each we have and so .
We now claim that for each , .
For the sake of contradiction, suppose that there exists with . Then This is a contradiction. Therefore for each we have and so .
The case when , and , is similar and will be omitted.

5. Numerical Examples

Example 5.1. Figure 1 shows that if , , () and , then the solution with initial conditions , , converges to zero.

930410.fig.001
Figure 1: The difference equation .

Example 5.2. Figure 2 shows that if , , () and , then the solution with initial conditions , , converges to zero.

930410.fig.002
Figure 2: The difference equation .

Example 5.3. Figure 3 shows that if , , () and , then the solution with initial conditions , is unbounded.

930410.fig.003
Figure 3: The difference equation .

Example 5.4. Figure 4 shows that if , , () and , then the solution with initial conditions , , , is unbounded.

930410.fig.004
Figure 4: The difference equation .

Acknowledgment

This paper was funded by the Deanship of the Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (15-662-D1432). The author, therefore, acknowledge with thanks DSR technical and financial support.

References

  1. R. P. Agarwal, Difference Equations and Inequalities, vol. 155, Marcel Dekker, New York, NY, USA, 1st edition, 1992. View at Zentralblatt MATH
  2. V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
  3. V. Lj. Kocić and G. Ladas, “Global attractivity in a second-order nonlinear difference equation,” Journal of Mathematical Analysis and Applications, vol. 180, no. 1, pp. 144–150, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. R. S. Kulenović, G. Ladas, and N. R. Prokup, “A rational difference equation,” Computers & Mathematics with Applications, vol. 41, no. 5-6, pp. 671–678, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2007.
  6. E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005.
  7. 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.
  8. H. Sedaghat, Nonlinear Difference Equations, Theory and Applications to Social Science Models, vol. 15, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  9. M. Aloqeili, “Dynamics of a rational difference equation,” Applied Mathematics and Computation, vol. 176, no. 2, pp. 768–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. Alaa. E. Hamza and R. Khalaf-Allah, “Global behavior of a higher order difference equation,” Journal of Mathematics and Statistics, vol. 3, no. 1, pp. 17–20, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. Alaa. E. Hamza and R. Khalaf-Allah, “On the recursive sequence xn+1=(Ai=lkxn-2i-1)/(B+Ci=lk-1xn-2i),” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1726–1731, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. A. Al-Shabi and R. Abo-Zeid, “Global asymptotic stability of a higher order difference equation,” Applied Mathematical Sciences, vol. 4, no. 17-20, pp. 839–847, 2010. View at Zentralblatt MATH
  13. R. Khalaf-Allah, “Asymptotic behavior and periodic nature of two difference equations,” Ukrainian Mathematical Journal, vol. 61, no. 6, pp. 834–838, 2009. View at Publisher · View at Google Scholar
  14. R. Abo-Zeid and C. Cinar, “global behavior of the difference equation xn+1=Axn-1/(B-Cxnxn-2),” Bulletin of Parana's Mathematical Society, vol. 31, no. 1, pp. 43–49, 2013.