Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 746738, 20 pages

doi:10.1155/2012/746738

Research Article

Global Attractivity and Periodic Character of Difference Equation of Order Four

M. A. Obaid,¹ E. M. Elsayed,1,2 and M. M. El-Dessoky1,2

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

²Department 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.

Figure 1

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

Figure 2

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

Figure 3

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

Figure 4

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

Figure 5

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