Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity

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:

(1.1)

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

(1.2)

possess distinct eigenvalues , where , that is, the integer part of .

For with , define . Now, we suppose that

(f₂), and there exists some such that

(1.3)

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

(1.4)

whose solvability has been studied in [2], where is a differentiable function satisfying

(1.5)

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

(1.6)

(f₄±)for some ,

(1.7)

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

(2.1)

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])

(2.2)

For , we call the Betti numbers of and define the Morse-type numbers of the pair by

(2.3)

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

(2.4)

The following results were due to Su [5].

(2.h)Assume that has a local linking at with respect to and . Then,

(2.5)

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

(3.1)

Equipped with the inner product and norm as follows:

(3.2)

is linearly homeomorphic to . Throughout this paper, we always identify with .

Define the operator by and denote , , where is the identity operator. Set

(3.3)

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

(3.4)

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

(3.5)

where and

(3.6)

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

(3.7)

where is the derivative of with respect to .

Let , and consist of satisfying

(3.8)

Remark 3.3.

is the solution space of the system , , where

(3.9)

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

(3.10)

Comparing the above two equalities, we get

(3.11)

which, by , , implies that

(3.12)

On the other hand, by the definition of and , we have

(3.13)

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

(3.14)

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,

(3.15)

Proof.

From (f₂), we have

(3.16)

where the limitation is uniformly in . It follows that for any , there exists such that

(3.17)

Thus, there exists such that

(3.18)

By the assumption on , we have . It follows from (3.5) that

(3.19)

which, combining with (3.18), implies that

(3.20)

By using, Holder inequality on the above two summations, we get

(3.21)

which leads to

(3.22)

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

(3.23)

Proof.

Since , , and are invariant with respect to , we have

(3.24)

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

(3.25)

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),

(3.26)

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

(3.27)

Proof.

Since as , we can assume (by passing to a subsequence if necessary) that

(3.28)

Thus, (3.16) implies

(3.29)

which implies that there exists a subsequence of , still called , and , such that

(3.30)

Let , then , and, by Lemma 3.6, there is a convergent subsequence of , call it again, such that

(3.31)

To prove (3.27), we only need to show that or . For every , we have as , that is,

(3.32)

If as for , then we can rewrite (3.32) as

(3.33)

Letting in (3.33) and using (3.30) and (3.31), we get

(3.34)

Since for , by setting for , we rewrite (3.34) as

(3.35)

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

(3.36)

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

(3.37)

If , then and are bounded for and . It follows that as for and

(3.38)

By (f₃(i)), there exist and such that and for and . Then, for , and ,

(3.39)

Choose such that for and . It follows that

(3.40)

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),

(3.41)

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

(3.42)

Proof.

First we have the following claim:

Claim 1.

For any sequences andifas, 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

(3.43)

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

(3.44)

Claim 2.

There exist such that

(3.45)

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

(3.46)

The chain rule for differentiation reads . Thus,

(3.47)

and , , which implies that , . Then, the flow is well defined on and is a homeomorphism of to and (see [11])

(3.48)

On the other hand,

(3.49)

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

(3.50)

The proof is completed.

Proof of Theorem 1.3.

By lemma 3.9, we get (3.42) which, by , implies that there exists with

(3.51)

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 (),

(3.52)

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 (),

(3.53)

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 (),

(3.54)

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

(3.55)

a contradiction. If , then and, by (),

(3.56)

Note that , we have . Thus, () with and with reads

(3.57)

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

(3.58)

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.