- 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

## 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

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