On the Recursive Sequence

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.

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.

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.

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.

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.

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

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