#### Abstract

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: , , where the parameters and the initial conditions . We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

#### 1. Introduction and Preliminaries

Our aim in this paper is to study the dynamical behavior of the following rational difference equation

where , , and the initial conditions .

When , (1.1) reduces to

In [1] (see also [2]), the authors investigated the global convergence of solutions to (1.2) and they obtained the following result.

Theorem 1.1. *Let , and be positive numbers. Then every solution of (1.2) converges to the unique equilibrium or to a prime-two solution.*

The main purpose of this paper is to further consider the global attractivity of all positive solutions of (1.1). That is to say, we will prove that the unique positive equilibrium of (1.1) is a global attractor under certain conditions (see Theorem 4.10).

For the general theory of difference equations, one can refer to the monographes [3] and [2]. For other related results on nonlinear difference equations, see, for example, [1–18].

For the sake of convenience, we firstly present some definitions and known results which will be useful in the sequel.

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

has a unique solution .

A point is called an equilibrium of (1.3) if

That is, for is a solution of (1.3), or equivalently, is a fixed point of .

An interval is called an invariant interval of (1.3) if

That is, every solution of (1.3) with initial conditions in *J* remains in *J*.

Let

denote the partial derivatives of evaluated at an equilibrium of (1.3). Then the linearized equation associated with (1.3) about the equilibrium is

and its characteristic equation is

Lemma 1.2 (see [3]). *Assume that and . Then
**
is a sufficient condition for asymptotic stability of the difference equation (1.7). Suppose in addition that one of the following two cases holds: *(i)* odd and ,*(ii)* even and .**Then (1.9) is also a necessary condition for the asymptotic stability of the difference equation (1.7).*

The following result will be useful in establishing the global attractivity character of the equilibrium of (1.1), and it is a reformulation of [2, 7].

Lemma 1.3. *Suppose that a continuous function satisfies one of (i)–(iii):*

(Note that for odd this is equivalent to (1.3) having no prime period-two solution)

Then (1.3) has a unique equilibrium in and every solution with initial values in converges to the equilibrium.

This work is organized as follows. In Section 2, the local stability and periodic character are discussed. In Section 3, the boundedness, invariant intervals of (1.1) are presented. Our main results are formulated and proved in Section 4, where the global attractivity of (1.1) is investigated.

#### 2. Local Stability and Period-Two Solutions

The unique positive equilibrium of (1.1) is

The linearized equation associated with (1.1) about is

and its characteristic equation is

From this and Lemma 1.2, we have the following result.

Theorem 2.1. *Assume that and initial conditions . Then the following stataments are true. *(i)*If
then the unique positive equilibrium of (1.1) is locally asymptotically stable;*(ii)* If
then the unique positive equilibrium of (1.1) is locally asymptotically stable. In particular, if is even, then the equilibrium is locally asymptotically stable if and only if (2.5) holds;*(iii)*If
then the unique positive equilibrium of (1.1) is locally asymptotically stable.*

In the following, we will consider the period-two solutions of (1.1).

Let

be a period-two solution of (1.1), where and are two arbitrary positive real numbers.

If is even, then , and and satisfy the following system:

then , we have , which is a contradiction.

If is odd, then , and and satisfy the following system:

then , . By calculating, (1.1) has prime period-two solution if and only if

From the above discussion, we have the following result.

Theorem 2.2. *Equation (1.1) has a positive prime period-two solution
**
if and only if
**
Furthermore, if (2.12) holds, then the prime period-two solution of (1.1) is “unique” and the values of and are the positive roots of the quadratic equation
*

#### 3. Boundedness and Invariant Intervals

In this section, we discuss the boundedness, invariant intervals of (1.1).

##### 3.1. Boundedness

Theorem 3.1. *All positive solutions of (1.1) are bounded.*

*Proof. *Equation (1.1) can be written as
for all . We denote
Then
for all . The proof is complete.

Let be a positive solution of (1.1). Then the following identities are easily established:

When , the unique positive equilibrium of (1.1) is , (3.4) becomes

When , the unique positive equilibrium is , (3.5) becomes

and (3.8) becomes

Set

Lemma 3.2. *Assume that is defined in (3.12). Then the following statements are true: *(i)*assume . Then is strictly decreasing in and increasing in for ; and it is strictly decreasing in each of its arguments for ;*(ii)*assume . Then is increasing in and strictly decreasing in for ; and it is strictly decreasing in each of its arguments for .*

*Proof. *By calculating the partial derivatives of the function , we have
from which these statements easily follow.

##### 3.2. Invariant Interval

In this subsection, we discuss the invariant interval of (1.1).

###### 3.2.1. The Case

Lemma 3.3. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)* for all ;*(ii)*If for some , , then ;*(iii)*If for some , , then ;*(iv)*If for some , , then ;*(v)*If , then (1.1) possesses an invariant interval and ;*(vi)*If , then (1.1) possesses an invariant interval and ;*(vii)*If , then (1.1) possesses an invariant interval and .*

*Proof. *The proofs of (i)–(iv) are straightforward consequences of the identities (3.5) and (3.4). So we only prove (v)–(vii). By the condition (i) of Lemma 3.2, the function is strictly decreasing in and increasing in for ; and it is strictly decreasing in both arguments for .

(v) Using the decreasing character of , we obtain

The inequalities
are equivalent to the inequality .

On the other hand, is the unique positive root of quadratic equation

Since
then we have that .

(vi) By using the monotonic character of , we obtain

The inequalities
follow from the inequality .

On the other hand, similar to (v) it can be proved that .

(vii) In this case note that holds, and using the monotonic character of , we obtain

Furthermore, similar to (v) it follows . The proof is complete.

When , (3.9) implies that the following result holds.

Lemma 3.4. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)*If for some , , then ;*(ii)*If for some , , then ;*(iii)*If for some , , then .*

###### 3.2.2. The Case

Lemma 3.5. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)* for all ;*(ii)*If for some , , then ;*(iii)*If for some , , then ;*(iv)*If for some , , then ;*(v)*If , then (1.1) possesses an invariant interval and ;*(vi)*If , then (1.1) possesses an invariant interval and ;*(vii)*If , then (1.1) possesses an invariant interval and .*

*Proof. *The proofs of (i)–(iv) are direct consequences of the identities (3.4) and (3.5). So we only give the proofs (v)–(vii). By Lemma 3.2 (ii), the function is increasing in and strictly decreasing in for ; and it is strictly decreasing in each of its arguments for .

(v) Using the decreasing character of , we obtain

The inequalities
are equivalent to the inequality . That is, is an invariant interval of (1.1).

On the other hand, similar to Lemma 3.3 (v), it can be proved that .

(vi) By using the monotonic character of , we obtain

The inequalities
are equivalent to the inequality .

On the other hand, similar to Lemma 3.3 (v) it can be proved that .

(vii) In this case note that holds. By the monotonic character of , we have

The inequalities
are equivalent to the inequality .

Furthermore, similar to Lemma 3.3 (v), it follows . The proof is complete.

Lemma 3.6. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)*If for some , , then ;*(ii)*If for some , , then ;*(iii)*If for some , , then ;*(iv)*If for some , , then ;*(v)*If for some , , then .*

*Proof. *In this case, we have that . These results follow from the identities (3.10) and (3.11) and the details are omitted.

#### 4. Global Attractivity

In this section, we discuss the global attractivity of the positive equilibrium of (1.1). We show that the unique positive equilibrium of (1.1) is a global attractor when or and or .

##### 4.1. The Case

In this subsection, we discuss the behavior of positive solutions of (1.1) when .

Theorem 4.1. *Assume that holds, and is a positive solution of (1.1). Then the unique positive equilibrium of (1.1) is a global attractor.*

*Proof. *By the change of variables
Equation (1.1) reduces to the difference equation
The unique positive equilibrium of (4.2) is
Applying Lemma 1.3 in interval , then every positive solution of (1.1) converges to . That is, is a global attractor. So, is a global attractor.

##### 4.2. The Case

In this subsection, we present global attractivity of (1.1) when .

The following result is straightforward consequence of the identity (3.7).

Lemma 4.2. *Assume that holds, and is a positive solution of (1.1). Then the following statements are true: *(i)*Suppose that . If for some , , then ;*(ii)*Suppose that . If for some , , then .*

Theorem 4.3. *Assume that , and hold. Let be a positive solution of (1.1). Then the following statements hold true: *(i)*Suppose . If , then for . Furthermore, every positive solution of (1.1) lies eventually in the interval .*(ii)*Suppose . If , then for . Furthermore, every positive solution of (1.1) lies eventually in the interval .*

*Proof. *We only give the proof of (i), the proof of (ii) is similar and will be omitted. First, note that in this case holds.

If , then by Lemma 3.3 (iv), we have that , and by Lemma 4.2 (i), we obtain that , which implies that , by induction, we have , for .

Now, to complete the proof it remains to show that when , there exists such that .

If , then we have the following two cases to be considered:

(a);(b).

Case (a). From Lemma 3.3 (ii), we see that . Thus, in the sequel, we only consider case (b).

If , then by Lemma 3.3 (iv), we have , and from Lemma 4.2 (i), we have . So and . By induction, there exists exactly one term greater than 1, which is followed by exactly one term less than , which is followed by exactly one term greater than 1, and so on. If for some , , then the former assertion implies that the result is true.

So assume for the sake of contradiction, that for all , never enter the interval , then the sequence will oscillate relative to the interval with semicycles of length one. Consider the subsequence and of solution , we have

Let
which in view of Theorem 3.1 exist as finite numbers, such that
From (4.6), we have , which implies that . Also, from (4.7), we have , which implies that . Thus and , hold, from which it follows that and exist.

Set

then , and , satisfies the system
which implies that , is a period-two solution of (1.1). Furthermore, in view of Theorem 2.2, (1.1) has no period-two solution when and hold. This is a contradiction, as desired. The proof is complete.

Theorem 4.4. *Assume that , and hold. Then the unique positive equilibrium of (1.1) is a global attractor.*

*Proof. *To complete the proof, there are four cases to be considered.

Case (i). .

By Theorem 4.3 (i), we know that all solutions of (1.1) lies eventually in the invariant interval . Furthermore, the function is non-increasing in each of its arguments in the interval . Thus, applying Lemma 1.3, every solution of (1.1) converges to , that is, is a global attractor.

Case (ii). .

In this case, the only positive equilibrium is . In view of Lemma 3.4, we see that, after the first semicycle, the nontrivial solution oscillates about with semicycles of length one. Consider the subsequences and of any nontrivial solution of (1.1). We have

or vice versa. Here, we may assume, without loss of generality, that and , for .

Let

which, in view of Theorem 3.1, exist. Then as the same argument in Theorem 4.3, we can see that and exist.

Set

then , . If , then, also as the same argument in Theorem 4.3, we can see that , is a period-two solution of (1.1), which contradicts Theorem 2.2. Thus , from which it follows that , which implies that is a global attractor.

Case (iii). .

By Theorem 4.3 (ii), we know that all solutions of (1.1) lies eventually in the invariant interval . Furthermore, the function decreases in and increases in in the interval . Thus, applying Lemma 1.3, every solution of converges to , that is, is a global attractor.

Case (iv). .

In this case, we note that holds. From Theorem 4.3 (ii) and Lemma 3.3 (i), we know that all solutions of (1.1) eventually enter the invariant interval . Hence, by using the same argument in (iii), is a global attractor. The proof is complete.

##### 4.3. The Case

In this subsection, we discuss the global behavior of (1.1) when .

The following three results are the direct consequences of equations (3.4), (3.5), (3.6), and (3.8).

Lemma 4.5. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)*If for some , , then ;*(ii)*If for some , , then ;*(iii)*If for some , , then .*

Lemma 4.6. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)*If for some , , then ;*(ii)*If for some , , then ;*(iii)*If for some , , then .*

Lemma 4.7. *Assume that , and is a positive solution of (1.1). Then the following statements are true: *(i)* for ;*(ii)*If for some , , then and ;*(iii)*If for some , , then .*

Theorem 4.8. *Assume that holds, and let be a positive solution of (1.1). Then the following statements hold true: *(i)*If , then every positive solution of (1.1) lies eventually in the interval .*(ii)*If , then every positive solution of (1.1) lies eventually in the interval .*(iii)*If , then every positive solution of (1.1) lies eventually in the interval .*

*Proof. *We only give the proof of (i), the proofs of (ii) and (iii) are similar and will be omitted.

When , recall that from Lemma 3.5, is an invariant interval and so it follows that every solution of (1.1) with consecutive values in , lies eventually in this interval. If the solution is not eventually in , there are three cases to be considered.

Case (i). If for some , , then there are two cases to be considered. If for every , then by Lemma 4.5, we have

hence, the subsequence is strictly monotonically decreasing convergent and its limit satisfies . Taking limit on both sides of (3.8), we obtain a contradiction. If for some , , then by Lemma 4.5 we obtain that is eventually in the interval .

Case (ii). If for some , , then there are two cases to be considered. If for every , then by Lemma 4.5 we obtain

which implies that the subsequence is convergent. Then as the same argument in case (i), obtain a contradiction. If for some , , then by Lemma 4.5 we have that is eventually in the interval .

Case (iii). If for some , then by Lemma 4.5 it follows that . Assume that there is a subsequence such that , or , for every . Then its limit satisfies , or . Taking limit on both sides of (3.8), obtain a contradiction. Hence, for all the subsequences are eventually in the interval .

Theorem 4.9. *Assume that and hold. Then the unique positive equilibrium of (1.1) is a global attractor.*

*Proof. *The proof will be accomplished by considering the following four cases.

Case (i). .

By part (i) of Theorem 4.8, we know that all positive solutions of (1.1) lie eventually in the invariant interval . Furthermore, the function is nonincreasing in each of its arguments in the interval . Thus, applying Lemma 1.3, every solution of (1.1) converges to , that is, is a global attractor.

Case (ii). .

In this case, the only positive equilibrium of (1.1) is . From Lemma 3.6 and (3.11), we know that each of the subsequences

of any solution of (1.1) is either identically equal to or strictly monotonically convergent and its limit is greater than zero. Set
Then, clearly,
is a period solution of (1.1) with period . By applying (3.11) to the solution (4.17) and using the fact for , we see that
and so
which implies that is a global attractor.

Case (iii). .

By Theorem 4.8 (ii), all positive solutions of (1.1) eventually enter the invariant interval . Furthermore, the function increases in and decreases in in the interval . Thus, applying Lemma 1.3 and assumption , every solution of (1.1) converges to . So, is a global attractor.

Case (iv). .

In this case, we note that holds. In view of Theorem 4.8 (iii), we obtain that all solutions of (1.1) eventually enter the invariant interval . Furthermore, the function increases in and decreases in in the interval . Then using the same argument in case (iii), every solution of (1.1) converges to . Thus the equilibrium is a global attractor. The proof is complete.

Finally, we summarize our results and obtain the following theorem, which shows that is a global attractor in three cases.

Theorem 4.10. *The unique positive equilibrium of (1.1) is a global attractor, when one of the following three cases holds: *(i)*;*(ii)* and ;*(iii)*.*