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On Boundedness of Solutions of the Difference Equation for
Advances in Difference Equations volume 2009, Article number: 463169 (2009)
Abstract
We study the boundedness of the difference equation , where and the initial values . We show that the solution of this equation converges to if or for all ; otherwise is unbounded. Besides, we obtain the set of all initial values such that the positive solutions of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenović and Ladas (2002).
1. Introduction
In this paper, we study the following difference equation:
where with and the initial values .
The global behavior of (1.1) for the case is certainly folklore. It can be found, for example, in [1] (see also a precise result in [2,]).
The global stability of (1.1) for the case follows from the main result in [3] (see also Lemma 1 in Stević's paper [4]). Some generalizations of Copson's result can be found, for example, in papers [5–8]. Some more sophisticated results, such as finding the asymptotic behavior of solutions of (1.1) for the case (even when ) can be found, for example, in papers [4](see also [8–11]). Some other properties of (1.1) have been also treated in [4].
The case was treated for the first time by Stević's in paper [12]. The main trick from [12] has been later used with a success for many times; see, for example, [13–15].
Some existing results for (1.1) are summarized as follows[16].
Theorem 1 A.
If , then the zero equilibrium of (1.1) is globally asymptotically stable.
If , then the equilibrium of (1.1) is globally asymptotically stable.
If , then every positive solution of (1.1) converges to the positive equilibrium .
If , then every positive solution of (1.1) converges to a periodtwo solution.
If , then (1.1) has unbounded solutions.
In [16], Kulenović and Ladas proposed the following open problem.
Open problem B (see Open problem 6.10.12of [16])
Assume that .

(a)
Find the set of all initial conditions such that the solutions of (1.1) are bounded.

(b)
Let . Investigate the asymptotic behavior of .
In this paper, we will obtain the following results: let with , and let be a positive solution of (1.1) with the initial values . If for all (or for all ), then converges to . Otherwise is unbounded.
2. Some Definitions and Lemmas
In this section, let and be the positive equilibrium of (1.1). Write and define by, for all ,
It is easy to see that if is a solution of (1.1), then for any . Let
Then . The proof of Lemma 2.1 is quite similar to that of Lemma 1 in [35] and hence is omitted.
Lemma 2.1.
The following statements are true.

(1)
is a homeomorphism.

(2)
and .

(3)
and .

(4)
and .

(5)
and .
Lemma 2.2.
Let , and let be a positive solution of (1.1).

(1)
If and , then .

(2)
If and , then .
Proof.
We show only (1) because the proof of (2) follows from (1) by using the change and the fact that (1) is autonomous. Since and , by (1.1) we have
Also it follows from (1.1) that
from which we have and . This completes the proof.
Lemma 2.3.
Let , and let be a positive solution of (1.1) with the initial values . If there exists some such that , then .
Proof.
Since , it follows from Lemma 2.1 that for any . Without loss of generality we may assume that , that is, . Now we show Suppose for the sake of contradiction that , then
By (2.5) we have
and by (2.6) we get
Claim 1.
If , then
Proof of Claim 1
Let , then we have
Since , it follows
This completes the proof of Claim 1.
By (2.8), we have
or
Claim 2.
We have
Proof of Claim 2
Since
we have
The proof of (2.14) is completed.
Now we show (2.15). Let
Note that ; it follows that if , then
which implies that is decreasing for . Since and
it follows that
Thus
This implies that
Finally we have
The proof of (2.15) is completed.
Note that since . By (2.12), (2.13), (2.14), and (2.15), we see which contradicts to (2.7). The proof of Lemma 2.3 is completed.
3. Main Results
In this section, we investigate the boundedness of solutions of (1.1). Let , and let be a positive solution of (1.1) with the initial values , then we see that for some or for all or for all .
Theorem 3.1.
Let , and let be a positive solution of (1.1) such that for all or for all , then converges to .
Proof.
Case 1.
for any . If for some , then
If for some , then
which implies that and
Thus for any . In similar fashion, we can show for any . Let and , then
which implies .
Case 2.
for any . Since is decreasing in , it follows that for any
In similar fashion, we can show that . This completes the proof.
Lemma 3.2 (see [20, Theorem 5]).
Let be a set, and let be a function which decreases in and increases in , then for every positive solution of equation , and do exactly one of the following.

(1)
They are both monotonically increasing.

(2)
They are both monotonically decreasing.

(3)
Eventually, one of them is monotonically increasing, and the other is monotonically decreasing.
Remark 3.3.
Using arguments similar to ones in the proof of Lemma 3.2, Stevi proved Theorem 2 in [25]. Beside this, this trick have been used by Stević in [18, 28, 29].
Theorem 3.4.
Let , and let be a positive solution of (1.1) such that for some , then is unbounded.
Proof.
We may assume without loss of generality that and (the proof for is similar). From Lemma 2.1 we see for all .If is eventually increasing, then it follows from Lemma 2.3 that is eventually increasing. Thus and , it follows from Lemma 2.2 that .
If is not eventually increasing, then there exists some such that
from which we obtain , since and .
Since is increasing in and is decreasing in , we have that for any . It follows from Lemma 3.2 that is eventually decreasing. Thus and . It follows from Lemma 2.2 that . This completes the proof.
By Theorems 3.1 and 3.4 we have the following.
Corollary 3.5.
Let , and let be a positive bounded solution of (1.1), then for all or for all .
Now one can find out the set of all initial values such that the positive solutions of (1.1) are bounded. Let For any let
It follows from Lemma 2.1 that , which implies
for any .
Let be the set of all initial values such that the positive solutions of (1.1) are bounded. Then we have the following theorem.
Theorem 3.6.
.
Proof.
Let be a positive solution of (1.1) with the initial values .
If , then for any , which implies for any . It follows from Theorem 3.1 that .
If , then , which implies for any . It follows from Theorem 3.1 that .
Now assume that is a positive solution of (1.1) with the initial values .
If , then it follows from Lemma 2.1 that , which along with Theorem 3.4 implies that is unbounded.
If , then there exists such that . Thus . By Lemma 2.1, we obtain and , which along with Theorem 3.4 implies that is unbounded.
If , then there exists such that and . Again by Lemma 2.1 and Theorem 3.4, we have that is unbounded. This completes the proof.
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Project Supported by NNSF of China (10861002) and NSF of Guangxi (0640205, 0728002).
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Xi, H., Sun, T., Yu, W. et al. On Boundedness of Solutions of the Difference Equation for . Adv Differ Equ 2009, 463169 (2009). https://doi.org/10.1155/2009/463169
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Keywords
 Differential Equation
 Partial Differential Equation
 Ordinary Differential Equation
 Functional Analysis
 Asymptotic Behavior