Discrete Dynamics in Nature and Society

Volume 2012, Article ID 746738, 20 pages

http://dx.doi.org/10.1155/2012/746738

## Global Attractivity and Periodic Character of Difference Equation of Order Four

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

Received 14 July 2012; Revised 2 September 2012; Accepted 17 September 2012

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2012 M. A. Obaid 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 investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence , where the parameters , andare positive real numbers and the initial conditions , and are positive real numbers.

#### 1. Introduction

Our goal in this paper is to investigate the global stability character, boundedness, 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.

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 [1–15].

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 [3] Elabbasy et al. investigated the global stability, periodicity character and gave the solution of special case of the following recursive sequence: In [5] Elabbasy and Elsayed investigated the global stability character and boundedness of solutions of the recursive sequence Elsayed [11] investigated the global character of solutions of the nonlinear, fourth-order, rational difference equation Saleh and Aloqeili [16] investigated the difference equation Yang et al. [17] investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequence For some related work see [16–26].

Here, we recall some basic definitions and some theorems that we need in the sequel.

Let be some interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation has a unique solution .

*Definition 1.1 (Equilibrium Point). *A point is called an equilibrium point of (1.8) if
That is, for is a solution of (1.8), or equivalently, is a fixed point of .

*Definition 1.2 (Periodicity). *A sequence is said to be periodic with period if for all .

*Definition 1.3 (Stability). *(i) The equilibrium point of (1.8) is locally stable if for every , there exists such that for all , , ,with
we have

(ii) The equilibrium point of (1.8) is locally asymptotically stable if is locally stable solution of (1.8) and there exists , such that for all , with
we have

(iii) The equilibrium point of (1.8) is a global attractor if for all we have

(iv) The equilibrium point of (1.8) is globally asymptotically stable if is locally stable, and is also a global attractor of (1.8).

(v) The equilibrium point of (1.8) is unstable if is not locally stable.

The linearized equation of (1.8) about the equilibrium is the linear difference equation

Theorem A (see [18]). *Assume that and. Then
**
is a sufficient condition for the asymptotic stability of the difference equation
*

Theorem B (see [19]). *Let be a continuous function, where is a positive integer, and where is an interval of real numbers. Consider the difference equation
**
Suppose that satisfies the following conditions.*(1)*For each integer with ; the function is weakly monotonic in for fixed .*(2)*If is a solution of thmhje system
**then , where for each , we set
**
Then there exists exactly one equilibrium of (1.18), and every solution of (1.18) converges to .**The paper proceeds as follows. In Section 2, we show that when
**
then the equilibrium point of (1.1) is locally asymptotically stable. In Section 3 we prove that the solution is bounded and persists when and the solution of (1.1) is unbounded if . In Section 4 we prove that there exists a period two solution of (1.1). In Section 5 we prove that the equilibrium point of (1.1) is global attractor. Finally we give a numerical examples of some special cases of (1.1) and draw it by using Matlab.*

#### 2. Local Stability of the Equilibrium Point of (1.1)

This section deals with study of the local stability character of the equilibrium point of (1.1).

Equation (1.1) has equilibrium point and is given by If , then the only positive equilibrium point of (1.1) is given by Let be a continuously differentiable function defined by Therefore, it follows that Then we see that Then the linearized equation of (1.1) about is whose characteristic equation is

Theorem 2.1. *Assume that
**
Then the positive equilibrium point of (1.1) is locally asymptotically stable. *

* Proof. * It is follows by Theorem A that (2.6) is asymptotically stable if all roots of (2.7) lie in the open disc , that is, if
and so
Dividing the denominator and numerator by gives
Suppose that
We consider the following cases.

, , and . In this case we see from (2.11) that
if and only if
which is always true.

, , and It follows from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . We see from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . It follows from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . We see from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . It follows from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . We see from (2.11) that
if and only if
which is satisfied by (2.8).

, , and . It follows from (2.11) that
if and only if
which is always true. The proof is complete.

#### 3. Boundedness of Solutions of (1.1)

Here we study the boundedness nature and persistence of solutions of (1.1).

Theorem 3.1. *Every solution of (1.1) is bounded and persists if .*

* Proof. *Let be a solution of (1.1). It follows from (1.1) that
Then
By using a comparison, we see that
Thus the solution is bounded.

Now we wish to show that there exists such that
The transformation
will reduce (1.1) to the equivalent form
or
It follows that
Thus we obtain
From (3.3) and (3.9), we see that
Therefore, every solution of (1.1) is bounded and persists.

Theorem 3.2. *Every solution of (1.1) is unbounded if .*

* Proof. *Let be a solution of (1.1). Then from (1.1) we see that
We see that the right-hand side can be written as follows:
and this equation is unstable because , and . Then by using ratio test is unbounded from above.

#### 4. Existence of Periodic Solutions

In this section we study the existence of periodic solutions of (1.1). The following theorem states the necessary and sufficient conditions that this equation has periodic solutions of prime period two.

Theorem 4.1. * Equation (1.1) has positive prime period two solutions if and only if *(i)*. *

* Proof. *First suppose that there exists a prime period two solution
of (1.1). We will prove that condition (i) holds.

We see from (1.1) that
Then
Subtracting (4.3) from (4.4) gives
Since , it follows that
Again, adding (4.3) and (4.4) yields
or
It follows by (4.6), (4.8) and the relation
that
Thus
Now it is clear from (4.6) and (4.11) that and are the two distinct roots of the quadratic equation
and so
or
Therefore, inequality (i) holds.

Second, suppose that inequality (i) is true. We will show that (1.1) has a prime period two solution.

Assume that
where .

We see from inequality (i) that
which is equivalent to
Therefore, and are distinct real numbers.

Set
We wish to show that
It follows from (1.1) that
Dividing the denominator and numerator by gives
Multiplying the denominator and numerator of the right side by gives
where denotes to and denotes to .

Multiplying the denominator and numerator of the right side by , we obtain
Similarly as before one can easily show that
Then it follows by induction that
Thus (1.1) has the prime period two solution
where and are the distinct roots of the quadratic equation (4.13) and the proof is complete.

#### 5. Global Attractor of the Equilibrium Point of (1.1)

In this section, we investigate the global asymptotic stability of (1.1).

Lemma 5.1. * For any values of the quotient , andthe function defined by (2.3) has the monotonicity behavior in its three arguments. *

* Proof. *The proof follows by some computations and it will be omitted.

*Remark 5.2. *It follows from (1.1), when, that

Whenever the quotients , and are not equal, we get the following result.

Theorem 5.3. *The equilibrium point is a global attractor of (1.1) if one of the following statements holds:
*

* Proof. *Let be a solution of (1.1) and again let be a function defined by (2.3).

We will prove the theorem when case (1.1) is true, and the proof of the other cases is similar and so will be omitted.

Assume that (5.2) is true, then it is easy from the equations after (2.3) to see that the function is nondecreasing in and nonincreasing in and it is not clear what is going on with . So we consider the following two cases.

*Case 1. *Assume that the function is nondecreasing in .

Suppose that is a solution of the system and . Then from (1.1), we see that
or
Then
Subtracting this two equations, we obtain
under the conditions , we see that
It follows by Theorem B that is a global attractor of (1.1) and then the proof is complete.

*Case 2. *Assumes that the function is nonincreasing in .

Suppose that is a solution of the system and . Then from (1.1), we see that
or
Then
Subtracting these two equations we obtain
Under the conditions , we see that
It follows by Theorem B that is a global attractor of (1.1) and then the proof is complete.

#### 6. Numerical Examples

For confirming the results of this paper, we consider numerical examples which represent different types of solutions to (1.1).

*Example 6.1. *We assume , , and . See Figure 1.

*Example 6.2 (see [Figure 2]). *Since , , and .

*Example 6.3. *We consider , , , , , and . See Figure 3.

*Example 6.4 (see [Figure 4]). *Since ,.

*Example 6.5 (see [Figure 5]). *It shows the solutions when,

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