- Research Article
- Open Access

- C. J. Schinas
^{1}, - G. Papaschinopoulos
^{1}Email author and - G. Stefanidou
^{1}

**2009**:327649

https://doi.org/10.1155/2009/327649

© C. J. Schinas et al. 2009

**Received:**11 June 2009**Accepted:**21 September 2009**Published:**11 October 2009

## Abstract

In this paper we study the boundedness, the persistence, the attractivity and the stability of the positive solutions of the nonlinear difference equation , where and . Moreover we investigate the existence of a prime two periodic solution of the above equation and we find solutions which converge to this periodic solution.

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Population Dynamic
- Positive Constant

## 1. Introduction

Difference equations have been applied in several mathematical models in biology, economics, genetics, population dynamics, and so forth. For this reason, there exists an increasing interest in studying difference equations (see [1–28] and the references cited therein).

where and , , was proposed by Stević at numerous conferences. For some results in the area see, for example, [3–5, 8, 11, 12, 19, 22, 24, 25, 28].

where are positive constants, and the initial conditions are positive numbers (see also [5] for more results on this equation).

where are positive constants and the initial conditions are positive numbers.

where are positive constants and the initial values are positive real numbers.

Finally equations, closely related to (1.4), are considered in [1–11, 14, 16–23, 26, 27], and the references cited therein.

## 2. Boundedness and Persistence

The following result is essentially proved in [22]. Hence, we omit its proof.

Proposition 2.1.

then every positive solution of (1.4) is bounded and persists.

In the next proposition we obtain sufficient conditions for the existence of unbounded solutions of (1.4).

Proposition 2.2.

then there exist unbounded solutions of (1.4).

Proof.

So is unbounded. This completes the proof of the proposition.

## 3. Attractivity and Stability

In the following proposition we prove the existence of a positive equilibrium.

Proposition 3.1.

holds, then (1.4) has a unique positive equilibrium .

Proof.

So if (3.1) holds, we get that (1.4) has a unique equilibrium in .

Suppose now that (3.2) holds. We observe that and since from (3.2) and (3.4) , we have that is decreasing in . Thus from (3.5) we obtain that (1.4) has a unique equilibrium in . The proof is complete.

In the sequel, we study the global asymptotic stability of the positive solutions of (1.4).

Proposition 3.2.

hold. Then the unique positive equilibrium of (1.4) is globally asymptotically stable.

Proof.

First we prove that every positive solution of (1.4) tends to the unique positive equilibrium of (1.4).

which contradicts to (3.6). So which implies that tends to the unique positive equilibrium .

Then arguing as above we can prove that tends to the unique positive equilibrium .

which implies that . So every positive solution of (1.4) tends to the unique positive equilibrium of (1.4).

which implies that (3.21) is true. So in this case the unique positive equilibrium of (1.4) is locally asymptotically stable.

Finally suppose that (3.1) and (3.7) are satisfied. Then we can prove that (3.23) is satisfied, and so the unique positive equilibrium of (1.4) satisfies (3.24). Therefore (3.21) hold. This implies that the unique positive equilibrium of (1.4) is locally asymptotically stable. This completes the proof of the proposition.

## 4. Study of 2-Periodic Solutions

Motivated by [5, Lemma 1], in this section we show that there is a prime two periodic solution. Moreover we find solutions of (1.4) which converge to a prime two periodic solution.

Proposition 4.1.

Then (1.4) has a periodic solution of prime period two.

Proof.

Hence, if , , then the solution with initial values , is a prime 2-periodic solution.

In the sequel, we shall need the following lemmas.

Lemma 4.2.

Let be a solution of (1.4). Then the sequences and are eventually monotone.

Proof.

Then using (4.17) and arguing as in [5, Lemma 2] (see also in [20, Theorem 2]) we can easily prove the lemma.

Lemma 4.3.

Proof.

Working inductively we can easily prove relations (4.20). Similarly if (4.19) is satisfied, we can prove that (4.21) holds.

Proposition 4.4.

Then every solution of (1.4) with initial values which satisfy either (4.18) or (4.19), converges to a prime two periodic solution.

Proof.

In addition from Lemma 4.3 we have that either or belongs to the interval . Furthermore from Proposition 3.1 we have that (1.4) has a unique equilibrium such that . Therefore from (4.23) we have that . So converges to a prime two-period solution. This completes the proof of the proposition.

## Declarations

### Acknowledgment

The authors would like to thank the referees for their helpful suggestions.

## Authors’ Affiliations

## References

- 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, Foley JD, Stević S:
**The global attractivity of the rational difference equation .***Proceedings of the American Mathematical Society*2007,**135**(4):1133–1140. 10.1090/S0002-9939-06-08580-7MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Foley JD, Stević S:
**The global attractivity of the rational difference equation .***Proceedings of the American Mathematical Society*2008,**136**(1):103–110. 10.1090/S0002-9939-07-08860-0MATHMathSciNetView ArticleGoogle Scholar - Berenhaut KS, Stević S:
**A note on positive non-oscillatory solutions of the difference equation .***Journal of Difference Equations and Applications*2006,**12**(5):495–499. 10.1080/10236190500539543MATHMathSciNetView 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 - Berg L:
**On the asymptotics of nonlinear difference equations.***Zeitschrift für Analysis und ihre Anwendungen*2002,**21**(4):1061–1074.MATHView 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 - El-Owaidy HM, Ahmed AM, Mousa MS:
**On asymptotic behaviour of the difference equation .***Journal of Applied Mathematics & Computing*2003,**12**(1–2):31–37. 10.1007/BF02936179MATHMathSciNetView ArticleGoogle Scholar - Grove EA, Ladas G:
*Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications*.*Volume 4*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2005:xiv+379.Google Scholar - Gutnik L, Stević S:
**On the behaviour of the solutions of a second-order difference equation.***Discrete Dynamics in Nature and Society*2007,**2007:**-14.Google Scholar - Hamza AE, Morsy A:
**On the recursive sequence .***Applied Mathematics Letters*2009,**22**(1):91–95. 10.1016/j.aml.2008.02.010MATHMathSciNetView ArticleGoogle Scholar - Iričanin B, Stević S:
**On a class of third-order nonlinear difference equations.***Applied Mathematics and Computation*2009,**213**(2):479–483. 10.1016/j.amc.2009.03.039MATHMathSciNetView ArticleGoogle Scholar - Kocić VL, Ladas G:
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications*.*Volume 256*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+228.Google Scholar - Kulenović MRS, Ladas G, Overdeep CB:
**On the dynamics of with a period-two coefficient.***Journal of Difference Equations and Applications*2004,**10**(10):905–914. 10.1080/10236190410001731434MATHMathSciNetView ArticleGoogle Scholar - Kulenović MRS, Ladas G:
*Dynamics of Second Order Rational Difference Equations*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218.MATHGoogle Scholar - Papaschinopoulos G, Schinas CJ:
**On a -th order difference equation with a coefficient of period ().***Journal of Difference Equations and Applications*2005,**11**(3):215–225. 10.1080/10236190500035310MATHMathSciNetView ArticleGoogle Scholar - Papaschinopoulos G, Schinas CJ:
**On a nonautonomous difference equation with bounded coefficient.***Journal of Mathematical Analysis and Applications*2007,**326**(1):155–164. 10.1016/j.jmaa.2006.02.081MATHMathSciNetView ArticleGoogle Scholar - Papaschinopoulos G, Schinas CJ, Stefanidou G:
**On a difference equation with 3-periodic coefficient.***Journal of Difference Equations and Applications*2005,**11**(15):1281–1287. 10.1080/10236190500386317MATHMathSciNetView ArticleGoogle Scholar - Papaschinopoulos G, Schinas CJ, Stefanidou G:
**Boundedness, periodicity and stability of the difference equation .***International Journal of Dynamical Systems and Differential Equations*2007,**1**(2):109–116. 10.1504/IJDSDE.2007.016513MATHMathSciNetView ArticleGoogle 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:
**A note on periodic character of a difference equation.***Journal of Difference Equations and Applications*2004,**10**(10):929–932. 10.1080/10236190412331272616MATHMathSciNetView ArticleGoogle 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:
**Asymptotics of some classes of higher-order difference equations.***Discrete Dynamics in Nature and Society*2007,**2007:**-20.Google Scholar - Stević S:
**On the recursive sequence .***Discrete Dynamics in Nature and Society*2007,**2007:**-9.Google Scholar - Stević S:
**On the recursive sequence .***Discrete Dynamics in Nature and Society*2007,**2007:**-9.Google 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 - Stević S:
**Boundedness character of a class of difference equations.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(2):839–848. 10.1016/j.na.2008.01.014MATHMathSciNetView 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.