- Research Article
- Open Access
Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity
© X. Zhang and D.Wang. 2011
- Received: 7 February 2010
- Accepted: 6 November 2010
- Published: 19 January 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.
- Periodic Solution
- Double Resonance
- Convergent Subsequence
- Critical Group
- Morse Index
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 .
possess distinct eigenvalues , where , that is, the integer part of .
For with , define . Now, we suppose that
(f2) , and there exists some such that
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 ,
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.
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 .
Assume that (f1) and (f3) hold. If there exists with such that , then problem (1.1) has at least two nontrivial -periodic solutions.
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.
is called the th critical group of at , where .
The following facts are derived from [6, Chapter 8].
(2.a)If for some , then there exists such that ,
(2.b)If , then ,
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 .
The following results were due to Su .
(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.
is linearly homeomorphic to . Throughout this paper, we always identify with .
then has the decomposition . In the rest of this paper, the expression for always means , .
From the discussion in [1, Section 2], we see that , , for and if is even or if is odd.
where is the derivative of with respect to .
Thus, since possesses of non-degenerate order submatrixes.
If , and satisfies (3.8), where and , then either or .
where . There are two cases to be considered.
for . Then by (3.12), and for , that is, .
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 .
Note that is arbitrarily small, we get (3.15), and the proof is complete.
This contradict to (3.15) and the proof is complete.
Obviously, if , (3.35) still holds. By Lemma 3.4, or and the proof is complete.
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.
By (f3(i)), there exist and such that and for and . Then, for , and ,
Obviously, if , the above inequality still holds.
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.
First we have the following claim:
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 .
Note that , , it follows that . This contradiction proves Claim 1.
In fact, if Claim 2 is not true, there exists and such that and as , which contradict Claim 1.
The proof is completed.
Proof of Theorem 1.3.
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 .
from which, ( ) reads , a contradiction.
If , then and . Since , we have . Thus, ( ) with reads , also a contradiction.
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 () and is omitted.
Let satisfy or . Then has a local linking at with respect to the decomposition , where (or , respectively).
Proof of Theorem 1.5.
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 , 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.
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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