Advances in Difference Equations volume 2011, Article number: 806458 (2011)
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.
Denote by 
the set of integers. For a given positive integer 
, consider the following periodic problem on difference equation:
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
possess 
distinct eigenvalues 
, where 
, that is, the integer part of 
.
For 
with 
, define 
. Now, we suppose that
(f2)
, and there exists some 
such that
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
whose solvability has been studied in [2], where 
is a differentiable function satisfying
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
(f4±)for some 
,
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:
and
for 
,
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:
and
with 
,
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.
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
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])
For 
, we call 
the Betti numbers of 
and define the Morse-type numbers of the pair 
by
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
The following results were due to Su [5].
(2.h)Assume that 
has a local linking at 
with respect to 
and 
. Then,
In this section, we will establish the variational structure relative to problem (1.1) and prove the main results via Morse theory.
Denote 
and
Equipped with the inner product 
and norm 
as follows:
is linearly homeomorphic to 
. Throughout this paper, we always identify 
with 
.
Define the operator 
by 
and denote 
, 
, where 
is the identity operator. Set
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
where 
, 
. Then, the Fréchet derivative of 
at 
, denoted by 
, can be described as (see [1])
where 
and
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
where 
is the derivative of 
with respect to 
.
Let 
, 
and 
consist of 
satisfying
Remark 3.3.
is the solution space of the system 
, 
, where
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
Comparing the above two equalities, we get
which, by 
, 
, implies that
On the other hand, by the definition of 
and 
, we have
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
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,
Proof.
From (f2), we have
where the limitation is uniformly in 
. It follows that for any 
, there exists 
such that
Thus, there exists 
such that
By the assumption on 
, we have 
. It follows from (3.5) that
which, combining with (3.18), implies that
By using, Holder inequality on the above two summations, we get
which leads to
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
Proof.
Since 
, 
, and 
are invariant with respect to 
, we have
If, for the contradiction, (3.23) is false, then there is a subsequence of 
, called 
again, and a number 
, such that 
, 
. Then,
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),
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
Proof.
Since 
as 
, we can assume (by passing to a subsequence if necessary) that
Thus, (3.16) implies
which implies that there exists a subsequence of 
, still called 
, and 
, such that
Let 
, then 
, and, by Lemma 3.6, there is a convergent subsequence of 
, call it 
again, such that
To prove (3.27), we only need to show that 
or 
. For every 
, we have 
as 
, that is,
If 
as 
for 
, then we can rewrite (3.32) as
Letting 
in (3.33) and using (3.30) and (3.31), we get
Since 
for 
, by setting 
for 
, we rewrite (3.34) as
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
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
If 
, then 
and 
are bounded for 
and 
. It follows that 
as 
for 
and
By (f3(i)), there exist 
and 
such that 
and 
for 
and 
. Then, for 
, 
and 
,
Choose 
such that 
for 
and 
. It follows that
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),
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
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
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
Claim 2.
There exist 
such that
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
The chain rule for differentiation reads 
. Thus,
and 
, 
, which implies that 
, 
. Then, the flow 
is well defined on 
and 
is a homeomorphism of 
to 
and (see [11])
On the other hand,
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
The proof is completed.
Proof of Theorem 1.3.
By lemma 3.9, we get (3.42) which, by 
, implies that there exists 
with
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 (
),
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 (
),
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 (
),
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
a contradiction. If 
, then 
and, by (
),
Note that 
, we have 
. Thus, (
) with 
and with 
reads
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
where 
or 
corresponding to the case (
) or the case (
), respectively. The rest of the proof is similar and is omitted. The proof is complete.
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.
Yu J, Long Y, Guo Z: Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation. Journal of Dynamics and Differential Equations 2004,16(2):575-586. 10.1007/s10884-004-4292-2
Su J, Zhao L: Multiple periodic solutions of ordinary differential equations with double resonance. Nonlinear Analysis: Theory, Methods & Applications 2009,70(4):1520-1527. 10.1016/j.na.2008.02.031
Fabry C, Fonda A: Periodic solutions of nonlinear differential equations with double resonance. Annali di Matematica Pura ed Applicata 1990, 157: 99-116. 10.1007/BF01765314
Robinson S: Double resonance in semilinear elliptic boundary value problems over bounded and unbounded domains. Nonlinear Analysis: Theory, Methods & Applications 1993,21(6):407-424. 10.1016/0362-546X(93)90125-C
Su J: Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues. Nonlinear Analysis: Theory, Methods & Applications 2002,48(6):881-895. 10.1016/S0362-546X(00)00221-2
Guo Z, Yu J: The existence of periodic and subharmonic solutions of subquadratic second order difference equations. Journal of the London Mathematical Society 2003,68(2):419-430. 10.1112/S0024610703004563
Guo Z, Yu J: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Science in China A 2003,46(4):506-515.
Zhu B, Yu J: Multiple positive solutions for resonant difference equations. Mathematical and Computer Modelling 2009,49(9-10):1928-1936. 10.1016/j.mcm.2008.09.009
Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.
Bartsch T, Li S: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Analysis: Theory, Methods & Applications 1997,28(3):419-441. 10.1016/0362-546X(95)00167-T
Perera K, Schechter M: Nontrivial solutions of elliptic semilinear equations at resonance. Manuscripta Mathematica 2000,101(3):301-311. 10.1007/s002290050217
Li SJ, Liu JQ: Nontrivial critical points for asymptotically quadratic function. Journal of Mathematical Analysis and Applications 1992,165(2):333-345. 10.1016/0022-247X(92)90044-E
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/806458