- Research Article
- Open Access

# On Boundedness of Solutions of the Difference Equation for

- Hongjian Xi
^{1, 2}, - Taixiang Sun
^{1}Email author, - Weiyong Yu
^{1}and - Jinfeng Zhao
^{1}

**2009**:463169

https://doi.org/10.1155/2009/463169

© Hongjian Xi et al. 2009

**Received:**4 February 2009**Accepted:**2 June 2009**Published:**5 July 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).

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Asymptotic Behavior

## 1. Introduction

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 period-two 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])

- (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

Then . The proof of Lemma 2.1 is quite similar to that of Lemma 1 in [35] and hence is omitted.

Lemma 2.1.

- (1)
is a homeomorphism.

- (2)
and .

- (3)
and .

- (4)
and .

- (5)
and .

Lemma 2.2.

- (1)
If and , then .

- (2)
If and , then .

Proof.

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.

Claim 1.

Proof of Claim 1

This completes the proof of Claim 1.

Claim 2.

Proof of Claim 2

The proof of (2.14) is completed.

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.

which implies .

Case 2.

In similar fashion, we can show that . This completes the proof.

Lemma 3.2 (see [20, Theorem 5]).

- (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 .

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 .

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.

## Declarations

### Acknowledgment

Project Supported by NNSF of China (10861002) and NSF of Guangxi (0640205, 0728002).

## Authors’ Affiliations

## References

- Tasković MR:
*Nonlinear Functional Analysis. Vol. I: Fundamental Elements of Theory*. Zavod, za udžbenike i nastavna sredstva, Beograd, Serbia; 1993.Google Scholar - Stević S:
**Behavior of the positive solutions of the generalized Beddington-Holt equation.***PanAmerican Mathematical Journal*2000,**10**(4):77-85.MATHMathSciNetGoogle Scholar - Copson ET:
**On a generalisation of monotonic sequences.***Proceedings of the Edinburgh Mathematical Society. Series 2*1970,**17:**159-164. 10.1017/S0013091500009433MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Asymptotic behavior of a sequence defined by iteration with applications.***Colloquium Mathematicum*2002,**93**(2):267-276. 10.4064/cm93-2-6MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**A note on bounded sequences satisfying linear inequalities.***Indian Journal of Mathematics*2001,**43**(2):223-230.MATHMathSciNetGoogle Scholar - Stević S:
**A generalization of the Copson's theorem concerning sequences which satisfy a linear inequality.***Indian Journal of Mathematics*2001,**43**(3):277-282.MATHMathSciNetGoogle Scholar - Stević S:
**A global convergence result.***Indian Journal of Mathematics*2002,**44**(3):361-368.MATHMathSciNetGoogle Scholar - Stević S:
**A note on the recursive sequence**.*Ukrainian Mathematical Journal*2003,**55**(4):691-697.MathSciNetView ArticleGoogle Scholar - Stević S:
**On the recursive sequence**.*Taiwanese Journal of Mathematics*2002,**6**(3):405-414.MATHMathSciNetGoogle Scholar - Stević S:
**Asymptotics of some classes of higher-order difference equations.***Discrete Dynamics in Nature and Society*2007,**2007:**-20.Google Scholar - Stević S:
**Existence of nontrivial solutions of a rational difference equation.***Applied Mathematics Letters*2007,**20**(1):28-31. 10.1016/j.aml.2006.03.002MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the recursive sequence**.*Applied Mathematics Letters*2002,**15:**305-308. 10.1016/S0893-9659(01)00135-5MathSciNetView ArticleGoogle Scholar - 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):183-189. 10.1080/10236190500539295MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Periodic character of a class of difference equation.***Journal of Difference Equations and Applications*2004,**10**(6):615-619. 10.1080/10236190410001682103MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the difference equation**.*Dynamics of Continuous, Discrete & Impulsive Systems*2007,**14**(3):459-463.MATHMathSciNetGoogle Scholar - 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.MATHGoogle Scholar - Amleh AM, Grove EA, Ladas G, Georgiou DA:
**On the recursive sequence**.*Journal of Mathematical Analysis and Applications*1999,**233**(2):790-798. 10.1006/jmaa.1999.6346MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Stević S:
**The behaviour of the positive solutions of the difference equation**.*Journal of Difference Equations and Applications*2006,**12**(9):909-918. 10.1080/10236190600836377MATHMathSciNetView ArticleGoogle Scholar - Camouzis E, Ladas G:
**When does periodicity destroy boundedness in rational equations?***Journal of Difference Equations and Applications*2006,**12**(9):961-979. 10.1080/10236190600822369MATHMathSciNetView ArticleGoogle Scholar - Camouzis E, Ladas G:
**When does local asymptotic stability imply global attractivity in rational equations?***Journal of Difference Equations and Applications*2006,**12**(8):863-885. 10.1080/10236190600772663MATHMathSciNetView ArticleGoogle Scholar - Devault R, Kocic VL, Stutson D:
**Global behavior of solutions of the nonlinear difference equation**.*Journal of Difference Equations and Applications*2005,**11**(8):707-719. 10.1080/10236190500137405MATHMathSciNetView ArticleGoogle Scholar - Feuer J:
**On the behavior of solutions of**.*Applicable Analysis*2004,**83**(6):599-606. 10.1080/00036810410001657260MATHMathSciNetView ArticleGoogle Scholar - Kulenović MRS, Ladas G, Prokup NR:
**On the recursive sequence**.*Journal of Difference Equations and Applications*2000,**6**(5):563-576. 10.1080/10236190008808246MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Asymptotic behavior of a nonlinear difference equation.***Indian Journal of Pure and Applied Mathematics*2003,**34**(12):1681-1687.MATHMathSciNetGoogle Scholar - Stević S:
**On the recursive sequence**.**II.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2003,**10**(6):911-916.MATHMathSciNetGoogle Scholar - Stević S:
**On the recursive sequence**.*Journal of Applied Mathematics & Computing*2005,**18**(1-2):229-234. 10.1007/BF02936567MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the recursive sequence**.*Journal of Difference Equations and Applications*2007,**13**(1):41-46. 10.1080/10236190601069325MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the difference equation**.*Computers & Mathematics with Applications*2008,**56**(5):1159-1171. 10.1016/j.camwa.2008.02.017MATHMathSciNetView ArticleGoogle Scholar - Stević S, Berenhaut KS:
**The behavior of positive solutions of a nonlinear second-order difference equation.***Abstract and Applied Analysis*2008,**2008:**-8.Google Scholar - Sun T, Xi H:
**Global asymptotic stability of a family of difference equations.***Journal of Mathematical Analysis and Applications*2005,**309**(2):724-728. 10.1016/j.jmaa.2004.11.040MATHMathSciNetView ArticleGoogle Scholar - 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):1165-1168. 10.1080/10236190500296516MATHMathSciNetView ArticleGoogle Scholar - Sun T, Xi H:
**On the system of rational difference equations**.*Advances in Difference Equations*2006,**2006:**-8.Google Scholar - Sun T, Xi H, Hong L:
**On the system of rational difference equations**.*Advances in Difference Equations*2006,**2006:**-7.Google Scholar - Xi H, Sun T:
**Global behavior of a higher-order rational difference equation.***Advances in Difference Equations*2006,**2006:**-7.Google Scholar - 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):945-952. 10.1080/10236190701388435MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.