Research Article | Open Access

# On the Recursive Sequence

**Academic Editor:**Guang Zhang

#### Abstract

This paper studies the dynamic behavior of the positive solutions to the difference equation , , where , and are positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation for .

#### 1. Introduction

In recent years, there has been intense interest in the dynamic behavior of the positive solutions to a class of difference equations of the form where and are positive real numbers. Now, let us make a brief review on the advances in this class of difference equations.

In 1999, Amleh et al. [1] studied the second-order rational difference equation

Later, Berenhaut and Stević [2], Stević [3], and El-Owaidy et al. [4] extended this work to the following more general second-order difference equation:

On the other hand, DeVault et al. [5] investigated the following higher-order version of (1.2):

By combining (1.3) and (1.4), Berenhaut and Stević [6] examined a larger class of difference equations, which are of the form

Very recently, Berenhaut et al. [7] studied the following generalization of (1.5):

For some related work, the interested reader is referred to [1, 3, 8–19].

Inspired by the previous work and by the work owing to Stević [15], this paper studies the behavior of the recursive equation We establish some interesting results regarding the stability and oscillation character of this equation for .

#### 2. Stability Character

In this section we investigate the stability character of the positive solutions to (1.7).

A point is an equilibrium point of (1.7) if and only if it is a root for the function that is,

Lemma 2.1. *Let ,
then (1.7) has a unique
equilibrium point .*

*Proof**Case 1. *. Then .*Case 2. *. Then
defined by (2.1) is
decreasing on and increasing on
.
Since
and ,
then
has a unique zero .*Case 3. *. Since
is increasing on
and ,
then
has a unique zero .

Lemma 2.2. *Let .
Assume that
is the equilibrium point of (1.7). If ,
then
is locally asymptotically stable.*

*Proof. *By the Linearized Stability Theorem [11],
is locally asymptotically stable if and only if .
A simple calculations shows that
where is
defined by (2.1). Then since
,
we have and
.
The proof is complete.

Lemma 2.3. *If , then every positive solution to
(1.7) is bounded.*

*Proof. *Note that each
can be written in the form
for some
and . From (1.7) and since
for every ,
we have that
for every
and . Let
be the solution to the difference equation

From (2.4) and by induction we
see that .
Hence it is enough to prove that the sequences are bounded.

Since the function is
increasing and concave for , it follows that there is a
unique fixed point
of the equation
and that the function
satisfies

Using this fact it is easy to see that if , the
sequence is nondecreasing and bounded from above by ,
and if ,
it is nonincreasing and bounded from below by .
Hence for every , each of the
sequences is bounded. The claimed
result follows.

Lemma 2.4 (see [18]). *Let
be distinct nonnegative integers. Consider the difference equation **
Suppose
satisfies the following
conditions.** is a
continuous function that is nondecreasing in the first argument and
is nonincreasing in the second
argument.**The system
has a unique solution .**Then
is the global attractor of all solutions to (2.7).*

Theorem 2.5. *Let ,
then the unique equilibrium
to (1.7) is globally
asymptotically stable.*

*Proof. *By Lemma 2.3, there must exist
positive constants
and
such that .
Let , it is easy
to verify that holds. In
addition, if
then
Assume that ,
then
or .

In case ,
we have ,
which contradicts with (2.10).

In case ,
we have ,
again a contradiction.

Thus .
By Lemma 2.4, the required result
follows.

Theorem 2.6. *Let
and .
Then every positive solution to (1.7) converges to the unique equilibrium .*

*Proof. *By Lemma 2.3, every positive
solution to (1.7) is bounded, which implies
that there are finite
and .
Assume that . Taking the
and
in (1.7), it follows that
From this and , it follows that
yielding

Define function . Since
we deduce that
is increasing, and thus (2.13)
cannot hold. Therefore we have ,
which implies the result.

Theorem 2.7. *Let ,
and .
Then every positive solution to (1.7) converges to the unique equilibrium .*

*Proof. *From (2.11) we have
Consequently, we obtain . Suppose that
,
we get
where , leading to

This implies that ,
which is a contradiction. Hence, .

#### 3. Oscillation Character

In this section we investigate the oscillation character of the positive solutions to (1.7).

Theorem 3.1. *Let
be a positive solution to (1.7).
Then either
consists of a single semicycle or
oscillates about the equilibrium
with semicycles having at most
terms.*

*Proof. *Suppose that
has at least two semicycles. Then there exists
such that either
or .
Assume that .
(The argument for the case
is similar and is omitted). Now suppose that the positive semicycle beginning
with the term
has
terms. Then
and so
This completes the proof.

Theorem 3.2. *Suppose that
is even and let
be a solution to (1.7), which
has
consecutive semicycles of length one, then every semicycle after this point is
of length one.*

*Proof. *There exists
such that either
or
We prove the former case. The proof for the latter is similar and is omitted.
Now, we have

The result then follows by induction.

Lemma 3.3. *Let .
Then (1.7) has no nontrivial
periodic solutions of (not necessarily prime) period .*

*Proof. *Suppose that
is a positive solution to (1.7)
satisfying
for all ,
then
implies that
for all .
The proof is complete.

Theorem 3.4. *Assume that .
Let
be a positive solution to (1.7),
which consists of a single semicycle, then
converges to the equilibrium .*

*Proof. *Suppose
(the case for
is similar and is omitted) for all ,
then
implying that
and so
From here it is clear that for
there exists
such that
But then
is a periodic solution of (not necessarily prime) period
.
By Lemma 3.3 the result holds.

#### Acknowledgments

The author is grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by the Natural Science Foundation of China (no. 10771227) and the Project for New Century Excellent Talents of Educational Ministry of China (no. NCET-05-0759).

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#### Copyright

Copyright © 2009 Fangkuan Sun et al. 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.