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On the Recursive Sequence
Advances in Difference Equations volume 2009, Article number: 327649 (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.
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).
The investigation of positive solutions of the following equation
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].
In [22] the author studied the boundedness, the global attractivity, the oscillatory behavior, and the periodicity of the positive solutions of the equation
where are positive constants, and the initial conditions are positive numbers (see also [5] for more results on this equation).
In [11] the authors obtained boundedness, persistence, global attractivity, and periodicity results for the positive solutions of the difference equation
where are positive constants and the initial conditions are positive numbers.
Motivating by the above papers, we study now the boundedness, the persistence, the existence of unbounded solutions, the attractivity, the stability of the positive solutions, and the two-period solutions of the difference equation
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.
If
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.
If
then there exist unbounded solutions of (1.4).
Proof.
Let be a solution of (1.4) with initial values such that
Then from (1.4), (2.2), and (2.3) we have
Moreover from (1.4), and (2.3) we have
Then using (1.4), and (2.3)–(2.5) and arguing as above we get
Therefore working inductively we can prove that for
which implies that
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.
If either
or
holds, then (1.4) has a unique positive equilibrium .
Proof.
A point will be an equilibrium of (1.4) if and only if it satisfies the following equation
Suppose that (3.1) is satisfied. Since (3.1) holds and
we have that is increasing in and is decreasing in . Moreover and
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.
Consider (1.4). Suppose that either
or (3.1) and
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).
Assume first that (3.6) is satisfied. Let be a positive solution of (1.4). From (3.6) and Proposition 2.1 we have
Then from (1.4) and (3.8) we get,
and so
Thus,
This implies that
Suppose for a while that . We shall prove that . Suppose on the contrary that . If we consider the function , then there exists a such that
Then from (3.12) and (3.13) we obtain
or
Moreover, since from (1.4),
from (3.6) and (3.15) we get
which contradicts to (3.6). So which implies that tends to the unique positive equilibrium .
Suppose that . Then from (3.12) and arguing as above we get
Then arguing as above we can prove that tends to the unique positive equilibrium .
Assume now that (3.7) holds. From (3.7) and (3.12) we obtain
which implies that . So every positive solution of (1.4) tends to the unique positive equilibrium of (1.4).
It remains to prove now that the unique positive equilibrium of (1.4) is locally asymptotically stable. The linearized equation about the positive equilibrium is the following:
Using [13, Theorem 1.3.4] the linear (3.20) is asymptotically stable if and only if
First assume that (3.6) holds. Since (3.6) holds, then we obtain that
From (3.6) and (3.22) we can easily prove that
Therefore
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.
Consider (1.4) where
Assume that there exists a sufficient small positive real number , such that
Then (1.4) has a periodic solution of prime period two.
Proof.
Let be a positive solution of (1.4). It is obvious that if
then is periodic of period two. Consider the system
Then system (4.5) is equivalent to
and so we get the equation
We obtain
and so from (4.1)
Moreover from (4.3) we can show that
Therefore the equation has a solution , where , in the interval . We have
We consider the function
Since from (4.1) and we have
From (4.2) we have , so from (4.13)
which implies that
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.
We define the sequence and the function as follows:
Then from (1.4) for we get
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.
Consider (1.4) where (4.1) and (4.3) hold. Let be a solution of (1.4) such that either
or
Then if (4.18) holds, one has
and if (4.19) is satisfied, one has
Proof.
Suppose that (4.18) is satisfied. Then from (1.4) and (4.3) we have
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.
Consider (1.4) where (4.1), (4.2), and (4.3) hold. Suppose also that
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.
Let be a solution with initial values which satisfy either (4.18) or (4.19). Using Proposition 2.1 and Lemma 4.2 we have that there exist
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.
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The authors would like to thank the referees for their helpful suggestions.
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Schinas, C.J., Papaschinopoulos, G. & Stefanidou, G. On the Recursive Sequence . Adv Differ Equ 2009, 327649 (2009). https://doi.org/10.1155/2009/327649
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DOI: https://doi.org/10.1155/2009/327649