Advances in Difference Equations

, 2011:806458 | Cite as

Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity

Open Access
Research Article

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.

Keywords

Periodic Solution Double Resonance Convergent Subsequence Critical Group Morse Index 

1. Introduction

Denote by Open image in new window the set of integers. For a given positive integer Open image in new window , consider the following periodic problem on difference equation:

where Open image in new window is the forward difference operator defined by Open image in new window and Open image in new window for Open image in new window . In this paper, we always assume that

(f1) Open image in new window is Open image in new window -differentiable with respect to the second variable and satisfies Open image in new window for Open image in new window and Open image in new window for Open image in new window .

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 Open image in new window distinct eigenvalues Open image in new window , where Open image in new window , that is, the integer part of Open image in new window .

For Open image in new window with Open image in new window , define Open image in new window . Now, we suppose that

(f2) Open image in new window , and there exists some Open image in new window 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 Open image in new window is a differentiable function satisfying

for some Open image in new window and uniformly for a.e. Open image in new window .

Recently, many authors have studied the boundary value problems on nonlinear differential equations with double resonance(see [2, 3, 4, 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, 7, 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 Open image in new window and Open image in new window :

Open image in new window , and there exists some Open image in new window such that

(f4±)for some Open image in new window ,

Remark 1.2.

The assumption Open image in new window implies Open image in new window and will be employed to control the resonance at infinity. We will need Open image in new window 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 Open image in new window -periodic solutions in each of the following two cases:

Theorem 1.4.

Assume that (f1) and (f3) hold. If there exists Open image in new window with Open image in new window such that Open image in new window , then problem (1.1) has at least two nontrivial Open image in new window -periodic solutions.

Theorem 1.5.

Assume that (f1) and (f3) hold. If there exists Open image in new window such that Open image in new window for Open image in new window . Then problem (1.1) has at least two nontrivial Open image in new window -periodic solutions in each of the following two cases:

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 Open image in new window be a Hilbert space and Open image in new window be a functional satisfying the compactness condition (PS), that is, every sequence Open image in new window such that Open image in new window is bounded and that Open image in new window as Open image in new window contains a convergent subsequence. Denote by Open image in new window the Open image in new window th singular relative homology group of the topological pair Open image in new window with integer coefficients. Let Open image in new window be an isolated critical point of Open image in new window with Open image in new window , Open image in new window , and Open image in new window be a neighborhood of Open image in new window . For Open image in new window , the group

is called the Open image in new window th critical group of Open image in new window at Open image in new window , where Open image in new window .

If the set of critical points of Open image in new window , denoted by Open image in new window , is finite and Open image in new window , the critical groups of Open image in new window at infinity are defined by (see [10])
For Open image in new window , we call Open image in new window the Betti numbers of Open image in new window and define the Morse-type numbers of the pair Open image in new window by

The following facts Open image in new window are derived from [6, Chapter 8].

(2.a)If Open image in new window for some Open image in new window , then there exists Open image in new window such that Open image in new window ,

(2.b)If Open image in new window , then Open image in new window ,

(2.c) Open image in new window ,

(2.d) Open image in new window .

If Open image in new window and Open image in new window is a Fredholm operator and the Morse index Open image in new window and nullity Open image in new window of Open image in new window are finite, then we have

(2.e) Open image in new window for Open image in new window ,

(2.f)If Open image in new window then Open image in new window and if Open image in new window then Open image in new window ,

(2.g) If Open image in new window , then Open image in new window when Open image in new window is local minimum of Open image in new window , while Open image in new window when Open image in new window is the local maximum of Open image in new window .

We say that Open image in new window has a local linking at Open image in new window if there exist the direct sum decompositions Open image in new window and Open image in new window such that

The following results were due to Su [5].

(2.h)Assume that Open image in new window has a local linking at Open image in new window with respect to Open image in new window and Open image in new window . Then,

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.

Equipped with the inner product Open image in new window and norm Open image in new window as follows:

Open image in new window is linearly homeomorphic to Open image in new window . Throughout this paper, we always identify Open image in new window with Open image in new window .

then Open image in new window has the decomposition Open image in new window . In the rest of this paper, the expression Open image in new window for Open image in new window always means Open image in new window , Open image in new window .

Remark 3.1.

From the discussion in [1, Section 2], we see that Open image in new window , Open image in new window , for Open image in new window and Open image in new window if Open image in new window is even or Open image in new window if Open image in new window is odd.

where Open image in new window , Open image in new window . Then, the Fréchet derivative of Open image in new window at Open image in new window , denoted by Open image in new window , can be described as (see [1])

Remark 3.2.

From (3.5) with Open image in new window , we know by computation(or see [1]) that Open image in new window is a critical point of Open image in new window if and only if Open image in new window is a Open image in new window -periodic solution of problem (1.1). Moreover, Open image in new window is Open image in new window differentiable and

where Open image in new window is the derivative of Open image in new window with respect to Open image in new window .

Remark 3.3.

Thus, Open image in new window since Open image in new window possesses of non-degenerate Open image in new window order submatrixes.

Lemma 3.4.

If Open image in new window , Open image in new window and Open image in new window satisfies (3.8), where Open image in new window and Open image in new window , then either Open image in new window or Open image in new window .

Proof.

Setting Open image in new window and Open image in new window , respectively, in (3.8), we have
Comparing the above two equalities, we get
On the other hand, by the definition of Open image in new window and Open image in new window , we have

where Open image in new window . There are two cases to be considered.

Case 1.

Open image in new window for Open image in new window . Then by (3.12), Open image in new window and Open image in new window for Open image in new window , that is, Open image in new window .

Case 2.

There exists Open image in new window such that Open image in new window . By (3.13), we have

If Open image in new window , then Open image in new window which, by (3.13), implies that Open image in new window for Open image in new window , that is, Open image in new window . If Open image in new window , then Open image in new window . This, by (3.12), implies Open image in new window and Open image in new window . Thus, by (3.13), Open image in new window for Open image in new window , that is Open image in new window . The proof is complete.

Set Open image in new window and Open image in new window . The following Lemmas 3.5–3.7 benefit from [4].

Lemma 3.5.

Proof.

From (f2), we have
where the limitation is uniformly in Open image in new window . It follows that for any Open image in new window , there exists Open image in new window such that
Thus, there exists Open image in new window such that
By the assumption on Open image in new window , we have Open image in new window . 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 Open image in new window 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.

If, for the contradiction, (3.23) is false, then there is a subsequence of Open image in new window , called Open image in new window again, and a number Open image in new window , such that Open image in new window , Open image in new window . Then,

where Open image in new window .

By the fact that Open image in new window and Open image in new window are two consecutive eigenvalues of Open image in new window with corresponding eigenspace Open image in new window and Open image in new window , we have Open image in new window and then, the function Open image in new window is strictly decreasing on Open image in new window with Open image in new window as Open image in new window . Besides, Open image in new window . 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 Open image in new window , still called Open image in new window , such that

Proof.

Since Open image in new window as Open image in new window , we can assume (by passing to a subsequence if necessary) that
Thus, (3.16) implies
which implies that there exists a subsequence of Open image in new window , still called Open image in new window , and Open image in new window , such that
Let Open image in new window , then Open image in new window , and, by Lemma 3.6, there is a convergent subsequence of Open image in new window , call it Open image in new window again, such that
Letting Open image in new window in (3.33) and using (3.30) and (3.31), we get

Obviously, if Open image in new window , (3.35) still holds. By Lemma 3.4, Open image in new window or Open image in new window and the proof is complete.

Lemma 3.8.

Proof.

As that in the above proof, we can assume that Open image in new window satisfies (3.28). Noticing that (f3) implies (f2) and by Lemma 3.7, we have two cases to be considered.

Case 1.

By (f3(i)), there exist Open image in new window and Open image in new window such that Open image in new window and Open image in new window for Open image in new window and Open image in new window . Then, for Open image in new window , Open image in new window and Open image in new window ,

where Open image in new window . Since Open image in new window is a finite dimensional vector space and possesses another norm defined by Open image in new window , Open image in new window , which is equivalent to Open image in new window , there exists a positive constant Open image in new window such that Open image in new window , Open image in new window . Thus, by (3.37)–(3.40),

Obviously, if Open image in new window , the above inequality still holds.

Case 2.

Open image in new window as Open image in new window . By using Open image in new window , we can show that Open image in new window in the same way. The proof is complete.

In the rest of this section, we will use the facts ( Open image in new window )–( Open image in new window ) stated in Section 2 to complete the proofs.

Lemma 3.9.

Let Open image in new window satisfy (f1) and (f3). Then, for every Open image in new window , Open image in new window satisfies the (PS) condition and

Proof.

First we have the following claim:

Claim 1.

For any sequences Open image in new window and Open image in new window if Open image in new window as Open image in new window , then Open image in new window is bounded.

In fact, if Open image in new window is unbounded, there exists a subsequence, still called Open image in new window , such that Open image in new window as Open image in new window . By Lemma 3.8, there exists a subsequence, still called Open image in new window , such that Open image in new window or Open image in new window .

Note that Open image in new window , Open image in new window , it follows that Open image in new window . This contradiction proves Claim 1.

Setting Open image in new window in Claim 1, we see that Open image in new window satisfies (PS) condition. Now, we start to prove (3.42). Define a functional Open image in new window as

Claim 2.

In fact, if Claim 2 is not true, there exists Open image in new window and Open image in new window such that Open image in new window and Open image in new window as Open image in new window , which contradict Claim 1.

The chain rule for differentiation reads Open image in new window . Thus,
On the other hand,
Note that Open image in new window is the unique critical point of Open image in new window with Morse index Open image in new window (see Remark 3.1) and nullity Open image in new window . 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 Open image in new window , implies that there exists Open image in new window with

Since Open image in new window , we have Open image in new window . Denote by Open image in new window and Open image in new window the Morse index and nullity of Open image in new window . By Open image in new window , we get Open image in new window .

Denote Open image in new window . Then, from (3.7) and Remark 3.3, we see that Open image in new window .

which, by comparing with (3.51), implies that Open image in new window . Besides, Open image in new window since Open image in new window . Assume, for the contradiction, that Open image in new window is the unique nontrivial critical point of Open image in new window , then Open image in new window . If Open image in new window or Open image in new window , we have, by ( Open image in new window ),

from which, ( Open image in new window ) reads Open image in new window , a contradiction.

If Open image in new window , then Open image in new window and Open image in new window . Since Open image in new window , we have Open image in new window . Thus, ( Open image in new window ) with Open image in new window reads Open image in new window , also a contradiction.

which, by comparing with (3.51), implies that Open image in new window . Besides, Open image in new window since Open image in new window . Assume, for the contradiction, that Open image in new window is the unique nontrivial critical point of Open image in new window , then Open image in new window . If Open image in new window or Open image in new window , then (3.53) holds, from which, Open image in new window ) reads

respectively, which implies that Open image in new window . Then, ( Open image in new window ) reads (3.55), also a contradiction. The proof is complete.

Proof of Theorem 1.4.

As above, there exists Open image in new window with the Morse index Open image in new window , and nullity Open image in new window satisfying Open image in new window , Open image in new window , and (3.51) holds.

On the other hand, Open image in new window is a nondegenerate critical point of Open image in new window with Morse index, denoted by Open image in new window . Thus, Open image in new window and Open image in new window since Open image in new window , which, by comparing with (3.51), implies that Open image in new window .

Assume for the contradiction, that Open image in new window is unique nontrivial critical point of Open image in new window , then Open image in new window . If Open image in new window or Open image in new window , then (3.53) holds and ( Open image in new window ) reads the contradicition Open image in new window .

Now, we consider the case Open image in new window where we have Open image in new window and Open image in new window with (3.56). Since Open image in new window , we know that either Open image in new window or Open image in new window . If Open image in new window , ( Open image in new window ) with Open image in new window reads contradiction Open image in new window . If Open image in new window , by similar argument, we can get (3.57). Thus Open image in new window and ( Open image in new window ) reads the contradiction Open image in new window . The proof is complete.

The proof of the following lemma is similar to that of ([12]) and is omitted.

Lemma 3.10.

Let Open image in new window satisfy Open image in new window Open image in new window or Open image in new window . Then Open image in new window has a local linking at Open image in new window with respect to the decomposition Open image in new window , where Open image in new window (or Open image in new window , respectively).

Proof of Theorem 1.5.

Now Open image in new window . Thus, Open image in new window is a degenerate critical point of Open image in new window . Let Open image in new window and Open image in new window denote the Morse index and nullity of 0. By Lemma 3.10 and ( Open image in new window ), we have

where Open image in new window or Open image in new window corresponding to the case ( Open image in new window ) or the case ( Open image in new window ), 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, 3, 4, 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 Open image in new window and Open image in new window , 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.

Notes

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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Copyright information

© X. Zhang and D.Wang. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.School of Mathematical SciencesSouth China Normal UniversityGuangzhouChina

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