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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.
References
 1.
Tasković MR: Nonlinear Functional Analysis. Vol. I: Fundamental Elements of Theory. Zavod, za udžbenike i nastavna sredstva, Beograd, Serbia; 1993.
 2.
Stević S: Behavior of the positive solutions of the generalized BeddingtonHolt equation. PanAmerican Mathematical Journal 2000,10(4):7785.
 3.
Copson ET: On a generalisation of monotonic sequences. Proceedings of the Edinburgh Mathematical Society. Series 2 1970, 17: 159164. 10.1017/S0013091500009433
 4.
Stević S: Asymptotic behavior of a sequence defined by iteration with applications. Colloquium Mathematicum 2002,93(2):267276. 10.4064/cm9326
 5.
Stević S: A note on bounded sequences satisfying linear inequalities. Indian Journal of Mathematics 2001,43(2):223230.
 6.
Stević S: A generalization of the Copson's theorem concerning sequences which satisfy a linear inequality. Indian Journal of Mathematics 2001,43(3):277282.
 7.
Stević S: A global convergence result. Indian Journal of Mathematics 2002,44(3):361368.
 8.
Stević S:A note on the recursive sequence . Ukrainian Mathematical Journal 2003,55(4):691697.
 9.
Stević S:On the recursive sequence . Taiwanese Journal of Mathematics 2002,6(3):405414.
 10.
Stević S: Asymptotics of some classes of higherorder difference equations. Discrete Dynamics in Nature and Society 2007, 2007:20.
 11.
Stević S: Existence of nontrivial solutions of a rational difference equation. Applied Mathematics Letters 2007,20(1):2831. 10.1016/j.aml.2006.03.002
 12.
Stević S:On the recursive sequence . Applied Mathematics Letters 2002, 15: 305308. 10.1016/S08939659(01)001355
 13.
Berenhaut KS, Dice JE, Foley JD, Iričanin B, Stević S:Periodic solutions of the rational difference equation . Journal of Difference Equations and Applications 2006,12(2):183189. 10.1080/10236190500539295
 14.
Stević S: Periodic character of a class of difference equation. Journal of Difference Equations and Applications 2004,10(6):615619. 10.1080/10236190410001682103
 15.
Stević S:On the difference equation . Dynamics of Continuous, Discrete & Impulsive Systems 2007,14(3):459463.
 16.
Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures. Chapman & Hall/CRC Press, Boca Raton, Fla, USA; 2002:xii+218.
 17.
Amleh AM, Grove EA, Ladas G, Georgiou DA:On the recursive sequence . Journal of Mathematical Analysis and Applications 1999,233(2):790798. 10.1006/jmaa.1999.6346
 18.
Berenhaut KS, Stević S:The behaviour of the positive solutions of the difference equation . Journal of Difference Equations and Applications 2006,12(9):909918. 10.1080/10236190600836377
 19.
Camouzis E, Ladas G: When does periodicity destroy boundedness in rational equations? Journal of Difference Equations and Applications 2006,12(9):961979. 10.1080/10236190600822369
 20.
Camouzis E, Ladas G: When does local asymptotic stability imply global attractivity in rational equations? Journal of Difference Equations and Applications 2006,12(8):863885. 10.1080/10236190600772663
 21.
Devault R, Kocic VL, Stutson D:Global behavior of solutions of the nonlinear difference equation . Journal of Difference Equations and Applications 2005,11(8):707719. 10.1080/10236190500137405
 22.
Feuer J:On the behavior of solutions of . Applicable Analysis 2004,83(6):599606. 10.1080/00036810410001657260
 23.
Kulenović MRS, Ladas G, Prokup NR:On the recursive sequence . Journal of Difference Equations and Applications 2000,6(5):563576. 10.1080/10236190008808246
 24.
Stević S: Asymptotic behavior of a nonlinear difference equation. Indian Journal of Pure and Applied Mathematics 2003,34(12):16811687.
 25.
Stević S:On the recursive sequence . II. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2003,10(6):911916.
 26.
Stević S:On the recursive sequence . Journal of Applied Mathematics & Computing 2005,18(12):229234. 10.1007/BF02936567
 27.
Stević S:On the recursive sequence . Journal of Difference Equations and Applications 2007,13(1):4146. 10.1080/10236190601069325
 28.
Stević S:On the difference equation . Computers & Mathematics with Applications 2008,56(5):11591171. 10.1016/j.camwa.2008.02.017
 29.
Stević S, Berenhaut KS: The behavior of positive solutions of a nonlinear secondorder difference equation. Abstract and Applied Analysis 2008, 2008:8.
 30.
Sun T, Xi H: Global asymptotic stability of a family of difference equations. Journal of Mathematical Analysis and Applications 2005,309(2):724728. 10.1016/j.jmaa.2004.11.040
 31.
Sun T, Xi H, Chen Z: Global asymptotic stability of a family of nonlinear recursive sequences. Journal of Difference Equations and Applications 2005,11(13):11651168. 10.1080/10236190500296516
 32.
Sun T, Xi H:On the system of rational difference equations . Advances in Difference Equations 2006, 2006:8.
 33.
Sun T, Xi H, Hong L:On the system of rational difference equations . Advances in Difference Equations 2006, 2006:7.
 34.
Xi H, Sun T: Global behavior of a higherorder rational difference equation. Advances in Difference Equations 2006, 2006:7.
 35.
Sun T, Xi H:On the basin of attraction of the two cycle of the difference equation . Journal of Difference Equations and Applications 2007,13(10):945952. 10.1080/10236190701388435
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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