On the Recursive Sequence

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(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.

If

(2.2)

then there exist unbounded solutions of (1.4).

Proof.

Let be a solution of (1.4) with initial values such that

(2.3)

Then from (1.4), (2.2), and (2.3) we have

(2.4)

Moreover from (1.4), and (2.3) we have

(2.5)

Then using (1.4), and (2.3)–(2.5) and arguing as above we get

(2.6)

Therefore working inductively we can prove that for

(2.7)

which implies that

(2.8)

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

(3.1)

or

(3.2)

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

(3.3)

Suppose that (3.1) is satisfied. Since (3.1) holds and

(3.4)

we have that is increasing in and is decreasing in . Moreover and

(3.5)

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

(3.6)

or (3.1) and

(3.7)

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

(3.8)

Then from (1.4) and (3.8) we get,

(3.9)

and so

(3.10)

Thus,

(3.11)

This implies that

(3.12)

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

(3.13)

Then from (3.12) and (3.13) we obtain

(3.14)

or

(3.15)

Moreover, since from (1.4),

(3.16)

from (3.6) and (3.15) we get

(3.17)

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

(3.18)

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

(3.19)

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:

(3.20)

Using [13, Theorem  1.3.4] the linear (3.20) is asymptotically stable if and only if

(3.21)

First assume that (3.6) holds. Since (3.6) holds, then we obtain that

(3.22)

From (3.6) and (3.22) we can easily prove that

(3.23)

Therefore

(3.24)

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

(4.1)

Assume that there exists a sufficient small positive real number , such that

(4.2)

(4.3)

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

(4.4)

then is periodic of period two. Consider the system

(4.5)

Then system (4.5) is equivalent to

(4.6)

and so we get the equation

(4.7)

We obtain

(4.8)

and so from (4.1)

(4.9)

Moreover from (4.3) we can show that

(4.10)

Therefore the equation has a solution , where , in the interval . We have

(4.11)

We consider the function

(4.12)

Since from (4.1) and we have

(4.13)

From (4.2) we have , so from (4.13)

(4.14)

which implies that

(4.15)

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:

(4.16)

Then from (1.4) for we get

(4.17)

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

(4.18)

or

(4.19)

Then if (4.18) holds, one has

(4.20)

and if (4.19) is satisfied, one has

(4.21)

Proof.

Suppose that (4.18) is satisfied. Then from (1.4) and (4.3) we have

(4.22)

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

(4.23)

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

(4.24)

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.