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Dynamics for Nonlinear Difference Equation ![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_IEq1_HTML.gif)
Advances in Difference Equations volume 2009, Article number: 235691 (2009)
Abstract
We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.
1. Introduction
Consider the following higher order difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ1_HTML.gif)
where , the parameters
and
are nonnegative real numbers and the initial conditions
and
are nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ2_HTML.gif)
It is easy to see that if one of the parameters is zero, then the equation is linear. If
, then (1.1) can be reduced to a linear one by the change of variables
. So in the sequel we always assume that the parameters
and
are positive real numbers.
The change of variables reduces (1.1) into the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ3_HTML.gif)
where .
Note that is always an equilibrium point of (1.3). When
, (1.3) also possesses the unique positive equilibrium
.
The linearized equation of (1.3) about the equilibrium point is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ4_HTML.gif)
so, the characteristic equation of (1.3) about the equilibrium point is either, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ5_HTML.gif)
or, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ6_HTML.gif)
The linearized equation of (1.3) about the positive equilibrium point has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ7_HTML.gif)
with the characteristic equation either, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ8_HTML.gif)
or, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ9_HTML.gif)
When ,
, (1.1) has been investigated in [1–4]. When
,
, (1.1) reduces to the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ10_HTML.gif)
El-Owaidy et al. [3] investigated the global asymptotical stability of zero equilibrium, the periodic character and the existence of unbounded solutions of (1.10).
On the other hand, when , (1.1) is just the discrete delay logistic model investigated in [4,
]. Therefore, it is both theoretically and practically meaningful to study (1.1).
Our aim in this paper is to extend and generalize the work in [3]. That is, we will investigate the global behavior of (1.1), including the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence and asymptotic behavior of nonoscillatory solutions of the equation. Our results extend and generalize the corresponding ones of [3].
For the sake of convenience, we now present some definitions and known facts, which will be useful in the sequel.
Consider the difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ11_HTML.gif)
where is a positive integer, and the function
has continuous partial derivatives.
A point is called an equilibrium of (1.11) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ12_HTML.gif)
That is, for
is a solution of (1.11), or equivalently,
is a fixed point of
.
The linearized equation of (1.11) associated with the equilibrium point is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ13_HTML.gif)
We need the following lemma.
-
(i)
If all the roots of the polynomial equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ14_HTML.gif)
lie in the open unit disk , then the equilibrium
of (1.11) is locally asymptotically stable.
-
(ii)
If at least one root of (1.11) has absolute value greater than one, then the equilibrium
of (1.11) is unstable.
For the related investigations for nonlinear difference equations, see also [7–11] and the references cited therein.
2. Global Asymptotic Stability of Zero Equilibrium
In this section, we investigate global asymptotic stability of zero equilibrium of (1.3). We first have the following results.
Lemma 2.1.
The following statements are true.
(a)If , then the equilibrium point
of (1.3) is locally asymptotically stable.
(b)If , then the equilibrium point
of (1.3) is unstable. Moreover, for
, (1.3) has a
-dimension local stable manifold and a
-dimension local unstable manifold.
(c)If ,
is odd and
is even, then the positive equilibrium point
of (1.3) is unstable.
Proof.
-
(a)
When
, it is clear from (1.5) and (1.6) that every characteristic root
satisfies
or
, and so, by Lemma 1.1(i),
is locally asymptotically stable.
-
(b)
When
, if
, then it is clear from (1.5) that every characteristic root
satisfies
, and so, by Lemma 1.1(ii),
is unstable. If
, then (1.6) has
characteristic roots
satisfying
, which corresponds to a
-dimension local stable manifold of (1.3), and
characteristic roots
satisfying
, which corresponds to a
-dimension local unstable manifold of (1.3).
-
(c)
If
is odd and
is even, then, regardless of
or
, correspondingly, the characteristic equation (1.8) or (1.9) always has one characteristic root
lying the interval
. It follows from Lemma 1.1(ii) that
is unstable.
Remark 2.2.
Lemma 2.1(a) includes and improves [3, Theorem  3.1(i)]. Lemma 2.1(b) and (c) include and generalize [3, Theorem 3.1(ii) and (iii)], respectively.
Now we state the main results in this section.
Theorem 2.3.
Assume that , then the equilibrium point
of (1.3) is globally asymptotically stable.
Proof.
We know from Lemma 2.1 that the equilibrium point of (1.3) is locally asymptotically stable. It suffices to show that
for any nonnegative solution
of (1.3).
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ15_HTML.gif)
converges for any
. Let
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ16_HTML.gif)
Thereout, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ17_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ18_HTML.gif)
which implies . The proof is over.
Remark 2.4.
Theorem 2.3 includes [3, Theorem  3.3].
3. Existence of Eventual Period Two Solution
In this section, one studies the eventual nonnegative prime period two solutions of (1.3). A solution of (1.3) is said to be eventual prime periodic two solution if there exists an
such that
for
and
holds for all
.
Theorem 3.1.
-
(a)
Assume
is odd and
is even, then (1.3) possesses eventual prime period two solutions if and only if
.
-
(b)
Assume
is odd and
is odd, then (1.3) possesses eventual prime period two solutions if and only if
.
-
(c)
Assume
is even and
is even. Then the necessary condition for (1.3) to possess eventual prime period two solutions is
and
.
-
(d)
Assume
is even and
is odd. Then, (1.3) has no eventual prime period two solutions.
Proof.
-
(a)
If (1.3) has the eventual nonnegative prime period two solution
then, we eventually have
and
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ19_HTML.gif)
If , then we can derive from (3.1) that
if
or vice versa, which contradicts the assumption that
is the eventual prime period two solution of (1.3). So,
. Accordingly,
and
, which indicate that
when
or that
and
do not exist when
, which are also impossible. Therefore,
.
Conversely, if , then choose the initial conditions such as
and
, or such as
and
. We can see by induction that
is the prime period two solution of (1.3).
-
(b)
Let
be the eventual prime period two solution of (1.3), then, it holds eventually that
and
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ20_HTML.gif)
If , then
. This is impossible. So
. Moreover,
and
or
and
, that is,
is the prime period two solution of (1.3).
-
(c)
Assume that (1.3) has the eventual nonnegative prime period two solution
then eventually
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ21_HTML.gif)
Obviously, implies
or vice versa. This is impossible. So
. It is easy to see from (3.3) that
and
satisfy the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ22_HTML.gif)
that is, and
are two distinct positive roots of
. From (3.4) we can see that
does not have two distinct positive roots at all when
, alternatively, (1.3) does not have the nonnegative prime period two solution at all when
. Therefore, we assume
in the following.
Let in (3.4), then the equation
has at least two distinct positive roots.
By simple calculation, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ23_HTML.gif)
If , we can see
for all
. This means that
is strictly increasing in the interval
and hence the equation,
cannot have two distinct positive roots. So, next we consider
, which implies
. Denote
. We need to discuss several cases, respectively, as follows.
Case 1.
It holds that . Then
for all
, hence,
is convex. Again,
. So it is impossible for
to have two distinct positive roots.
Case 2.
It holds that and
. Then, for
,
and so
; for
,
and so
. At this time, one always has
. Then
cannot have two distinct positive roots.
Case 3.
It holds that ,
and
. Then, for
,
and so
and hence
, that is,
has no solutions for
; for
,
, that is,
is convex for
. Noticing
, it is also impossible for
to have two distinct positive roots for
.
Case 4.
It holds that ,
and
. This is only case where
could have two distinct positive roots, which implies
and
.
-
(d)
Let …, φ, ψ, φ, ψ,… be the eventual nonnegative prime period two solution of (1.3), then, it is eventually true that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ24_HTML.gif)
It is easy to see that and
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ25_HTML.gif)
that is, and
are two distinct positive roots of
. Obviously, when
, the
has no positive roots.
Now let and set
. Then the function,
has at least two distinct positive roots. However,
for any
, which indicates that
is strictly increasing in the interval
. This implies that the function
does not have two distinct positive roots at all in the interval
. In turn, (1.3) does not have the prime period two solution when
.
4. Existence of Oscillatory Solution
For the oscillatory solution of (1.3), we have the following results.
Theorem 4.1.
Assume ,
is odd and
is even. Then, there exist solutions
of (1.3) which oscillate about
with semicycles of length one.
Proof.
We only prove the case where (the proof of the case where
is similar and will be omitted). Choose the initial values of (1.3) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ26_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ27_HTML.gif)
We will only prove the case where (4.1) holds. The case where (4.2) holds is similar and will be omitted. According to (1.3), one can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ28_HTML.gif)
So, the proof follows by induction.
5. Existence of Unbounded Solution
With respect to the unbounded solutions of (1.3), the following results are derived.
Theorem 5.1.
Assume ,
is odd, and
is even, then (1.3) possesses unbounded solutions.
Proof.
We only prove the case where (the proof of the case where
is similar and will be omitted). Choose the initial values of (1.3) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ29_HTML.gif)
In the following, assume . From the proof of Theorem 4.1, one can see that
when
is odd and that
for
even. Together with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ30_HTML.gif)
It is derived that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ31_HTML.gif)
So, is decreasing for
odd whereas
is increasing for
even. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ32_HTML.gif)
then one has
(1) for
odd and
for
even,
(2).
Now, either for some even
in which case the proof is complete, or
for all even
. We shall prove that this latter is impossible. In fact, we prove that
for all even
.
Assume for some even
, then one has, by (5.2),
. Noticing (1), one hence further gets
. However
is odd, according to (1),
. This is a contradiction.
Therefore, for any even
. Accordingly,
are unbounded subsequences of this solution
of (1.3) for even
. Simultaneously, for odd
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ33_HTML.gif)
The proof is complete.
Remark 5.2.
Theorem 5.1 includes and generalizes [3, Theorem  3.5].
6. Existence and Asymptotic Behavior of Nonoscillatory Solution
In this section, we consider the existence and asymptotic behavior of nonoscillatory solution of (1.3). Because all solutions of (1.3) are nonnegative, relative to the zero equilibrium point , every solution of (1.3) is a positive semicycle, a trivial nonoscillatory solution! Thus, it suffices to consider the positive equilibrium
when studying the nonoscillatory solutions of (1.3). At this time,
.
Firstly, we have the following results.
Theorem 6.1.
Every nonoscillatory solution of (1.3) with respect to approaches
.
Proof.
Let be any one nonoscillatory solution of (1.3) with respect to
. Then, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ34_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ35_HTML.gif)
We only prove the case where (6.1) holds. The proof for the case where (6.2) holds is similar and will be omitted. According to (6.1), for , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ36_HTML.gif)
So, is decreasing for
with upper bound
. Hence,
exists and is finite. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ37_HTML.gif)
Then . Taking limits on both sides of (1.3), we can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ38_HTML.gif)
which shows and completes this proof.
A problem naturally arises: are there nonoscillatory solutions of (1.3)? Next, we will positively answer this question. Our result is as follows.
Theorem 6.2.
However (1.3) possesses asymptotic solutions with a single semicycle (positive semicycle or negative semicycle).
The main tool to prove this theorem is to make use of Berg' inclusion theorem [12]. Now, for the sake of convenience of statement, we first state some preliminaries. For this, refer also to [13]. Consider a general real nonlinear difference equation of order with the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ39_HTML.gif)
where ,
. Let
and
be two sequences satisfying
and
as
. Then (maybe under certain additional conditions), for any given
, there exist a solution
of (6.6) and an
such
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ40_HTML.gif)
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ41_HTML.gif)
which is called an asymptotic stripe. So, if , then it is implied that there exists a real sequence
such that
and
for
.
We now state the inclusion theorem [12].
Lemma 6.3.
Let be continuously differentiable when
, for
, and
. Let the partial derivatives of
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ42_HTML.gif)
as uniformly in
for
,
, as far as
. Assume that there exist a sequence
and constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ43_HTML.gif)
for as
, and suppose there exists an integer
, with
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ44_HTML.gif)
Then, for sufficiently large , there exists a solution
of (6.6) satisfying (6.7).
Proof of Theorem 6.2.
We only prove the case where (the proof of the case where
is similar and will be omitted). Put
(
is denoted into
for short). Then (1.3) is transformed into
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ45_HTML.gif)
An approximate equation of (6.12) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ46_HTML.gif)
provided that as
. The general solution of (6.13) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ47_HTML.gif)
where and
are the
roots of the polynomial
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ48_HTML.gif)
Obviously, . So,
has a positive root
lying in the interval (0, 1). Now, choose the solution
for this
. For some
, define the sequences
and
, respectively, as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ49_HTML.gif)
Obviously, and
as
.
Now, define again the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ50_HTML.gif)
Then the partial derivatives of w.r.t.
, respectively, are
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ51_HTML.gif)
When ,
. So,
,
, as
uniformly in
for
,
.
Moreover, from the definition of the function and (6.17) and (6.18), after some calculation, we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ52_HTML.gif)
Now choose . Noting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ53_HTML.gif)
we have Again,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ55_HTML.gif)
Therefore, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F235691/MediaObjects/13662_2009_Article_1173_Equ56_HTML.gif)
Up to here, all conditions of Lemma 6.3 with and
are satisfied. Accordingly, we see that, for arbitrary
and for sufficiently large
, say
, (6.12) has a solution
in the stripe
, where
and
are as defined in (6.16). Because
,
for
. Thus, (1.3) has a solution
satisfying
for
. Since (1.3) is an autonomous equation,
still is its solution, which evidently satisfies
for
. Therefore, the proof is complete.
Remark 6.4.
If we take in (6.16), then
. At this time, (1.3) possesses solutions
which remain below its equilibrium for all
, that is, (1.3) has solutions with a single negative semicycle.
Remark 6.5.
The appropriate equation (6.12) is just the linearized equation of (1.3) associated with .
Remark 6.6.
The existence and asymptotic behavior of nonoscillatory solution of special cases of (1.3) has not been found to be considered in the known literatures.
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Acknowledgments
This work of the second author is partly supported by NNSF of China (Grant: 10771094) and the Foundation for the Innovation Group of Shenzhen University (Grant: 000133). Y. Wang work is supported by School Foundation of JiangSu Polytechnic University(Grant: JS200801).
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Chen, D., Li, X. & Wang, Y. Dynamics for Nonlinear Difference Equation .
Adv Differ Equ 2009, 235691 (2009). https://doi.org/10.1155/2009/235691
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DOI: https://doi.org/10.1155/2009/235691