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

1School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

2School of Mathematical Sciences, Peking University, Beijing 100875, China

3School 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 byZ the set of integers. For a given positive integerp, consider the following periodic problem on difference equation:

−Δ²xk−1 fk, xk, x

kp

xk, k∈Z, 1.1

whereΔis the forward difference operator defined byΔxk xk1−xkandΔ²xk ΔΔxkfork∈Z. In this paper, we always assume that

f1f :Z×R → R isC¹-differentiable with respect to the second variable and satisfies fkp, t fk, tfork, t∈Z×R andfk,0≡0 fork∈Z.

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 knownsee1 that the eigenvalue problem

−Δ²xk−1 λxk, x kp

xk, 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∈Z witha < 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 ≤λ_h1 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

−zt ¨ gt, z,

z0−z2π z0˙ −z2π˙ 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 ≤h1², 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.

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 works2,4 .

We need the following assumptionsf3andf4: f3p >3, and there exists someh∈Z0, p1−1such that

i lim inf

|t| → ∞ |t|

fk, t t −λ_h

>0, ii lim sup

|t| → ∞ |t|

fk, t t −λ_h1

<0,

fork∈Z 1, p

1.6

f4±for somem∈Z0, p1,

± _t

0

fk, s−λ_ms

ds≥0 for|t|>0 small, k∈Z 1, 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 λ_m < fk,0 < λ_m1 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≡λ_m 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

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 monograph9 and the references cited therein. LetHbe a Hilbert space andΦ∈C²H,R

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 byH_qX, 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:HqΦ^c∩U,Φ^c∩U\ {u0} 2.1 is called theqth critical group ofΦatu₀, whereΦ^c{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

C_qΦ,∞:H_qH,Φ^a, q∈N∪ {0}. 2.2 Forq∈N∪ {0}, we callβq :dimCqΦ,∞the Betti numbers ofΦand define the Morse-type numbers of the pairH,Φ^aby

M_q:M_qH,Φ^a

u∈K

dimC_qΦ, 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}, thenC_qΦ,∞∼C_qΦ, x0, 2.cq

j0−1^q−jM_j ≥q

j0−1^q−jβ_j forq∈N∪ {0}, 2.d_∞

j0−1^jMj_∞

j0−1^jβj.

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,μ0Z and ifCμ0ν0Φ, x0 0 thenCqΦ, x0 ∼ δ_q,μ0ν0Z,

2.gIfm:dimH <∞, thenC_qΦ, x0∼δ_q,0Z whenx₀is local minimum ofΦ, while CqΦ, x0∼δq,mZ whenx0is 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 The following results were due to Su5 .

2.hAssume thatΦhas a local linking at x0 ∈ K with respect toH H ⊕H⁻ and kdimH⁻<∞. Then,

CqΦ, x0∼δq,μ0Z, ifkμ0,

C_qΦ, x0∼δ_q,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∈R fork∈Z}and E: x∈X:x

kp

xkfork∈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 toR^p. Throughout this paper, we always identifyx∈ E withx x1, x2, . . . , xp^T∈R^p.

Define the operator−Δ² : E→ Eby−Δ²x {−Δ²xk−1}, x ∈ Eand denoteE ^l ker−Δ²−λlI,l0,1, . . . , p1, whereIis the identity operator. Set

E⁻⊕^h−1_l0E^l, E

⊕^h1_l0E^l_⊥

, E^vE⁻⊕E, 3.3

then Ehas the decompositionE E^h⊕E^h1⊕E^v. In the rest of this paper, the expression xx^hx^h1x^vforx∈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 dimE^p11 ifpis even or dimE^p1 2 ifpis odd.

Define a family of functionalsJ_s: E → R,s∈0,1 by

J_sx −1 2

Δ²x, x

−1−s

4 λhλ_h1x²−s p k1

Fk, xk forx∈E, 3.4

whereFk, t _t

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

J_sx, y −

Δ²x, y

−^p

k1

gsk, xkyk fory∈E, 3.5

wheres∈0,1 and

g_sk, t sfk, t 1−s

2 λhλ_h1t fork, t∈Z×R. 3.6 Remark 3.2. From3.5withs1, we know by computationor see1 thatx∈Eis a critical point ofJ1 if and only if{xk}_k∈Z is ap-periodic solution of problem1.1. Moreover,J1is C²−differentiable and

J₁ xy, z −

Δ²y, z

− p k1

f_tk, xkykzk, ∀y, z∈E, 3.7

whereftk, tis the derivative offk, twith respect tot.

Letα_k∈R,k∈Z1, pandE0consist ofw∈Esatisfying −Δ²w, z

p k1

αkwkzk forz∈E. 3.8

Remark 3.3. E0is the solution space of the systemBx0,x∈E, where

B

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

2−α₁ −1 0 . . . 0 0 −1

−1 2−α2 −1 . . . 0 0 0 . . . . . . . . . .

0 0 0 . . . −1 2−α_p−1 −1

−1 0 0 . . . 0 −1 2−α_p

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

p×p

. 3.9

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

Lemma 3.4. Ifλh ≤αk≤λ_h1,k∈Z1, pandww^hw^h1satisfies3.8, wherew^h∈E^hand w^h1∈E^h1, then eitherw^h0 orw^h10.

Proof. Settingzw^handzw^h1, respectively, in3.8, we have

λ_h p k1

w^hk₂

p k1

α_k

w^hk₂

p k1

α_kw^h1kw^hk,

λ_h1 p k1

w^h1k₂

p k1

α_k

w^h1k₂

p k1

α_kw^hkw^h1k.

3.10

Comparing the above two equalities, we get p

k1

λ_h1−α_k

w^h1k2

^p

k1

λh−α_k

w^hk2

, 3.11

which, byαk∈λh, λ_h1 ,k∈Z1, p, implies that

αk−λhw^hk≡0, αk−λ_h1w^h1k≡0 for k∈Z 1, p

. 3.12

On the other hand, by the definition ofw^handw^h1, we have λ_†w^†k 2w^†k−w^†k−1−w^†k1,

w^† kp

w^†k fork∈Z 1, p

, 3.13

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

Case 1. w^hk/0 fork∈Z1, p. Then by3.12,α_kλ_handw^h1k 0 fork∈Z1, p, that is,w^h10.

Case 2. There existsk^∗∈Z1, psuch thatw^hk^∗ 0. By3.13, we have

w^hk^∗1 −w^hk^∗−1. 3.14

Ifw^hk^∗1 0, thenw^hk−1 0 which, by3.13, implies thatw^hk 0 fork ∈Z1, p, that is,w^h0. Ifw^hk^∗1/0, thenw^hk^∗−1/0. This, by3.12, impliesα_k∗−1 α_k∗1λ_h and w^h1k^∗ −1 w^h1k^∗1 0. Thus, by 3.13,w^h1k 0 fork ∈ Z1, p, that is w^h10. 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 → 0 asn → ∞. Then,

lim sup

n→ ∞ Axnxn⁻¹≤γ₁. 3.15

Proof. Fromf2, we have λ_h≤lim inf

|t|γ→ ∞

g_sk, t

t ≤lim sup

|t| → ∞

g_sk, t

t ≤λ_h1 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∈Z 1, p

, s∈0,1 . 3.17 Thus, there existsη >0 such that

gsk, t−γ2t≤ γ1

|t|η

fork∈Z 1, p

, s∈0,1 . 3.18

By the assumption on{xn}, we haveJ_snxn, Axn/Axn → 0 as n → ∞. It follows from3.5that

Axn − Axn⁻¹ p k1

gsnk, xnk−γ2xnk

Axnk−→0 asn−→ ∞, 3.19

which, combining with3.18, implies that lim sup

n→ ∞

Axn − γ₁ Axn

_p

k1

|xnkAxnk|η p k1

|Axnk|

≤0. 3.20

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

n→ ∞ Axn − γ₁

xn −η p

≤0, 3.21

which leads to

lim sup

n→ ∞

Axn

xn ≤γ₁. 3.22

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 x^v_n

xn −→0 as n−→ ∞. 3.23 Proof. SinceE^h,E^h1, andE^vare invariant with respect toA, we have

Axn²!!!Ax^h_n!!!²!!!Ax^h1_n !!!²Ax^v_n² γ₁²!!!x^h_nx^h1_n !!!²Ax_n^v2.

3.24

If, for the contradiction,3.23is false, then there is a subsequence of{xn}, called{xn} again, and a numberδ >0, such thatx^v_n/xn ≥δ,n1,2, . . .. Then,

γ₁⁻²Axn¹

xn² !!x^h_nx^h1_n !!/x^v_n 2γ₁⁻²Ax^v_n /x_n^{v 2}

!!!xn^hx^h1_n !!!/xn^v₂ 1

≥ x^h_nx^h1_n /x^v_n² θ/γ1²

!!!x_n^hx^h1_n !!!/x^vn₂ 1

,

3.25

whereθinf{Ax^v/x^v:x^v∈E^v\ {0}}.

By the fact that−γ1andγ1are two consecutive eigenvalues ofAwith corresponding eigenspaceE^handE^h1, we haveθ/γ₁>1 and then, the functionφt t θ/γ1²/t1 is strictly decreasing on 0,∞ with φt → 1 as t → ∞. Besides, x^h_n x_n^h1/x^v_n ≤ xn/x^v_n ≤1/δ. So, by3.25,

γ₁⁻²Axn² 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_n^h1!!

xn −→1 as n−→ ∞. 3.27 Proof. Sincexn → ∞asn → ∞, we can assumeby passing to a subsequence if necessary that

for someK⊂Z 1, p

withK /∅, lim

n→ ∞xnk ∞ fork∈K, ifK^c≡Z

1, p

\K /∅,{xnk}is bounded fork∈K^c.

3.28

Thus,3.16implies

λh≤lim inf

n→ ∞

gsnk, xnk

x_nk ≤lim sup

n→ ∞

gsnk, xnk

x_nk ≤λ_h1 fork∈K, 3.29 which implies that there exists a subsequence of{xn}, still called{xn}, andαk∈λh, λ_h1 , k∈ K, such that

n→ ∞lim

gsnk, xnk

xnk αk fork∈K. 3.30

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

w_n−→w∈E^h⊕E^h1 as n−→ ∞. 3.31 To prove3.27, we only need to show thatw^h 0 orw^h1 0. For everyy ∈E, we haveJ_snxn, y/xn → 0 asn → ∞, that is,

−Δ²wn, y

−^p

k1

gsnk, xnk

xn yk−→0 as n−→ ∞. 3.32

IfK^c/∅, gsnk, xnk/xn → 0 asn → ∞fork∈K^c, then we can rewrite3.32as −Δ²w_n, y

−

k∈K

g_snk, xnk xnk

x_nk

xnyk−→0 asn−→ ∞. 3.33

Lettingn → ∞in3.33and using3.30and3.31, we get −Δ²w, y

k∈K

αkwkyk fory∈E. 3.34

Sincewk 0 fork∈K^c, by settingαkλhfork∈K^c, we rewrite3.34as −Δ²w, y

p k1

α_kwkyk fory∈E. 3.35

Obviously, ifK^c ∅,3.35still holds. By Lemma 3.4,w^h 0 orw^h1 0 and the proof is complete.

Lemma 3.8. Assume thatf1andf3hold. Let{sn} ⊂0,1 and{xn} ⊂Esatisfyxn → ∞and J_snxn → 0 asn → ∞. Then, there exists a subsequence of{xn}, still called{xn}, such that

either Γ1:lim sup

n→ ∞

p k1

gsnk, xnk−λhxnkx^h_nk

!!!x^h_n!!! >0 or Γ2:lim inf

n→ ∞

p k1

g_snk, xnk−λ_h1x_nkx^h1_n k

!!!x^h1_n !!! <0.

3.36

Proof. As that in the above proof, we can assume that{xn}satisfies3.28. Noticing thatf3 impliesf2and by Lemma3.7, we have two cases to be considered.

Case 1. x^h_n/xn → 1 asn → ∞. We havex^h_n → ∞asn → ∞and

nlim→ ∞

!!x_n^h1!!

!!!xn^h!!! 0, lim

n→ ∞

x^v_n

!!!x^hn!!! 0. 3.37

IfK^c/∅, then{xnk}and{gsnk, xnk−λhx_nk}are bounded fork∈K^candn∈N.

It follows thatxnk/x^h_n → 0 asn → ∞fork∈K^cand

nlim→ ∞ gsnk, xnk−λhxnkx_n^hk

!!!x^h_n!!! 0 for k∈K^c. 3.38

Byf3i, there existM >0 andξ >0 such that|t|fk, t/t−λh> ξand|t|λh1−λh> ξ for|t|> Mandk∈Z1, p. Then, for|t|> M,k∈Z1, pands∈0,1 ,

g_sk, t t −λ_h

|t|s

fk, t t −λ_h

|t|1−s

2 λ_h1−λ_h|t|

≥s ξ1−s 2 ξ≥ ξ

2.

3.39

ChooseN >0 such that|xnk|> Mfork∈Kandn > N. It follows that g_snk, xnk−λ_hx_nk

x^h_nk

"g_snk, xnk x_nk −λ_h

#

x_nkxnk−z_nk

≥

"gsnk, xnk x_nk −λh

#

|xnk||xnk| − |znk|

≥ ξ

2|xnk| − |znk| fork∈K, n > N,

3.40

where z_n x^h1_n x^v_n. Since Eis a finite dimensional vector space and possesses another norm defined byx₁≡p

k1|xk|,x∈E, which is equivalent to · , there exists a positive constantC >0 such thatx₁Cx,x∈E. Thus, by3.37–3.40,

Γ1≥lim sup

n→ ∞

ξ 2!!!x^h_n!!!

k∈K

|xnk| −

k∈K

|znk|

lim sup

n→ ∞

ξ 2!!!x^hn!!!

_p

k1

|xnk| − p k1

|znk|

≥lim sup

n→ ∞

ξ 2!!!x^hn!!!

Cxn − pzn Cξ

2 .

3.41

Obviously, ifK^c∅, the above inequality still holds.

Case 2. x_n^h1/xn → 1 asn → ∞. By usingf3ii, we can show thatΓ2 <0 in the same way. The proof is complete.

In the rest of this section, we will use the facts2.a–2.hstated in Section2to complete the proofs.

Lemma 3.9. Letfsatisfy (f₁) and (f₃). Then, for everys$∈0,1 ,Js_$satisfies the (PS) condition and

C_qJ1,∞∼δ_q,μZ, μ2h1. 3.42

Proof. First we have the following claim:

Claim 1. For any sequences{xn} ⊂ E and {sn} ⊂ 0,1 , if J_snxn → 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

%

−Δ²xn, x^†_n

!!!x^†_n!!!

&

−^p

k1

gsnk, xnkx^†_nk

!!!x^†_n!!! −→0 as n−→ ∞,†h, h1. 3.43

Note that−Δ²xn, x^†_nλ†xn, x^†_n,†h, h1, it follows thatΓ1 Γ20. This contradiction proves Claim1.

Settings_n ≡s, n$ 1,2, . . .in Claim1, we see thatJ_s$satisfiesPScondition. Now, we start to prove3.42. Define a functionalI:E→R as

Ix 1

4λ_h1λ_hx²− 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 → ∞ andJ_snxn → 0 asn → ∞, which contradict Claim1.

Noticing thata0 : inf{Jsx : s ∈ 0,1 ,x ≤ M} > −∞, we set a < a0. Then, x∈J₀^a{x∈E:J0≤a}impliesx> M. Consider the flowσ:0,1 ×E → Egenerated by

dσ

ds − Iσ

Jsσ²J_sσ, σ0, x x, x∈J₀^a. 3.46 The chain rule for differentiation readsdJ_sσ/dsJ_sσ, dσ/dsIσ. Thus,

dJ_sσ

ds −

% Iσ

Jsσ²J_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 onJ₀^aandσ1,·is a homeomorphism ofJ₀^atoJ₁^aandsee11

Hq

E, J₀^a∼Hq

E, J₁^a

. 3.48

On the other hand,

J₀x 1 2

−Δ²x, x

−1

4λhλ_h1x². 3.49

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

C_qJ1,∞∼C_qJ0,∞∼C_qJ0,0∼δ_q,μZ. 3.50 The proof is completed.

Proof of Theorem1.3. By lemma 3.9, 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.

Denoteα_k f_tk, 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 −1^μ −1^μ, 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,

C_qJ1,0∼δ_q,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

−1^p −1^μ −1^μ, 3.55

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

dimC_μJ1, x₁≥1, dimC_μJ1, 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, C_qJ1,0 ∼ δ_q,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

−1^μ0 −1^μ −1^μ.

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,μZ and2.dreads the contradiction−1^μ0 −1^μ −1^μ. 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 0 with respect to the decompositionEH⁻⊕E, whereE⁻:⊕l≤mE^lorE⁻ :⊕l<mE^l, respectively.

Proof of Theorem1.5. Nowfk,0 λ_m, 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

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 functionalsee2–5 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 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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