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Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity
Advances in Difference Equations volume 2011, Article number: 806458 (2011)
Abstract
By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity. To the best of our knowledge, investigations on double-resonant difference systems have not been seen in the literature.
1. Introduction
Denote by the set of integers. For a given positive integer
, consider the following periodic problem on difference equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ1_HTML.gif)
where is the forward difference operator defined by
and
for
. In this paper, we always assume that
(f1) is
-differentiable with respect to the second variable and satisfies
for
and
for
.
As a natural phenomenon, resonance may take place in the real world such as machinery, construction, electrical engineering, and communication. In a system described by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity. It is known (see [1]) that the eigenvalue problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ2_HTML.gif)
possess distinct eigenvalues
, where
, that is, the integer part of
.
For with
, define
. Now, we suppose that
(f2), and there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ3_HTML.gif)
Remark 1.1.
The assumption (f2) characterizes problem (1.1) as double resonant between two consecutive eigenvalues at infinity. Problem (1.1) is the discrete analogue of the differential equation with double resonance
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ4_HTML.gif)
whose solvability has been studied in [2], where is a differentiable function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ5_HTML.gif)
for some and uniformly for a.e.
.
Recently, many authors have studied the boundary value problems on nonlinear differential equations with double resonance(see [2–5]). It is well known that in different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, and economics, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. For this reason, in recent years the solvability of nonlinear difference equations have been extensively investigated(see [1, 6–8] and the references cited therein). However, to the best of our knowledge, investigations on double resonant difference systems have not been seen in the literature.
In this paper, several theorems on the multiplicity of the periodic solutions to the double resonant system (1.1) are obtained via variational methods and Morse theory. The research here was mainly motivated by the works [2, 4].
We need the following assumptions and
:
, and there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ6_HTML.gif)
(f4±)for some ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ7_HTML.gif)
Remark 1.2.
The assumption implies
and will be employed to control the resonance at infinity. We will need
in the case that (1.1) is also resonant at the origin.
Now, the main results of this paper are stated as follows.
Theorem 1.3.
Assume that (f1) and (f3) hold. Then, problem (1.1) has at least two nontrivial -periodic solutions in each of the following two cases:
-
(i)
and
for
,
-
(ii)
and
for
.
Theorem 1.4.
Assume that (f1) and (f3) hold. If there exists with
such that
, then problem (1.1) has at least two nontrivial
-periodic solutions.
Theorem 1.5.
Assume that (f1) and (f3) hold. If there exists such that
for
. Then problem (1.1) has at least two nontrivial
-periodic solutions in each of the following two cases:
-
(i)
and
with
,
-
(ii)
and
with
.
In Section 3, we will prove the main results, before which some preliminary results on Morse theory will be collected in Section 2. Some fundamental facts relative to (1.1) revealed here will benefit the further investigations in this direction, which will be remarked in Section 4.
2. Preliminary Results on Critical Groups
In this section, we recall some basic facts in Morse theory which will be used in the proof of the main results. For the systematic discussion on Morse theory, we refer the reader to the monograph [9] and the references cited therein. Let be a Hilbert space and
be a functional satisfying the compactness condition (PS), that is, every sequence
such that
is bounded and that
as
contains a convergent subsequence. Denote by
the
th singular relative homology group of the topological pair
with integer coefficients. Let
be an isolated critical point of
with
,
, and
be a neighborhood of
. For
, the group
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ8_HTML.gif)
is called the th critical group of
at
, where
.
If the set of critical points of , denoted by
, is finite and
, the critical groups of
at infinity are defined by (see [10])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ9_HTML.gif)
For , we call
the Betti numbers of
and define the Morse-type numbers of the pair
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ10_HTML.gif)
The following facts are derived from [6, Chapter 8].
(2.a)If for some
, then there exists
such that
,
(2.b)If , then
,
(2.c),
(2.d).
If and
is a Fredholm operator and the Morse index
and nullity
of
are finite, then we have
(2.e) for
,
(2.f)If then
and if
then
,
(2.g) If , then
when
is local minimum of
, while
when
is the local maximum of
.
We say that has a local linking at
if there exist the direct sum decompositions
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ11_HTML.gif)
The following results were due to Su [5].
(2.h)Assume that has a local linking at
with respect to
and
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ12_HTML.gif)
3. Proofs of Main Results
In this section, we will establish the variational structure relative to problem (1.1) and prove the main results via Morse theory.
Denote and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ13_HTML.gif)
Equipped with the inner product and norm
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ14_HTML.gif)
is linearly homeomorphic to
. Throughout this paper, we always identify
with
.
Define the operator by
and denote
,
, where
is the identity operator. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ15_HTML.gif)
then has the decomposition
. In the rest of this paper, the expression
for
always means
,
.
Remark 3.1.
From the discussion in [1, Section 2], we see that ,
, for
and
if
is even or
if
is odd.
Define a family of functionals ,
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ16_HTML.gif)
where ,
. Then, the Fréchet derivative of
at
, denoted by
, can be described as (see [1])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ17_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ18_HTML.gif)
Remark 3.2.
From (3.5) with , we know by computation(or see [1]) that
is a critical point of
if and only if
is a
-periodic solution of problem (1.1). Moreover,
is
differentiable and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ19_HTML.gif)
where is the derivative of
with respect to
.
Let ,
and
consist of
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ20_HTML.gif)
Remark 3.3.
is the solution space of the system
,
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ21_HTML.gif)
Thus, since
possesses of non-degenerate
order submatrixes.
Lemma 3.4.
If ,
and
satisfies (3.8), where
and
, then either
or
.
Proof.
Setting and
, respectively, in (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ22_HTML.gif)
Comparing the above two equalities, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ23_HTML.gif)
which, by ,
, implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ24_HTML.gif)
On the other hand, by the definition of and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ25_HTML.gif)
where . There are two cases to be considered.
Case 1.
for
. Then by (3.12),
and
for
, that is,
.
Case 2.
There exists such that
. By (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ26_HTML.gif)
If , then
which, by (3.13), implies that
for
, that is,
. If
, then
. This, by (3.12), implies
and
. Thus, by (3.13),
for
, that is
. The proof is complete.
Set and
. The following Lemmas 3.5–3.7 benefit from [4].
Lemma 3.5.
Assume that (f1) and (f2) hold. Let and
satisfy
and
as
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ27_HTML.gif)
Proof.
From (f2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ28_HTML.gif)
where the limitation is uniformly in . It follows that for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ29_HTML.gif)
Thus, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ30_HTML.gif)
By the assumption on , we have
. It follows from (3.5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ31_HTML.gif)
which, combining with (3.18), implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ32_HTML.gif)
By using, Holder inequality on the above two summations, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ33_HTML.gif)
which leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ34_HTML.gif)
Note that is arbitrarily small, we get (3.15), and the proof is complete.
Lemma 3.6.
Under the conditions of Lemma 3.5, one further has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ35_HTML.gif)
Proof.
Since ,
, and
are invariant with respect to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ36_HTML.gif)
If, for the contradiction, (3.23) is false, then there is a subsequence of , called
again, and a number
, such that
,
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ37_HTML.gif)
where .
By the fact that and
are two consecutive eigenvalues of
with corresponding eigenspace
and
, we have
and then, the function
is strictly decreasing on
with
as
. Besides,
. So, by (3.25),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ38_HTML.gif)
This contradict to (3.15) and the proof is complete.
Lemma 3.7.
Under the assumption of Lemma 3.5, there exists a subsequence of , still called
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ39_HTML.gif)
Proof.
Since as
, we can assume (by passing to a subsequence if necessary) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ40_HTML.gif)
Thus, (3.16) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ41_HTML.gif)
which implies that there exists a subsequence of , still called
, and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ42_HTML.gif)
Let , then
, and, by Lemma 3.6, there is a convergent subsequence of
, call it
again, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ43_HTML.gif)
To prove (3.27), we only need to show that or
. For every
, we have
as
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ44_HTML.gif)
If as
for
, then we can rewrite (3.32) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ45_HTML.gif)
Letting in (3.33) and using (3.30) and (3.31), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ46_HTML.gif)
Since for
, by setting
for
, we rewrite (3.34) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ47_HTML.gif)
Obviously, if , (3.35) still holds. By Lemma 3.4,
or
and the proof is complete.
Lemma 3.8.
Assume that and
hold. Let
and
satisfy
and
as
. Then, there exists a subsequence of
, still called
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ48_HTML.gif)
Proof.
As that in the above proof, we can assume that satisfies (3.28). Noticing that (f3) implies (f2) and by Lemma 3.7, we have two cases to be considered.
Case 1.
as
. We have
as
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ49_HTML.gif)
If , then
and
are bounded for
and
. It follows that
as
for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ50_HTML.gif)
By (f3(i)), there exist and
such that
and
for
and
. Then, for
,
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ51_HTML.gif)
Choose such that
for
and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ52_HTML.gif)
where . Since
is a finite dimensional vector space and possesses another norm defined by
,
, which is equivalent to
, there exists a positive constant
such that
,
. Thus, by (3.37)–(3.40),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ53_HTML.gif)
Obviously, if , the above inequality still holds.
Case 2.
as
. By using
, we can show that
in the same way. The proof is complete.
In the rest of this section, we will use the facts ()–(
) stated in Section 2 to complete the proofs.
Lemma 3.9.
Let satisfy (f1) and (f3). Then, for every
,
satisfies the (PS) condition and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ54_HTML.gif)
Proof.
First we have the following claim:
Claim 1.
For any sequences and
if
as
, then
is bounded.
In fact, if is unbounded, there exists a subsequence, still called
, such that
as
. By Lemma 3.8, there exists a subsequence, still called
, such that
or
.
On the other hand, as
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ55_HTML.gif)
Note that ,
, it follows that
. This contradiction proves Claim 1.
Setting in Claim 1, we see that
satisfies (PS) condition. Now, we start to prove (3.42). Define a functional
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ56_HTML.gif)
Claim 2.
There exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ57_HTML.gif)
In fact, if Claim 2 is not true, there exists and
such that
and
as
, which contradict Claim 1.
Noticing that , we set
. Then,
implies
. Consider the flow
generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ58_HTML.gif)
The chain rule for differentiation reads . Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ59_HTML.gif)
and ,
, which implies that
,
. Then, the flow
is well defined on
and
is a homeomorphism of
to
and (see [11])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ60_HTML.gif)
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ61_HTML.gif)
Note that is the unique critical point of
with Morse index
(see Remark 3.1) and nullity
. Then, by (2.b), (2.f) and (3.48), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ62_HTML.gif)
The proof is completed.
Proof of Theorem 1.3.
By lemma 3.9, we get (3.42) which, by , implies that there exists
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ63_HTML.gif)
Since , we have
. Denote by
and
the Morse index and nullity of
. By
, we get
.
Denote . Then, from (3.7) and Remark 3.3, we see that
.
In Case (i), is a local minimum of
, hence, by (
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ64_HTML.gif)
which, by comparing with (3.51), implies that . Besides,
since
. Assume, for the contradiction, that
is the unique nontrivial critical point of
, then
. If
or
, we have, by (
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ65_HTML.gif)
from which, () reads
, a contradiction.
If , then
and
. Since
, we have
. Thus, (
) with
reads
, also a contradiction.
In Case (ii), is a local maximum of
, hence, by (
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ66_HTML.gif)
which, by comparing with (3.51), implies that . Besides,
since
. Assume, for the contradiction, that
is the unique nontrivial critical point of
, then
. If
or
, then (3.53) holds, from which,
) reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ67_HTML.gif)
a contradiction. If , then
and, by (
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ68_HTML.gif)
Note that , we have
. Thus, (
) with
and with
reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ69_HTML.gif)
respectively, which implies that . Then, (
) reads (3.55), also a contradiction. The proof is complete.
Proof of Theorem 1.4.
As above, there exists with the Morse index
, and nullity
satisfying
,
, and (3.51) holds.
On the other hand, is a nondegenerate critical point of
with Morse index, denoted by
. Thus,
and
since
, which, by comparing with (3.51), implies that
.
Assume for the contradiction, that is unique nontrivial critical point of
, then
. If
or
, then (3.53) holds and (
) reads the contradicition
.
Now, we consider the case where we have
and
with (3.56). Since
, we know that either
or
. If
, (
) with
reads contradiction
. If
, by similar argument, we can get (3.57). Thus
and (
) reads the contradiction
. The proof is complete.
The proof of the following lemma is similar to that of ([12]) and is omitted.
Lemma 3.10.
Let satisfy
or
. Then
has a local linking at
with respect to the decomposition
, where
(or
, respectively).
Proof of Theorem 1.5.
Now . Thus,
is a degenerate critical point of
. Let
and
denote the Morse index and nullity of 0. By Lemma 3.10 and (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F806458/MediaObjects/13662_2010_Article_73_Equ70_HTML.gif)
where or
corresponding to the case (
) or the case (
), respectively. The rest of the proof is similar and is omitted. The proof is complete.
4. Conclusion and Future Directions
It is known that there have been many investigations on the solvability of elliptic equations with double-resonance via variational methods, where the so called unique continuation property of the Laplace operator, proved by Robinson [4], plays an important role in proving the compactness of the corresponding functional (see [2–5] and the references cited therein). In this paper, the solvability of the periodic problem on difference equations with double resonance is first studied and the "unique continuation property" of the second-order difference operator is derived by proving Lemma 3.4.
In addition, under the double resonance assumption and
, some fundamental facts relative to (1.1) are revealed in Lemmas 3.5–3.7, on which, further investigations, employing new restrictions different from (f3) and (f4), may be based.
On the observations as above, it is reasonable to believe that the research in this paper will benefit the future study in this direction.
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Acknowledgments
The authors are grateful for the referee's careful reviewing and helpful comments. Also the authors would like to thank Professor Su Jiabao for his helpful suggestions. This work is supported by NSFC(10871005) and BJJW(KM200610028001).
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Zhang, X., Wang, D. Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity. Adv Differ Equ 2011, 806458 (2011). https://doi.org/10.1155/2011/806458
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DOI: https://doi.org/10.1155/2011/806458