Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 258502 | 12 pages | https://doi.org/10.1155/2012/258502

Some Notes on the Difference Equation

Academic Editor: Garyfalos Papaschinopoulos
Received12 Apr 2012
Accepted20 May 2012
Published08 Aug 2012

Abstract

We investigate the behavior of the solutions of the recursive sequence, where , and the initial conditions are arbitrary positive numbers. Included are results that considerably improve those in the recently published paper by Hamza and Morsy (2009).

1. Introduction

Our aim in this paper is to give some remarks for the positive solutions of the difference equation where , , and the initial conditions are arbitrary positive numbers. Amleh et al. in [1] obtained important results for the difference equation which guide many authors. It was proved in [1] that, when , the equilibrium of (1.2) is globally asymptotically stable. When , every positive solution of (1.2) converges to a period-two solution. Every positive solution of (1.2) is bounded if and only if . Finally, when , the equilibrium is an unstable saddle point. Closely related equations to (1.1) are investigated by many authors, for example, [210].

In [4] the authors investigated the behavior of positive solutions of (1.1). It was proved in [4] that, when , every positive solution of (1.1) is bounded and when , the equilibrium of (1.1) is globally asymptotically stable. But in [4] the authors obtain some incorrect results for the boundedness character and the global stability of solutions of (1.1), and it is not shown that (1.1) has periodic solutions with conditions of and .

Our aim here is to improve and correct these results and extend some of the results in [4].

We say that the equilibrium point of the equation is the point that satisfies the condition A positive semicycle of a solution of (1.1) consists of a “string” of terms all greater than or equal to , with and , such that A negative semicycle of a solution of (1.1) consists of a “string” of terms all less than , with and , such that A solution of (1.1) is called nonoscillatory if there exists such that either or of (1.1) is called oscillatory if it is not nonoscillatory. We say that a solution of (1.1) is bounded and persists if there exist positive constants and such that for .

The linearized equation for (1.1) about the positive equilibrium is We need the following lemmas, which were given in [4].

Lemma 1.1. Let be the equilibrium point of (1.1).(i)If , then the equilibrium point of (1.1), is locally asymptotically stable.(ii)If , then the equilibrium point of (1.1) is unstable.

Lemma 1.2. The following statements are true.(i)If , then (1.1) has a unique equilibrium point .(ii)If , then (1.1) has a unique equilibrium point .

Lemma 1.3. Let be a solution of (1.1), which consists of at least two semicycles. Then, is oscillatory and, except possibly for the first semicycle, every semicycle is of length one.

The paper is organized as follows. In Section 2 we investigate the boundedness character of positive solutions of (1.1). We prove that if , then there exist unbounded solutions of (1.1) and when the cases either and or and , then every positive solution of (1.1) is unbounded. We show that when and , then every positive solution of (1.1) is bounded. Also we show that if , and , then the equilibrium point of (1.1) is globally asymptotically stable. Section 3 is devoted to the periodic character of the positive solutions of (1.1). Finally we show that a sufficient condition that every positive solution of (1.1) converges to a prime two periodic solution.

2. Boundedness and Global Stability of (1.1)

In this section, we present some results for the boundedness character of positive solutions and global stability of the equilibrium point of (1.1).

Theorem 2.1. Consider (1.1). Then, the following statements are true.(a)If and , then every positive solution of (1.1) is unbounded.(b)If and , then every positive solution of (1.1) is unbounded.

Proof. On the contrary, we assume that is a positive bounded solution of (1.1). Then, we have Thus, from (1.1) we get Let ; then we obtain which contradicts , so the proof is complete.
Again we assume that is a positive bounded solution of (1.1). Then, we have Thus, from (1.1), we have Let ; then which contradicts , so the proof is complete.

Now, we show that if , then there exist positive solutions of (1.1) that are unbounded.

Theorem 2.2. One has Then there exist positive solutions of (1.1) that are unbounded.

Proof. Assume that . Choose , and let be a solution of (1.1) with the initial conditions such that Then, Further we have Therefore, working inductively we can prove that for Hence, Since which implies that is unbounded. For , the proof is complete.
Now, we assume that and choose the initial conditions such that So, we have Further we have By induction we have Thus, This completes the proof.

The following theorem is given in [4].

Theorem 2.3. Suppose that ; then; every positive solution of (1.1) is bounded.

In [4] this result is not correct. So, we give the following theorem for the boundedness of (1.1).

Theorem 2.4. Suppose that , , and ; then every positive solution of (1.1) is bounded.

Proof . From (1.1), for . Thus, from (1.1), without loss of generality, we obtain for From (2.20) using induction, we obtain From which the proof follows.

Actually, Hamza and Morsy in [4] obtained global stability of the equilibrium point of (1.1). But the result does not include the case and some parts of its proof are incorrect. So, here we will obtain global stability of the equilibrium point of (1.1) when .

Theorem 2.5. Consider (1.1). Let and . Suppose that hold. Then, the unique positive equilibrium of (1.1) is globally asymptotically stable.

Proof. By Lemma 1.1, is locally asymptotically stable. Thus, it is enough for the proof that every positive solution of (1.1) tends to the unique positive equilibrium . Let be a solution of (1.1). By Theorem 2.4, is bounded. Thus, we have Then, from (2.23), we get We claim that , otherwise . From (2.24), we obtain And, from (2.25), Thus, Assume that . We consider with ; then there exists such that From (2.27) and (2.28), we obtain which is equivalent to Since we have , we get Since and , for some values and , (2.31) is not satisfied. This is a contradiction. Thus, we find .
Now, assume that . Then, from (2.27) and arguing as above, we get Furthermore, we have We consider the following difference equation: Every positive solution of the previous equation converges to . It follows that . Then, we obtain that Thus, Since and , for some values and , (2.36) is not satisfied. So , which implies that tends to the unique positive equilibrium. From which the proof follows.

Remark 2.6. Consider (1.1), where . Let be a solution of (1.1). If is bounded, then it is stable too.

3. Periodicity of the Solutions of (1.1)

In this section we investigate the periodicity of (1.1) when and .

We need the following lemma whose proof follows by simple computation and thus it will be omitted.

Lemma 3.1. Let be a solution of (1.1), and let . Then, the following statements are satisfied.(i) if and only if ,(ii) if and only if .

Theorem 3.2. Consider (1.1), where Assume that there exists a sufficient small positive number such that Then, (1.1) has a periodic solution of prime period two.

Proof. Let be a solution of (1.1). It is obvious that if hold, then is a periodic solution of period two. Consider the system Then this system is equivalent to and so we get the equation and obtain Thus, Moreover, from (3.2) we can show that Therefore, the equation has a solution , where , in the interval . So, we have We now consider the function Since from (3.1) we have From (3.3), we have , and thus , which implies that Hence, if ,  , then the solution with initial values is a prime -periodic solution. This completes the proof.

In the following theorem we will generalize the result due to Stević [6, Theorem ].

Theorem 3.3. For every positive solution of (1.1), the sequences and are eventually monotone.

Proof. We have If and , we obtain from (3.16) and consequently . By induction we obtain Similarly if and , using induction we obtain from (3.16) If , and , we can obtain from (3.16) Hence, we may assume that and . If further , then So we may assume that and . By induction we obtain the result in this case.
The cases and can be treated similarly.

Theorem 3.4. Consider (1.1) where (3.1), (3.2), and (3.3) hold. Then, every positive solution of (1.1) converges to a prime two periodic solution.

Proof. By Theorem 3.3, for every positive solution of (1.1) the sequences and are eventually monotone. By Theorem 2.4, the sequences and are bounded. Hence, the sequences and are convergent. Using Lemmas 1.3 and 3.1 the result follows.

References

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Copyright © 2012 Mehmet Gümüş and Özkan Öcalan. 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.


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