Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 495838, 6 pages

http://dx.doi.org/10.1155/2013/495838

## Qualitative Behavior of Rational Difference Equation of Big Order

^{1}Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 4 February 2013; Accepted 20 April 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 M. M. El-Dessoky. 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 investigate the global convergence, boundedness, and periodicity of solutions of the recursive sequence , where the parameters and are positive real numbers, and the initial conditions and are positive real numbers where .

#### 1. Introduction

Recently, there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations see for example, [1–22].

The study of the nonlinear rational difference equations of a higher order is quite challenging and rewarding, and the results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations of a big order; recently, many researchers have investigated the behavior of the solution of difference equations. For example, in [8]. Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence: Elabbasy et al. [9] investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the difference equation Elabbasy et al. [10] investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation Saleh and Aloqeili [23] investigated the difference equation Wang et al. [24] studied the global attractivity of the equilibrium point and the asymptotic behavior of the solutions of the difference equation In [25], Wang et al. investigated the asymptotic behavior of equilibrium point for a family of rational difference equation Yalçinkaya [26] considered the dynamics of the difference equation Zayed and El-Moneam [27, 28] studied the behavior of the following rational recursive sequences: For some related works see [29–39].

Our goal in this paper is to investigate the global stability character and the periodicity of solutions of the recursive sequence where the parameters , , , andare positive real numbers and the initial conditions and are positive real numbers where .

#### 2. Local Stability of the Equilibrium Point of (9)

This section deals with the local stability character of the equilibrium point of (9)

Equation (9) has equilibrium points given by then Then the equilibrium points of (9) are given by Let be a continuously differentiable function defined by Therefore, it follows that

Theorem 1. *The following statements are true. *(1)*If, then the only equilibrium point of (9) is locally stable.*(2)*If, then the positive equilibrium point of (9) is locally stable if .*

*Proof. * If, then we see from (14) that
Then, the linearized equation associated with (9) about is
whose characteristic equation is
Then, (16) is asymptotically stable if , and then the equilibrium point of (9) is locally stable.

If, then we see from (14) that
Then, the linearized equation of (9) about is
whose characteristic equation is
Then, (19) is asymptotically stable if all roots of (20) lie in the open disc , that is, if
which is true if
The proof is complete.

#### 3. Boundedness of the Solutions of (9)

Here, we study the boundedness nature of the solutions of (9).

Theorem 2. *Every solution of (9) is bounded if .*

*Proof. *Let be a solution of (9). It follows from (9) that
By using a comparison, we can write the right-hand side as follows:
and this equation is locally asymptotically stable if and converges to the equilibrium point .

Therefore,
Thus, the solution is bounded.

#### 4. Existence of Periodic Solutions

In this section, we study the existence of periodic solutions of (9). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two.

Theorem 3. *Equation (9) has a prime period two solutions if and only if one of the following statements holds:*(1)*, and , —odd,*(2)*, and —odd, —even,*(3)*, and —odd, —even.*

*Proof. *We will prove the theorem when condition is true, and the proof of the other cases is similar and so we will be omit it.

First suppose that there exists a prime period two solution
of (9). We will prove that Condition holds.

We see from (9) that
Then,
Subtracting (28) from (29) gives
Since , it follows that
Again, from (28) and (29)
and so
Therefore, inequality holds.

Second, suppose that inequality is true. We will show that (9) has a prime period two solution.

Assume that
We see from inequality that
Therefore,and are distinct real numbers.

Set
We wish to show that
It follows from (9) that
Similarly, we see that
Then, it follows by induction that
Thus, (9) has the prime period two solution
where and are distinct roots of a quadratic equation, and the proof is complete.

#### 5. Global Attractor of the Equilibrium Point of (9)

In this section, we investigate the global asymptotic stability of (9). If we take the function defined by (16), then we have four cases of the monotonicity behavior in its arguments (all of these cases we suppose that ).

Theorem 4. *If the function defined by (16) is nondecreasing (or nonincreasing) in , , then the positive equilibrium point is a global attractor of (9). *

*Proof. *Let be a solution of (9) and again let be a function defined by (16).

We will prove the theorem when is nondecreasing and the proof of the other cases is similar, and so we will omit it.

Suppose that is a solution of the systems and . Then, from (9), we see that
or
Subtracting these two equations, we obtain
Under the condition , we see that
It follows by Theorem 2 that is a global attractor of (9), and then the proof is complete.

Theorem 5. *If the function defined by (16) is nondecreasing in and nonincreasing in , then the positive equilibrium point is a global attractor of (9) if .*

*Proof. *Let be a solution of (9) and again let be a function defined by (16).

Suppose that is a solution of the systems and . Then, from (9), we see that
or
Subtracting these two equations, we obtain
Under the condition , we see that

It follows by Theorem 2 that is a global attractor of (9), and then the proof is complete.

Theorem 6. *If the function defined by (16) is nondecreasing in nonincreasing in . Then the positive equilibrium point is a global attractor of (9) if . *

*Proof. *The proof is similar to the previous Theorem and so we will be omit it.

Lemma 7. *When then the equilibrium point of (9) is global attractor. *

*Proof. *If , then the proof follows by Theorem 2.

#### 6. Numerical Examples

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to (9).

*Example 1. *We assume that , , , , , , , , and . See Figure 1.

*Example 2. *See Figure 2, since , , , , , , , , , and .

*Example 3. *Figure 3 shows the solutions when, , , , , , , and . (Since ).

*Example 4. *Figure 4 shows the solutions when, , , , , , , and . (Since ).

#### Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (130-056-D1433). The author, therefore, acknowledges with thanks to DSR technical and financial support.

#### References

- R. Abu-Saris, C. Çinar, and I. Yalçinkaya, “On the asymptotic stability of ${x}_{n+1}=a+{x}_{n}{x}_{n-k}/\left({x}_{n}+{x}_{n-k}\right)$,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1172–1175, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal,
*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 1st edition, 1992. View at MathSciNet - R. P. Agarwal,
*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at MathSciNet - R. P. Agarwal and E. M. Elsayed, “Periodicity and stability of solutions of higher order rational difference equation,”
*Advanced Studies in Contemporary Mathematics*, vol. 17, no. 2, pp. 181–201, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - N. Battaloglu, C. Cinar, and I. Yalçınkaya, “The dynamics of the difference equation,”
*Ars Combinatoria*, vol. 97, pp. 281–288, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Çinar, “On the positive solutions of the difference equation ${x}_{n+1}={ax}_{n-1}/\left(1+{bx}_{n}{x}_{n-1}\right)$,”
*Applied Mathematics and Computation*, vol. 156, no. 2, pp. 587–590, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation ${x}_{n+1}={ax}_{n}-\left({bx}_{n}/{cx}_{n}-{dx}_{n-1}\right)$,”
*Advances in Difference Equations*, vol. 2006, Article ID 82579, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation ${x}_{n+1}={ax}_{n-k}/\left(\beta +\gamma {\prod}_{i=0}^{k}{x}_{n-i}\right)$,”
*Journal of Concrete and Applicable Mathematics*, vol. 5, no. 2, pp. 101–113, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of higher order difference equation,”
*Soochow Journal of Mathematics*, vol. 33, no. 4, pp. 861–873, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kang and B. Shi, “Periodic solutions for a system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 760328, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Y. Őzban, “On the system of rational difference equations ${x}_{n}=a/{y}_{n-3}$, ${y}_{n}={by}_{n-3}/{x}_{n-q}{y}_{n-q}$,”
*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 - E. Camouzis and G. Papaschinopoulos, “Global asymptotic behavior of positive solutions on the system of rational difference equations ${x}_{n+1}=1+{x}_{n}/{y}_{n-m}$, ${y}_{n+1}=1+{y}_{n}/{x}_{n-m}$,”
*Applied Mathematics Letters*, vol. 17, no. 6, pp. 733–737, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy and E. M. Elsayed, “Global asymptotic behavior attractivity and periodic nature of a difference equation,”
*World Applied Sciences Journal*, vol. 12, no. 1, pp. 39–47, 2011. View at Google Scholar - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at MathSciNet - 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, 2001. View at MathSciNet - I. Yalçinkaya, “On the global asymptotic stability of a second-order system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 860152, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. El-Metwally, “Global behavior of an economic model,”
*Chaos, Solitons & Fractals*, vol. 33, no. 3, pp. 994–1005, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, “On the global asymptotic behavior of a system of two nonlinear difference equations,”
*Ars Combinatoria*, vol. 95, pp. 151–159, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, C. Çinar, and M. Atalay, “On the solutions of systems of difference equations,”
*Advances in Difference Equations*, vol. 2008, Article ID 143943, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Diblík, B. Iricanin, S. Stevic, and Z. Šmarda, “On some symmetric systems of difference equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 246723, 7 pages, 2013. View at Publisher · View at Google Scholar - E. M. Elsayed, “Dynamics of a recursive sequence of higher order,”
*Communications on Applied Nonlinear Analysis*, vol. 16, no. 2, pp. 37–50, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Saleh and M. Aloqeili, “On the difference equation ${y}_{n+1}=A+\left({y}_{n}/{y}_{n-k}\right)$ with $A<0$,”
*Applied Mathematics and Computation*, vol. 176, no. 1, pp. 359–363, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Wang, S. Wang, Z. Wang, H. Gong, and R. Wang, “Asymptotic stability for a class of nonlinear difference equation,”
*Discrete Dynamics in Natural and Society*, vol. 2010, Article ID 791610, 10 pages, 2010. View at Publisher · View at Google Scholar - C.-y. Wang, Q.-h. Shi, and S. Wang, “Asymptotic behavior of equilibrium point for a family of rational difference equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 505906, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, “On the difference equation ${x}_{n+1}=\alpha +\left({x}_{n-m}/{x}_{n}^{k}\right)$,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 805460, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequence ,”
*Communications on Applied Nonlinear Analysis*, vol. 15, no. 2, pp. 47–57, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed and M. A. EL-Moneam, “On the rational recursive sequence ${x}_{n+1}=\alpha +\beta {x}_{n}+\gamma {x}_{n-1}/\left(A+B{x}_{n}+C{x}_{n-1}\right)$,”
*Communications on Applied Nonlinear Analysis*, vol. 12, no. 4, pp. 15–28, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elsayed and M. M. El-Dessoky, “Dynamics and behavior of a higher order rational recursive sequence,”
*Advances in Difference Equations*, pp. 2012–69, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. Simsek, B. Demir, and C. Cinar, “On the solutions of the system of difference equations ${x}_{n+1}=\text{max}\{A/{x}_{n},{y}_{n}/{x}_{n}\}$, ${y}_{n+1}=\text{max}\{A/{y}_{n},{x}_{n}/{y}_{n}\}$,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 325296, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mansour, M. M. El-Dessoky, and E. M. Elsayed, “The form of the solutions and periodicity of some systems of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 406821, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,”
*Dynamics of Continuous, Discrete & Impulsive Systems A*, vol. 13, no. 3-4, pp. 499–507, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gelisken, C. Cinar, and I. Yalcinkaya, “On a max-type difference equation,”
*Advances in Difference Equations*, vol. 2010, Article ID 584890, 6 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Wang, S. Wang, L. Li, and Q. Shi, “Asymptotic behavior of equilibrium point for a class of nonlinear difference equation,”
*Advances in Difference Equations*, vol. 2009, Article ID 214309, 8 pages, 2009. View at Publisher · View at Google Scholar - C.-Y. Wang, S. Wang, Z.-w. Wang, F. Gong, and R.-f. Wang, “Asymptotic stability for a class of nonlinear difference equations,”
*Discrete Dynamics in Nature and Society. An International Multidisciplinary Research and Review Journal*, vol. 2010, Article ID 791610, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4680–4691, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations: ${x}_{n+1}={x}_{n-1}/\left({y}_{n}{x}_{n-1}-1\right)$, ${x}_{n+1}={y}_{n-1}/\left({x}_{n}{y}_{n-1}-1\right)$ and ${z}_{n+1}={z}_{n-1}/\left({y}_{n}{z}_{n-1}-1\right)$,”
*Discrete Dynamics in Nature and Society. An International Multidisciplinary Research and Review Journal*, vol. 2011, Article ID 932362, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional systems of rational difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 178483, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “On a system of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3372–3378, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet