Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity

doi:10.1155/2011/806458

Research Article

Multiple Periodic Solutions for Difference

Equations with Double Resonance at Infinity

Xiaosheng Zhang

_{and Duo Wang}

1_{School of Mathematical Sciences, Capital Normal University, Beijing 100048, China} 2_{School of Mathematical Sciences, Peking University, Beijing 100875, China}

3_{School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China}

Correspondence should be addressed to Xiaosheng Zhang,[email protected] Received 6 November 2010; Accepted 19 January 2011

Academic Editor: John Graef

Copyrightq2011 X. Zhang and D. Wang. 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.

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 byZthe set of integers. For a given positive integerp, consider the following periodic problem on difference equation:

−Δ2_{xk−1 fk, xk,}

xkpxk,

k∈Z, 1.1

whereΔis the forward difference operator defined byΔxk xk1−xkandΔ2_xk

ΔΔxkfork∈Z. In this paper, we always assume that

f1f :Z×R → RisC1-differentiable with respect to the second variable and satisfies

fkp, t fk, tfork, t∈Z×Randfk,0≡0 fork∈Z.

by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity. It is knownsee1 that the eigenvalue problem

−Δ2_{xk−1 λxk, xkpxk, k∈Z 1.2}

possessp11 distinct eigenvaluesλl 4sin2lπ/p, l 0,1,2, . . . , p1, wherep1 p/2 , that is, the integer part ofp/2.

Fora, b∈Zwitha < b, defineZa, b {a, a1, . . . b}. Now, we suppose that

f2p >3, and there exists someh∈Z0, p1−1such that

λh≤lim inf

|t| → ∞

fk, t

t ≤lim sup_{|t| → ∞} fk, t

t ≤λh1 fork∈Z

1, p. 1.3

Remark 1.1. The assumptionf2characterizes problem1.1as double resonant between two

consecutive eigenvalues at infinity. Problem1.1is the discrete analogue of the differential equation with double resonance

−z¨t gt, z,

z0−z2π z˙0−z˙2π 0, 1.4

whose solvability has been studied in 2 , where g : 0,2π ×R → R is a differentiable function satisfying

h2 ≤lim inf

|z| → ∞

gt, z

z ≤lim sup_{|z| → ∞} gt, z

z ≤h1

2_{, 1.5}

for someh∈N∪ {0}and uniformly for a.e.t∈0,2π .

Recently, many authors have studied the boundary value problems on nonlinear differential equations with double resonancesee 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 investigatedsee1,6–8 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.

We need the following assumptionsf3andf4:

f3p >3, and there exists someh∈Z0, p1−1such that

i lim inf

|t| → ∞ |t|

_{fk, t}

t −λh

>0,

ii lim sup

|t| → ∞ |

t|

_{fk, t}

t −λh1

<0,

fork∈Z1, p 1.6

f4±for somem∈Z0, p1,

±

_t

0

fk, s−λms

ds≥0 for|t|>0 small, k∈Z1, p. 1.7

Remark 1.2. The assumptionf3impliesf2and will be employed to control the resonance

at infinity. We will needf4in the case that1.1is 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 nontrivialp

-periodic solutions in each of the following two cases:

ih∈Z1, p1−1andfk,0< λ0fork∈Z1, p,

iih∈Z0, p1−2andfk,0> λp1fork∈Z1, p.

Theorem 1.4. Assume that (f1) and (f3) hold. If there existsm ∈Z0, p1−1withm /hsuch that

λm < fk,0 < λm1 for k ∈ Z1, p, then problem1.1has at least two nontrivialp-periodic

solutions.

Theorem 1.5. Assume that (f1) and (f3) hold. If there existsm∈Z0, p1−1such thatfk,0≡λm

fork ∈ Z1, p. Then problem 1.1 has at least two nontrivialp-periodic solutions in each of the following two cases:

ih∈Z0, p1−2andf₄withm /h,

iih∈Z1, p1−1andf₄−withm /h1.

In Section3, 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 Section4.

2. Preliminary Results on Critical Groups

be a functional satisfying the compactness conditionPS, that is, every sequence{un}such that{Φun}is bounded and thatΦun → 0 asn → ∞contains a convergent subsequence. Denote byHqX, Ytheqth singular relative homology group of the topological pairX, Y with integer coefficients. Letu0be an isolated critical point ofΦwithΦu0 c,c∈R, andU be a neighborhood ofu0. Forq∈N∪ {0}, the group

CqΦ, u0:HqΦc∩U,Φc∩U\ {u0} 2.1

is called theqth critical group ofΦatu0, whereΦc{u∈H: Φu≤c}.

If the set of critical points ofΦ, denoted byK : {u ∈ H : Φu 0}, is finite and

a <infΦK, the critical groups ofΦat infinity are defined bysee10

CqΦ,∞:HqH,Φa, q∈N∪ {0}. 2.2

Forq∈N∪ {0}, we callβq :dimCqΦ,∞the Betti numbers ofΦand define the Morse-type numbers of the pairH,Φa_by

Mq:MqH,Φa

u∈K

dimCqΦ, u. _2.3

The following facts2.a–2.gare derived from6, Chapter 8 .

2.aIfCμΦ,∞0 for someμ∈N∪{0}, then there existsx0∈ Ksuch thatCμΦ, x0 0,

2.bIfK{x0}, thenCqΦ,∞∼CqΦ, x0,

2.cq_j0−1q−jMj ≥ q

j0−1 q−j_β

j forq∈N∪ {0},

2.d∞_j0−1jMj∞j0−1jβj.

Ifx0 ∈ KandΦ x0is a Fredholm operator and the Morse indexμ0and nullityv0of

x0are finite, then we have

2.edimCqΦ, x0∼0 forq /∈Zμ0, μ0ν0,

2.fIfCμ0Φ, x00 thenCqΦ, x0∼ δq,μ0Zand ifCμ0ν0Φ, x0 0 thenCqΦ, x0 ∼

δq,μ0ν0Z,

2.gIfm:dimH <∞, thenCqΦ, x0∼δq,0Zwhenx0is local minimum ofΦ, while

CqΦ, x0∼δq,mZwhenx0is the local maximum ofΦ.

We say thatΦhas a local linking atx0∈ Kif there exist the direct sum decompositions

HH⊕H−and >0 such that

Φx>Φx0 ifx−x0∈H, 0<x−x0 ≤,

Φx≤Φx0 ifx−x0∈H−, x−x0 ≤.

2.4

2.hAssume thatΦhas a local linking at x0 ∈ K with respect toH H ⊕H− and

kdimH−<∞. Then,

CqΦ, x0∼δq,μ0Z, ifkμ0,

CqΦ, x0∼δq,μ0v0Z, ifkμ0v0.

2.5

3. Proofs of Main Results

In this section, we will establish the variational structure relative to problem1.1and prove the main results via Morse theory.

DenoteX:{x{xk}_k∈Z:xk∈Rfork∈Z}and

E: x∈X:xkpxkfork∈Z. 3.1

Equipped with the inner product·,·and norm · as follows:

x, y

p

k1

xkyk, x

_p

k1

|xk|2

1/2

, x, y∈E, 3.2

E,·,·is linearly homeomorphic toRp. Throughout this paper, we always identifyx∈ E

withx x1, x2, . . . , xpT∈Rp_.

Define the operator−Δ2 _{: E→ Eby−Δ}2_{x {−Δ}2_{xk−1}, x ∈ Eand denoteE} l ker−Δ2_−λ

lI,l0,1, . . . , p1, whereIis the identity operator. Set

E−⊕h_l−₀1El, E⊕h_l01El⊥, EvE−⊕E, 3.3

then Ehas the decompositionE Eh_⊕Eh1_⊕Ev_{. In the rest of this paper, the expression}

xxh_xh1_xv_{forx∈Ealways meansx}†_∈E†_{,†h, h1, v.}

Remark 3.1. From the discussion in1, Section 2, we see that dimE0 _{1, dimE}l _{2, for}

l1,2, . . . , p1−1 and dimEp11 ifpis even or dimEp1 2 ifpis odd. Define a family of functionalsJs: E → R,s∈0,1 by

Jsx − 1 2

Δ2_{x, x−}1−s

4 λhλh1x 2_−s

p

k1

Fk, xk forx∈E, 3.4

whereFk, t ₀tfk, ξdξ,k, t ∈Z1, p×R. Then, the Fr´echet derivative ofJs atx∈ E, denoted byJ_sx, can be described assee1

J_sx, y−Δ2x, y−

p

k1

wheres∈0,1 and

Thus, dimE0≤2 sinceBpossesses of non-degeneratep−2order submatrixes.

Lemma 3.4. Ifλh ≤αk≤λh1,k∈Z1, pandwwhwh1satisfies3.8, wherewh∈Ehand

Comparing the above two equalities, we get

which, byαk∈λh, λh1 ,k∈Z1, p, implies that

αk−λhwhk≡0, αk−λh1wh1k≡0 for k∈Z

1, p. 3.12

On the other hand, by the definition ofwhandwh1, we have

λ†w†k 2w†k−w†k−1−w†k1,

w†kpw†k fork∈Z1, p,

3.13

where†h, h1. There are two cases to be considered.

Case 1. wh_{k/0 fork∈Z1, p. Then by3.12,α}

kλhandwh1k 0 fork∈Z1, p, that is,wh1_0.

Case 2. There existsk∗∈Z1, psuch thatwh_k∗ _{0. By3.13, we have}

whk∗1 −whk∗−1. 3.14

Ifwh_k∗_{1 0, thenw}h_{k−1 0 which, by3.13, implies thatw}h_{k 0 fork ∈Z1, p,} that is,wh_{0. Ifw}h_k∗_{1/0, thenw}h_k∗_{−1/0. This, by3.12, impliesα}

k∗₋₁ α_k∗₁λ_h and wh1_k∗ _{−1 w}h1_k∗_{1 0. Thus, by 3.13,w}h1_{k 0 fork ∈ Z1, p, that is}

wh1_{0. The proof is complete.}

Setγ1 λh1−λh/2, γ2 λh1λh /2 andA−Δ2−γ2I. The following Lemmas

3.5–3.7benefit from4 .

Lemma 3.5. Assume that (f1) and (f2) hold. Let{sn} ⊂0,1 and{xn} ⊂Esatisfyxn → ∞and

J_snxn → 0asn → ∞. Then,

lim sup n→ ∞

Axnxn−1≤γ1. _3.15

Proof. Fromf2, we have

λh≤lim inf

|t|γ→ ∞

gsk, t

t ≤lim sup_{|t| → ∞} gsk, t

t ≤λh1 fork, t∈Z×R, 3.16

where the limitation is uniformly ins∈0,1 . It follows that for any >0, there existsR >0 such that

_{gsk, t−γ2t≤}

γ1

|t| for|t|> R, k∈Z1, p, s∈0,1 . 3.17

Thus, there existsη >0 such that _{gsk, t−γ2t≤}

γ1

By the assumption on{xn}, we haveJsnxn, Axn/Axn → 0 as n → ∞. It follows

which, combining with3.18, implies that

lim sup

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

lim sup

Note that >0 is arbitrarily small, we get3.15, and the proof is complete.

Lemma 3.6. Under the conditions of Lemma3.5, one further has

By the fact that−γ1andγ1are two consecutive eigenvalues ofAwith corresponding eigenspaceEh_andEh1_{, we haveθ/γ}

1>1 and then, the functionφt t θ/γ12/t1 is strictly decreasing on 0,∞ with φt → 1 as t → ∞. Besides, xh

n xnh1/xvn ≤

xn/xnv ≤1/δ. So, by3.25,

γ₁−2Axn

2

xn2

≥φ

1

δ2

>1. 3.26

This contradict to3.15and the proof is complete.

Lemma 3.7. Under the assumption of Lemma3.5, there exists a subsequence of{xn}, still called{xn},

such that

either !!x h n!!

xn −→

1 or !!x h1 n !!

xn −→

1 as n−→ ∞. 3.27

Proof. Sincexn → ∞asn → ∞, we can assumeby passing to a subsequence if necessary

that

for someK⊂Z1, p withK /∅, lim

n→ ∞xnk ∞ fork∈K,

ifKc≡Z1, p\K /∅,{xnk}is bounded fork∈Kc.

3.28

Thus,3.16implies

λh≤lim inf n→ ∞

gsnk, xnk

xnk ≤

lim sup n→ ∞

gsnk, xnk

xnk ≤

λh1 fork∈K, 3.29

which implies that there exists a subsequence of{xn}, still called{xn}, andαk∈λh, λh1 , k∈

K, such that

lim n→ ∞

gsnk, xnk

xnk αk fork∈K. 3.30

Let wn xn/xn, then wn 1, and, by Lemma 3.6, there is a convergent subsequence of{xn}, call it{xn}again, such that

wn−→w∈Eh⊕Eh1 as n−→ ∞. 3.31

To prove3.27, we only need to show thatwh _{0 orw}h1 _{0. For everyy ∈E, we} haveJ_snxn, y/xn → 0 asn → ∞, that is,

−Δ2_w n, y

−p

k1

gsnk, xnk

Byf3i, there existM >0 andξ >0 such that|t|fk, t/t−λh> ξand|t|λh1−λh> ξ

Obviously, ifKc_{∅, the above inequality still holds.}

Proof. First we have the following claim:

Claim 1. For any sequences{xn} ⊂ E and {sn} ⊂ 0,1 , if Jsnxn → 0 as n → ∞, then

{xn}is bounded.

In fact, if{xn}is unbounded, there exists a subsequence, still called {xn}, such that

xn → ∞asn → ∞. By Lemma3.8, there exists a subsequence, still called{xn}, such that

Γ1>0 orΓ2<0.

On the other hand,J_snxn, x†n/x†n → 0 as n → ∞, †h, h1, that is %

−Δ2_x n,

x†n !! !x†n!!!

&

−p

k1

gsnk, xnk

x†nk !! !x†n!!!

−→0 as n−→ ∞,†h, h1. _3.43

Note that−Δ2_x

n, x†nλ†xn, xn†,†h, h1, it follows thatΓ1 Γ20. This contradiction proves Claim1.

Settingsn ≡s, n$ 1,2, . . .in Claim1, we see thatJs$satisfiesPScondition. Now, we

start to prove3.42. Define a functionalI:E→Ras

Ix 1

4λh1λhx 2₋

p

k1

Fk, xk. 3.44

Claim 2. There existM >0 such that

inf !!J_sx!!:x> M, s∈0,1 >0. 3.45

In fact, if Claim2is not true, there exists{xn} ⊂Eand{sn} ⊂0,1 such thatxn → ∞ andJsnxn → 0 asn → ∞, which contradict Claim1.

Noticing thata0 : inf{Jsx : s ∈ 0,1 ,x ≤ M} > −∞, we set a < a0. Then,

x∈J₀a{x∈E:J0≤a}impliesx> M. Consider the flowσ:0,1 ×E → Egenerated by

dσ ds −

Iσ

Jsσ2

J_sσ, σ0, x x, x∈J₀a. 3.46

The chain rule for differentiation readsdJsσ/dsJsσ, dσ/dsIσ. Thus,

dJsσ

ds −

%

Iσ

Jsσ2

J_sσ, J_sσ

&

Iσ 0 for s∈0,1 , 3.47

andJsσs, x≡J0x≤a,s∈0,1 , which implies thatσs, x> M,s∈0,1 . Then, the flowσs, xis well defined onJa

0 andσ1,·is a homeomorphism ofJ0atoJ1aandsee11

Hq

E, J₀a∼Hq

On the other hand,

J0x 1 2

−Δ2_{x, x−}1

4λhλh1x

2_{. 3.49}

Note that x 0 is the unique critical point of J0 with Morse index μ : dimE−⊕Eh 2h1see Remark3.1and nullityν0. Then, by2.b,2.fand3.48, we have

CqJ1,∞∼CqJ0,∞∼CqJ0,0∼δq,μZ. 3.50

The proof is completed.

Proof of Theorem1.3. By lemma3.9, we get3.42 which, by 2.a, implies that there exists

x1∈ Kwith

CμJ1, x1/0, μ2h1. 3.51

Since 0≤h < p1, we have 1≤μ < p. Denote byμ1andν1the Morse index and nullity ofx1. By2.e, we getμ1≤μ≤μ1ν1.

Denoteαk ftk, x1k, k ∈ Z1, p. Then, from3.7and Remark3.3, we see that

ν1dim kerJ1 x1≤2.

In Casei,x0 is a local minimum ofJ1, hence, by2.g,

CqJ1,0∼δq,0Z, 3.52

which, by comparing with3.51, implies thatx1/0. Besides, 3≤μ < psinceh∈Z1, p1−1. Assume, for the contradiction, thatx1 is the unique nontrivial critical point ofJ1, thenK

{0, x1}. Ifμμ1orμμ1ν1, we have, by2.f,

CqJ1, x1 δq,μ1Z, orCqJ1, x1 δq,μ1ν1Z, 3.53

from which,2.dreads−10 −1μ −1μ, a contradiction.

Ifμ1< μ < μ1ν1, thenν12 andμμ11. Sinceμ≥3, we haveμ1≥2. Thus,2.c withq1 reads−1≥0, also a contradiction.

In Caseii,x0 is a local maximum ofJ1, hence, by2.g,

CqJ1,0∼δq,pZ, 3.54

which, by comparing with3.51, implies thatx1/0. Besides, 1≤μ < p−3 sinceh∈Z0, p1−2. Assume, for the contradiction, thatx1 is the unique nontrivial critical point ofJ1, thenK

{0, x1}. Ifμμ1orμμ1ν1, then3.53holds, from which,2.dreads

a contradiction. Ifμ1< μ < μ1ν1, thenν12, μμ11 and, by2.f,

Cμ1J1, x1∼Cμ1ν1J1, x1∼0. 3.56

Note thatμ ≤p−3, we haveμ11 < μ12 ≤p−2. Thus,2.cwithq μ11and with

qμ12 reads

dimCμJ1, x1≥1, dimCμJ1, x1≤1, 3.57

respectively, which implies that CqJ1, x1 ∼ δq,μZ. Then, 2.d reads 3.55, also a contradiction. The proof is complete.

Proof of Theorem1.4. As above, there existsx1 ∈ Kwith the Morse indexμ1, and nullityv1

satisfyingμ1≤μ≤μ1ν1, 0≤v1≤2, and3.51holds.

On the other hand,x 0 is a nondegenerate critical point ofJ1 with Morse index, denoted by μ0. Thus, CqJ1,0 ∼ δq,μ0Z, μ0 2m 1 and μ0/μ since m /h, which, by

comparing with3.51, implies thatx1/0.

Assume for the contradiction, that x1 is unique nontrivial critical point of J1, then

K {0, x1}. Ifμ μ1 orμ μ1ν1, then3.53 holds and2.d reads the contradicition

−1μ0 −₁μ −₁μ_.

Now, we consider the caseμ1 < μ < μ1ν1where we haveμμ11 andν12 with

3.56. Sincem /h, we know that either μ0 < μ−1 orμ0 > μ1. Ifμ0 < μ−1,2.cwith

q μ01 reads contradiction−1 ≥ 0. Ifμ0 > μ1, by similar argument, we can get3.57. ThusCqJ1, x1∼δq,μZand2.dreads the contradiction−1μ0 −1μ −1μ. The proof is complete.

The proof of the following lemma is similar to that of12 and is omitted.

Lemma 3.10. Letf satisfyf₄ orf₄−. ThenJ1 has a local linking atx 0with respect to the

decompositionEH−⊕E, whereE−:⊕l≤mElorE− :⊕l<mEl, respectively.

Proof of Theorem1.5. Nowfk,0 λm, k ∈Z1, p. Thus,x0 is a degenerate critical point

ofJ1. Letμ0 andν0 denote the Morse index and nullity of 0. By Lemma3.10and 2.h, we have

CqJ1,0∼δq,rZ, 3.58

whererμ0orμ0ν0corresponding to the casef−4or the casef4, respectively. The rest of the proof is similar and is omitted. The proof is complete.

4. Conclusion and Future Directions

double resonance is first studied and the “unique continuation property” of the second-order difference operator is derived by proving Lemma3.4.

In addition, under the double resonance assumptionf1andf2, some fundamental facts relative to 1.1 are revealed in Lemmas 3.5–3.7, on which, further investigations, employing new restrictions different fromf3andf4, 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.

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 NSFC10871005and BJJWKM200610028001.

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