Discrete Dynamics in Nature and Society

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

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

Research Article

Qualitative Behavior of Rational Difference Equation of Big Order

M. M. El-Dessoky1,2

¹Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Mathematics 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.

Figure 1: It shows the solution of (9) with , , , , , , , , and .

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

Figure 2: It shows the behavior of the solution of (9) with , , , , , , , , , and .

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

Figure 3: It shows the periodicity of the solution of (9) when , , , , , , , and .

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

Figure 4: It shows the periodicity of the solution of (9) when , , , , , , , and .

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.

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