# Periodic Solutions of Second-Order Difference Problem with Potential Indefinite in Sign

Authors: Honghua Bin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the periodic solutions of second-order difference problem with potential indefinite in sign. We consider the compactness condition of variational functional and local linking at 0 by introducing new numberλ∗. By using Morse theory, we obtain some new results concerning the existence of nontrivial periodic solution.


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## Body

## 1. Introduction

We consider the second-order discrete Hamiltonian systems(1)Δ2xn-1+W′(n,xn)=0,xn+T=xn,
where T≥2 is a given integer, n∈ℤ,xn∈ℝN, Δxn=xn+1-xn, Δ2xn=Δ(Δxn), W′ stands for the gradient of W with respect to the second variable. W∈C2(ℤ×ℝN,ℝ) is T-periodic in the first variable and has the form W(n,x)=(1/2)a|x|2+H(n,x), where a=4sin2(mπ/T) for some m∈Z[0,r], r=[T/2], [·] stands for the greatest-integer function. For integers a≤b, the discrete interval {a,a+1,…,b} is denoted by Z[a,b].In this paper we consider thatH is sign changing, that is,
(2)H(n,x)=b(n)(1s|x|s+G-s(n,x))≜1sb(n)|x|s+Gs(n,x),Ω+={n∈Z[1,T]|b(n)>0)}, Ω-={n∈Z[1,T]|b(n)<0)} are two nonempty subsets of Z[1,T], where s>1, b(·) is a T-periodic real function, Gs∈C1(ℤ×ℝN,ℝ), and Gs(n,0)=0.Consider the second-order Hamiltonian system(3)x¨(t)+W′(t,x)=0,x(0)=x(T),x˙(0)=x˙(T),
where W∈C2(ℝ×ℝN,ℝ) is T-periodic in t, W(t,x)=(1/2)(A(t)x,x)+H(t,x). Here A(·) is a continuous, T-periodic matrix-value function.Systems (1) and (3) have been investigated by many authors using various methods, see [1–5]. The dynamical behavior of differential and difference equations was studied by using various methods, and many interesting results have obtained, see [6–10] and references therein. The critical point theory [11–14] is a useful tool to investigate differential equations. Morse theory [15–19] has also been used to solve the asymptotically linear problem. By minimax methods in critical point theory, Tang and Wu [4], Antonacci [20, 21] considered the problem (3) with potential indefinite in sign, where H is superquadratic at zero and infinity. By using Morse theory, Zou and Li [10] study the existence of T-periodic solution of (3), where H is asymptotically superquadratic and sign changing. Moroz [19] studies system (3) where H is asymptotically subquadratic and sign changing. Motivated by [5, 10, 19], we investigate periodic solutions for asymptotically superquadratic or subquadratic discrete system (1).By expression ofH(n,x), system (1) possesses a trivial solution x=0. Here we are interested in finding the nonzero T-periodic solution of (1), and we divide the problem into two cases: s>2 and 1<s<2. For s=2, one can refer to [22].Case 1 (asymptotically superquadratic case:s>2).
In this case, we replacep with s in (2). Letting gp(n,x)=Gp′(n,x), we rewrite (1) as
(4)Δ2xn-1+axn+b(n)|xn|p-2xn+gp(n,xn)=0,xn+T=xn.
Furthermore, for all (n,x)∈ℤ×ℝN, we assume that gp satisfies(A1)
gp(n,x)=o(|x|)  as |x|→∞ uniformly in n,(A2)
gp(n,x)=o(|x|p-1) as |x|→0 uniformly in n.Case 2 (asymptotically subquadratic case:1<s<2).
Here we replaceq with s in (2). Letting gq(n,x)=Gq′(n,x), we rewrite (1) as
(5)Δ2xn-1+axn+b(n)|xn|q-2xn+gq(n,xn)=0,xn+T=xn.
For all (n,x)∈ℤ×ℝN, we assume that gq satisfies(B1)
gq(n,x)=o(|x|q-1) as |x|→∞ uniformly in n,(B2)
gq(n,x)=o(|x|) as |x|→0 uniformly in n.Before stating the main results, we introduce spaceET={x={xn}∈S|xn+T=xn,n∈ℤ}, where S={x={xn}|xn∈ℝN,n∈ℤ}. For any x,y∈S,a,b∈ℝ, we define ax+by={axn+byn}n∈ℤ. Then S is a linear space. Let 〈x,y〉ET=∑n=1T(xn,yn), ∥x∥ET=(∑n=1T|xn|2)1/2, forallx,y∈ET, where (·,·) and |·| are the usual inner product and norm in ℝN, respectively. Obviously, ET is a Hilbert space with dimension NT and homeomorphism to ℝNT. For r>1, let ∥x∥r=(∑n=1T|xn|r)1/r,x∈ET. Moreover, for simplicity, we write 〈x,y〉 and ∥x∥ instead of 〈x,y〉ET and ∥x∥ET, respectively.Lemma 1.
There exist positive numbersa1,a2, such that a1∥x∥r≤∥x∥≤a2∥x∥r.
Inspired by [10, 19], one introduces two numbers as follows:
(6)λ*(p)=inf∥x∥=1{∥Δx∥2|∑n=1T‍b(n)|xn|p=0},λ*(q)=inf∥x∥=1{∥Δx∥2|∑n=1T‍b(n)|xn|q=0}.Theorem 2.
Ifa<λ*(p), then (4) has a nonzero T-periodic solution.Theorem 3.
Ifa<λ*(q), then (5) has a nonzero T-periodic solution.This paper is divided into four sections. Section2 contains some preliminaries, and the proofs of Theorems 2 and 3 are given in Sections 3 and 4, respectively.


## 2. Preliminaries

### 2.1. Variational Functional and (PS) Condition

For seekingT-periodic solution of (1), we consider variational functional Jp associated with (4) as Jp(x)=(1/2)∑n=1T|Δxn|2-(1/2)a∑n=1T|xn|2-1/p∑n=1Tb(n)|xn|p-∑n=1TGp(n,xn), that is
(7)Jp(x)=12∥Δx∥2-12a∥x∥2-1p∑n=1T‍b(n)|xn|p-∑n=1T‍Gp(n,xn),x∈ET.
Moreover, T-periodic solution of (5) is associated with the critical point of functional
(8)Jq(x)=12∥Δx∥2-12a∥x∥2-1q∑n=1T‍b(n)|xn|q-∑n=1T‍Gq(n,xn),x∈ET.We say that aC1-functional φ on Hilbert space X satisfies the Palais-Smale (PS) condition if every sequence {x(j)} in X, such that {φ(x(j))}, is bounded and φ′(x(j))→0 as j→∞ contains a convergent subsequence.Lemma 4.
FunctionalJp satisfies (PS) condition if a<λ*(p).Proof.
Let{x(j)}⊂ET be the (PS) sequence for functional Jp, such that Jp(x(j)) is bounded, and Jp′(x(j))→0 as j→∞. Hence, for any ε>0, there exist Nε>0 and constant c1>0, such that
(9)|〈Jp′(x(j)),x(j)〉|≤ε∥x(j)∥forj≥Nε,|Jp(x(j))|≤c1.
To prove thatJp satisfies (PS) condition, it suffices to show that ∥x(j)∥ is bounded in ET. Suppose not that there exists a subsequence {x(jk)}, ∥x(jk)∥→∞ as k→∞. For simplicity, we write as {x(j)} instead of {x(jk)}. Without loss of generality, we assume that there exists k∈Z[1,T], such that
(10)|xn(j)|⟶∞asj⟶∞forn∈Z[1,k],xn(j)areboundedforn∈Z[k+1,T].
Therefore for all n∈[1,T], by assumption (A1), there exists c2>0 such that
(11)|Gp(n,xn(j))|≤ε|xn(j)|2+c2,|gp(n,xn(j))|≤ε|xn(j)|+c2
for large j. By the previous argument, it follows that
(12)|∑n=1T‍(gp(n,xn(j)),xn(j))|≤∑n=1T‍|gp(n,xn(j))||xn(j)|≤ε∥x(j)∥2+c2T∥x(j)∥.
By (7), we have
(13)pJp(x(j))-〈Jp′(x(j)),x(j)〉=(p2-1)(∥Δx(j)∥2-a∥x(j)∥2)-p∑n=1T‍Gp(n,xn(j))+∑n=1T‍(gp(n,xn(j)),xn(j)).
In terms of (9) and (11), for large j, it follows that
(14)(p2-1)(∥Δx(j)∥2-a∥x(j)∥2)≤pc1+ε∥x(j)∥+(p+1)ε∥x(j)∥2+pc2T+c2T∥x(j)∥.
Set yn(j)=xn(j)/∥x(j)∥. Dividing by ∥x(j)∥2 in the previous formula, it follows that
(15)∥Δy(j)∥2≤a+2p-2((p+1)ε+c2T+ε∥x(j)∥+pc2T+pc1∥x(j)∥2)
for large j. Therefore, by ε being chosen arbitrarily, there is a subsequence that converges to y0∈ET such that
(16)∥Δy0∥2≤a,∥y0∥=1.
On the other hand, we have(17)Jp(x(j))-12〈Jp′(x(j)),x(j)〉=(12-1p)∑n=1T‍b(n)|xn(j)|p-∑n=1T‍Gp(n,xn(j))+12∑n=1T‍(gp(n,xn(j)),xn(j)).
Then, by (9) and (11), for large j, we get
(18)|(12-1p)∑n=1T‍b(n)|xn(j)|p|=|Jp(x(j))-12〈Jp′(x(j)),x(j)〉+∑n=1T‍Gp(n,xn(j))-12∑n=1T‍(gp(n,xn(j)),xn(j))|≤c1+ε2∥x(j)∥+ε∥x(j)∥2+c2T+12(ε∥x(j)∥2+c2T∥x(j)∥).
By dividing by ∥x(j)∥p in the previous formula, then by p>2, we have ∑n=1Tb(n)|yn(j)|p→0 as j→∞, that is, ∑n=1Tb(n)|yn0|p=limj→∞∑n=1Tb(n)|yn(j)|p=0. By the definition of λ*(p), see (6), we have ∥Δy0∥2≥λ*(p). This contradicts with (16) and assumption a<λ*(p). The proof is completed.Lemma 5.
FunctionalJq satisfies (PS) condition if a<λ*(q).The proof is similar to that of Lemma4 and is omitted.


### 2.2. Eigenvalue Problem

Consider eigenvalue problem:(19)-Δ2xn-1=λxn,xn+T=xn,xn∈ℝN,
that is, xn+1+(λ-2)xn+xn-1=0, xn+T=xn. By the periodicity, the difference system has complexity solution xn=einθc for c∈ℂN, where θ=2kπ/T, k∈ℤ. Moreover, λ=2-e-iθ-eiθ=2(1-cosθ)=4sin2(kπ/T). Let ηk denote the real eigenvector corresponding to the eigenvalues λk=4sin2(kπ/T), where k∈Z[0,r] and r=[T/2]. Since a=4sin2(mπ/T) for some m∈Z[0,r], we can split space ET as follows:
(20)ET=W-⨁W0⨁W+,
where
(21)W-=span{ηk∣k∈Z[0,m-1]},W0=span{ηm},W+=span{ηk∣k∈Z[m+1,r]}.By means of eigenvalue problem, we have|Δxn|2-a|xn|2=(Δxn,Δxn)-a(xn,xn)=(-Δ2xn-1,xn)-a(xn,xn)=(λ-a)(xn,xn)=(λ-a)|xn|2. Let
(22)δ={min{4sin2(m+1)πT-4sin2mπT,min4sin2mπT-4sin2(m-1)πT},m∈Z[1,r],4sin2πT,m=0.
Then ±(∥Δx∥2-a∥x∥2)≥δ∥x∥2 for x∈W±.On the other hand, associating to numbersλ*(p) and λ*(q) (see (6)), we set
(23)Λ*(p)=∑n=1T‍b(n)|en|p,Λ*(q)=∑n=1T‍b(n)|en|q,
where en=u∈ℝN (n∈[1,T]) is the real eigenvector corresponding to eigenvalue λ0=0. e=(e1T,e2T,…,eNT)T=(uT,uT,…,uT)T∈ET, where  •T denotes the transpose of a vector or a matrix. Moreover, letting |u|=T-1/2, we have ∥e∥=1, ∥Δe∥=0. Therefore, by definition of λ*(p), if Λ*(p)=0 then λ*(p)=0.However, by assumptionλ*(p)>a=4sin2(mπ/T) for some m∈Z[0,r], thus λ*(p)>0. That is to say the equality Λ*(p)=0 cannot hold. Therefore our discussion will be distinguished in two cases: Λ*(p)>0 and Λ*(p)<0.


### 2.3. Preliminaries

LetX be a Hilbert space, and let φ∈C1(X,ℝ) be a functional satisfying the (PS) condition. Write crit(φ)={x∈X∣φ′(x)=0} for the set of critical points of functional φ and φc={x∈X∣φ(x)≤c} for the level set. Denote by Hk(A,B) the kth singular relative homology group with integer coefficients. Let x0∈crit(φ) be an isolated critical point with value c=φ(x0), c∈ℝ, the group Ck(φ,x0)=Hk(φc∩U,(φc∩U)∖{x0}), and k∈ℤ is called the kth critical group of φ at x0, where U is a closed neighbourhood of u. Due to the excision of homology [13], Ck(φ,x0) is dependent onU.Suppose thatφ(crit(φ)) is strictly bounded from below by a∈ℝ, then the critical groups of φ at infinity are formally defined [11] as Ck(φ,∞)=Hk(X,φa), k∈ℤ.Proposition 6 (Proposition 2.3, [11]).
Assume thatC2-functional φ satisfying (PS) condition has a local linking at 0 with respect to X=X0+⨁X0-; that is, there exists ρ>0 such that
(24)φ(x)≤φ(0)forx∈X0-and∥x∥≤ρ,φ(x)>φ(0)forx∈X0+and0<∥x∥≤ρ.
Then Ck(φ,0)≠0, k=dimX0-.By Propostion6, one proves the following lemmas with respect to ET=X+⨁X-.Lemma 7.
Ifa<λ*(p), then Ck(Jp,0)≠0, k=dimX-, where X-=W-⨁W0 as Λ*(p)>0, X-=W- as Λ*(p)<0. Λ*(p) is defined by (23).Proof.
We first consider the following.
Case
1
(
Λ
*
(
p
)
>
0 and X+=W+, X-=W-⨁W0). By p>2, |x|p=o(|x|2) as |x|→0, then there exists θ∈(0,1) suitably small, such that |x|p≤δ/3(b/p+ε)|x|2 as |x|<θ, where δ>0 see (22) and b=max{|b(1)|,…,|b(T)|}>0. By assumption (A2) and Gp(n,0)=0, for any given ε>0, there exists ρn∈(0,θ), such that |Gp(n,xn)|≤ε|xn|p as |xn|≤ρn, n∈Z[1,T]. Thus
(25)1p∑n=1T‍b(n)|xn|p+∑n=1T‍Gp(n,xn)≤(bp+ε)∑n=1T‍|xn|p≤13δ∥x∥2.
Let ρ=min{ρ1,…,ρT}. For 0<∥x∥≤ρ<1, it follows that
(26)Jp(x)≥12δ∥x∥2-13δ∥x∥2>0,x∈W+=X+.
We need to prove thatJp(x)≤0 for x∈X-=W-⨁W0, ∥x∥≤ρ. We first claim that
(27)∑n=1T‍b(n)|xn|p>0,∀x∈W-⨁W0,x≠0.
Indeed, by contradiction, assume that ∑n=1Tb(n)|xn|p≤0, for some x∈W-⨁W0, x≠0. Since Λ*(p)=∑n=1Tb(n)|en|p>0, where e=(e1T,e2T,…,eNT)T=(uT,uT,…,uT)T∈W-⨁W0, and (W-⨁W0)∖{0} is arcwise connected, then there exists a x0∈(W-⨁W0)∖{0}, such that ∑n=1Tb(n)|xn0|p=0. Thus ∥Δx0∥2≥λ*(p)∥x0∥2 by the definition of λ*(p). On the other hand, by the definition of W-⨁W0, we have ∥Δx0∥2≤a∥x0∥2. This is a contradiction with assumption a<λ*(p). So the claim (27) holds.
There existsc4>0 by (27), such that ∑n=1Tb(n)|xn|p≥c4∥x∥pp for all x∈W-⨁W0∖{0}, where ∥x∥p=(∑n=1T|xn|p)1/p. For x∈W-⨁W0, ∥x∥≤ρ, ε sufficiently small, we have
(28)Jp(x)≤-1p∑n=1T‍b(n)|xn|p-∑n=1T‍Gp(n,xn)≤-c4p∥x∥pp+ε∥x∥pp≤0.
Since Jp(0)=0 and Jp satisfies (PS) condition by Lemma 4, so by Proposition 6, we obtain that Ck(Jp,0)≠0 for k=dim(W-⨁W0).
Case2(Λ*(p)<0,X+=W+⨁W0,X-=W-). It is easy to see that Jp(x)≤0 by ∥Δx∥2-a∥x∥2≤-δ∥x∥2 and p>2, where x∈W- and ∥x∥≤ρ. We need to claim that Jp(x)>0, for x∈W+⨁W0, 0<∥x∥≤ρ.
Suppose not that there exists a sequence{x(j)}⊂ET such that
(29){x(j)}⊂W+⨁W0∖{0},0<∥x(j)∥≤ρ,Jp(x(j))≤0,
for large j. For ∥x(j)∥≤ρ, by Lemma 1, we get
(30)|∑n=1T‍[1pb(n)|xn(j)|p+Gp(n,xn(j))]|≤∑n=1T‍[bp|xn(j)|p+ε|xn(j)|p]≤(bp+ε)(1a1)p∥x(j)∥p.
Set yn(j)=xn(j)/∥x(j)∥. Then by (29) and the previous formula, we have
(31)0≥Jp(x(j))∥x(j)∥2≥12(∥Δy(j)∥2-a)-(bp+ε)(1a1)p∥x(j)∥p-2.
On the other hand,∥Δy(j)∥2≥a by the definition of W+⨁W0. Hence by p>2, there exists a subsequence converges to y0∈ET, such that ∥Δy0∥2=a, that is y0∈W0 and ∥y0∥=1. Since ∥Δx(j)∥2≥a∥x(j)∥2 for {x(j)}⊂W+⨁W0, it follows from Jp(x(j))≤0 that
(32)0≤1p∑n=1T‍b(n)|xn(j)|p+∑n=1T‍Gp(n,xn(j))≤1p∑n=1T‍b(n)|xn(j)|p+ε(1a1)p∥x(j)∥p.
Dividing by ∥x(j)∥p in the previous inequality, then ∑n=1Tb(n)|yn0|p=limj→∞∑n=1Tb(n)|yn(j)|p≥0.
Sincee,y0∈W-⨁W0, Λ*(p)=∑n=1Tb(n)|en|p<0 and (W-⨁W0)∖{0} is arcwise connected, then there exists a y-∈(W-⨁W0)∖{0} such that ∑n=1Tb(n)|y-n|p=0. Thus ∥Δx-∥2≥λ*(p)∥x-∥2 by the definition of λ*(p). On the other hand, ∥Δx-∥2≤a∥x-∥2 by the definition of W-⨁W0. This is a contradiction with assumption a<λ*(p). That is to say, the claim is valid.
By Proposition6, we obtain Ck(Jp,0)≠0, k=dimW-. The proof is completed.Lemma 8.
Ifa<λ*(q), then Ck(Jq,∞)≠0 for k=dimX-, where X-=W-⨁W0 as Λ*(q)>0, X-=W- as Λ*(q)<0.
The proof is similar to that of Lemma7 and is omitted.


## 2.1. Variational Functional and (PS) Condition

For seekingT-periodic solution of (1), we consider variational functional Jp associated with (4) as Jp(x)=(1/2)∑n=1T|Δxn|2-(1/2)a∑n=1T|xn|2-1/p∑n=1Tb(n)|xn|p-∑n=1TGp(n,xn), that is
(7)Jp(x)=12∥Δx∥2-12a∥x∥2-1p∑n=1T‍b(n)|xn|p-∑n=1T‍Gp(n,xn),x∈ET.
Moreover, T-periodic solution of (5) is associated with the critical point of functional
(8)Jq(x)=12∥Δx∥2-12a∥x∥2-1q∑n=1T‍b(n)|xn|q-∑n=1T‍Gq(n,xn),x∈ET.We say that aC1-functional φ on Hilbert space X satisfies the Palais-Smale (PS) condition if every sequence {x(j)} in X, such that {φ(x(j))}, is bounded and φ′(x(j))→0 as j→∞ contains a convergent subsequence.Lemma 4.
FunctionalJp satisfies (PS) condition if a<λ*(p).Proof.
Let{x(j)}⊂ET be the (PS) sequence for functional Jp, such that Jp(x(j)) is bounded, and Jp′(x(j))→0 as j→∞. Hence, for any ε>0, there exist Nε>0 and constant c1>0, such that
(9)|〈Jp′(x(j)),x(j)〉|≤ε∥x(j)∥forj≥Nε,|Jp(x(j))|≤c1.
To prove thatJp satisfies (PS) condition, it suffices to show that ∥x(j)∥ is bounded in ET. Suppose not that there exists a subsequence {x(jk)}, ∥x(jk)∥→∞ as k→∞. For simplicity, we write as {x(j)} instead of {x(jk)}. Without loss of generality, we assume that there exists k∈Z[1,T], such that
(10)|xn(j)|⟶∞asj⟶∞forn∈Z[1,k],xn(j)areboundedforn∈Z[k+1,T].
Therefore for all n∈[1,T], by assumption (A1), there exists c2>0 such that
(11)|Gp(n,xn(j))|≤ε|xn(j)|2+c2,|gp(n,xn(j))|≤ε|xn(j)|+c2
for large j. By the previous argument, it follows that
(12)|∑n=1T‍(gp(n,xn(j)),xn(j))|≤∑n=1T‍|gp(n,xn(j))||xn(j)|≤ε∥x(j)∥2+c2T∥x(j)∥.
By (7), we have
(13)pJp(x(j))-〈Jp′(x(j)),x(j)〉=(p2-1)(∥Δx(j)∥2-a∥x(j)∥2)-p∑n=1T‍Gp(n,xn(j))+∑n=1T‍(gp(n,xn(j)),xn(j)).
In terms of (9) and (11), for large j, it follows that
(14)(p2-1)(∥Δx(j)∥2-a∥x(j)∥2)≤pc1+ε∥x(j)∥+(p+1)ε∥x(j)∥2+pc2T+c2T∥x(j)∥.
Set yn(j)=xn(j)/∥x(j)∥. Dividing by ∥x(j)∥2 in the previous formula, it follows that
(15)∥Δy(j)∥2≤a+2p-2((p+1)ε+c2T+ε∥x(j)∥+pc2T+pc1∥x(j)∥2)
for large j. Therefore, by ε being chosen arbitrarily, there is a subsequence that converges to y0∈ET such that
(16)∥Δy0∥2≤a,∥y0∥=1.
On the other hand, we have(17)Jp(x(j))-12〈Jp′(x(j)),x(j)〉=(12-1p)∑n=1T‍b(n)|xn(j)|p-∑n=1T‍Gp(n,xn(j))+12∑n=1T‍(gp(n,xn(j)),xn(j)).
Then, by (9) and (11), for large j, we get
(18)|(12-1p)∑n=1T‍b(n)|xn(j)|p|=|Jp(x(j))-12〈Jp′(x(j)),x(j)〉+∑n=1T‍Gp(n,xn(j))-12∑n=1T‍(gp(n,xn(j)),xn(j))|≤c1+ε2∥x(j)∥+ε∥x(j)∥2+c2T+12(ε∥x(j)∥2+c2T∥x(j)∥).
By dividing by ∥x(j)∥p in the previous formula, then by p>2, we have ∑n=1Tb(n)|yn(j)|p→0 as j→∞, that is, ∑n=1Tb(n)|yn0|p=limj→∞∑n=1Tb(n)|yn(j)|p=0. By the definition of λ*(p), see (6), we have ∥Δy0∥2≥λ*(p). This contradicts with (16) and assumption a<λ*(p). The proof is completed.Lemma 5.
FunctionalJq satisfies (PS) condition if a<λ*(q).The proof is similar to that of Lemma4 and is omitted.


## 2.2. Eigenvalue Problem

Consider eigenvalue problem:(19)-Δ2xn-1=λxn,xn+T=xn,xn∈ℝN,
that is, xn+1+(λ-2)xn+xn-1=0, xn+T=xn. By the periodicity, the difference system has complexity solution xn=einθc for c∈ℂN, where θ=2kπ/T, k∈ℤ. Moreover, λ=2-e-iθ-eiθ=2(1-cosθ)=4sin2(kπ/T). Let ηk denote the real eigenvector corresponding to the eigenvalues λk=4sin2(kπ/T), where k∈Z[0,r] and r=[T/2]. Since a=4sin2(mπ/T) for some m∈Z[0,r], we can split space ET as follows:
(20)ET=W-⨁W0⨁W+,
where
(21)W-=span{ηk∣k∈Z[0,m-1]},W0=span{ηm},W+=span{ηk∣k∈Z[m+1,r]}.By means of eigenvalue problem, we have|Δxn|2-a|xn|2=(Δxn,Δxn)-a(xn,xn)=(-Δ2xn-1,xn)-a(xn,xn)=(λ-a)(xn,xn)=(λ-a)|xn|2. Let
(22)δ={min{4sin2(m+1)πT-4sin2mπT,min4sin2mπT-4sin2(m-1)πT},m∈Z[1,r],4sin2πT,m=0.
Then ±(∥Δx∥2-a∥x∥2)≥δ∥x∥2 for x∈W±.On the other hand, associating to numbersλ*(p) and λ*(q) (see (6)), we set
(23)Λ*(p)=∑n=1T‍b(n)|en|p,Λ*(q)=∑n=1T‍b(n)|en|q,
where en=u∈ℝN (n∈[1,T]) is the real eigenvector corresponding to eigenvalue λ0=0. e=(e1T,e2T,…,eNT)T=(uT,uT,…,uT)T∈ET, where  •T denotes the transpose of a vector or a matrix. Moreover, letting |u|=T-1/2, we have ∥e∥=1, ∥Δe∥=0. Therefore, by definition of λ*(p), if Λ*(p)=0 then λ*(p)=0.However, by assumptionλ*(p)>a=4sin2(mπ/T) for some m∈Z[0,r], thus λ*(p)>0. That is to say the equality Λ*(p)=0 cannot hold. Therefore our discussion will be distinguished in two cases: Λ*(p)>0 and Λ*(p)<0.


## 2.3. Preliminaries

LetX be a Hilbert space, and let φ∈C1(X,ℝ) be a functional satisfying the (PS) condition. Write crit(φ)={x∈X∣φ′(x)=0} for the set of critical points of functional φ and φc={x∈X∣φ(x)≤c} for the level set. Denote by Hk(A,B) the kth singular relative homology group with integer coefficients. Let x0∈crit(φ) be an isolated critical point with value c=φ(x0), c∈ℝ, the group Ck(φ,x0)=Hk(φc∩U,(φc∩U)∖{x0}), and k∈ℤ is called the kth critical group of φ at x0, where U is a closed neighbourhood of u. Due to the excision of homology [13], Ck(φ,x0) is dependent onU.Suppose thatφ(crit(φ)) is strictly bounded from below by a∈ℝ, then the critical groups of φ at infinity are formally defined [11] as Ck(φ,∞)=Hk(X,φa), k∈ℤ.Proposition 6 (Proposition 2.3, [11]).
Assume thatC2-functional φ satisfying (PS) condition has a local linking at 0 with respect to X=X0+⨁X0-; that is, there exists ρ>0 such that
(24)φ(x)≤φ(0)forx∈X0-and∥x∥≤ρ,φ(x)>φ(0)forx∈X0+and0<∥x∥≤ρ.
Then Ck(φ,0)≠0, k=dimX0-.By Propostion6, one proves the following lemmas with respect to ET=X+⨁X-.Lemma 7.
Ifa<λ*(p), then Ck(Jp,0)≠0, k=dimX-, where X-=W-⨁W0 as Λ*(p)>0, X-=W- as Λ*(p)<0. Λ*(p) is defined by (23).Proof.
We first consider the following.
Case
1
(
Λ
*
(
p
)
>
0 and X+=W+, X-=W-⨁W0). By p>2, |x|p=o(|x|2) as |x|→0, then there exists θ∈(0,1) suitably small, such that |x|p≤δ/3(b/p+ε)|x|2 as |x|<θ, where δ>0 see (22) and b=max{|b(1)|,…,|b(T)|}>0. By assumption (A2) and Gp(n,0)=0, for any given ε>0, there exists ρn∈(0,θ), such that |Gp(n,xn)|≤ε|xn|p as |xn|≤ρn, n∈Z[1,T]. Thus
(25)1p∑n=1T‍b(n)|xn|p+∑n=1T‍Gp(n,xn)≤(bp+ε)∑n=1T‍|xn|p≤13δ∥x∥2.
Let ρ=min{ρ1,…,ρT}. For 0<∥x∥≤ρ<1, it follows that
(26)Jp(x)≥12δ∥x∥2-13δ∥x∥2>0,x∈W+=X+.
We need to prove thatJp(x)≤0 for x∈X-=W-⨁W0, ∥x∥≤ρ. We first claim that
(27)∑n=1T‍b(n)|xn|p>0,∀x∈W-⨁W0,x≠0.
Indeed, by contradiction, assume that ∑n=1Tb(n)|xn|p≤0, for some x∈W-⨁W0, x≠0. Since Λ*(p)=∑n=1Tb(n)|en|p>0, where e=(e1T,e2T,…,eNT)T=(uT,uT,…,uT)T∈W-⨁W0, and (W-⨁W0)∖{0} is arcwise connected, then there exists a x0∈(W-⨁W0)∖{0}, such that ∑n=1Tb(n)|xn0|p=0. Thus ∥Δx0∥2≥λ*(p)∥x0∥2 by the definition of λ*(p). On the other hand, by the definition of W-⨁W0, we have ∥Δx0∥2≤a∥x0∥2. This is a contradiction with assumption a<λ*(p). So the claim (27) holds.
There existsc4>0 by (27), such that ∑n=1Tb(n)|xn|p≥c4∥x∥pp for all x∈W-⨁W0∖{0}, where ∥x∥p=(∑n=1T|xn|p)1/p. For x∈W-⨁W0, ∥x∥≤ρ, ε sufficiently small, we have
(28)Jp(x)≤-1p∑n=1T‍b(n)|xn|p-∑n=1T‍Gp(n,xn)≤-c4p∥x∥pp+ε∥x∥pp≤0.
Since Jp(0)=0 and Jp satisfies (PS) condition by Lemma 4, so by Proposition 6, we obtain that Ck(Jp,0)≠0 for k=dim(W-⨁W0).
Case2(Λ*(p)<0,X+=W+⨁W0,X-=W-). It is easy to see that Jp(x)≤0 by ∥Δx∥2-a∥x∥2≤-δ∥x∥2 and p>2, where x∈W- and ∥x∥≤ρ. We need to claim that Jp(x)>0, for x∈W+⨁W0, 0<∥x∥≤ρ.
Suppose not that there exists a sequence{x(j)}⊂ET such that
(29){x(j)}⊂W+⨁W0∖{0},0<∥x(j)∥≤ρ,Jp(x(j))≤0,
for large j. For ∥x(j)∥≤ρ, by Lemma 1, we get
(30)|∑n=1T‍[1pb(n)|xn(j)|p+Gp(n,xn(j))]|≤∑n=1T‍[bp|xn(j)|p+ε|xn(j)|p]≤(bp+ε)(1a1)p∥x(j)∥p.
Set yn(j)=xn(j)/∥x(j)∥. Then by (29) and the previous formula, we have
(31)0≥Jp(x(j))∥x(j)∥2≥12(∥Δy(j)∥2-a)-(bp+ε)(1a1)p∥x(j)∥p-2.
On the other hand,∥Δy(j)∥2≥a by the definition of W+⨁W0. Hence by p>2, there exists a subsequence converges to y0∈ET, such that ∥Δy0∥2=a, that is y0∈W0 and ∥y0∥=1. Since ∥Δx(j)∥2≥a∥x(j)∥2 for {x(j)}⊂W+⨁W0, it follows from Jp(x(j))≤0 that
(32)0≤1p∑n=1T‍b(n)|xn(j)|p+∑n=1T‍Gp(n,xn(j))≤1p∑n=1T‍b(n)|xn(j)|p+ε(1a1)p∥x(j)∥p.
Dividing by ∥x(j)∥p in the previous inequality, then ∑n=1Tb(n)|yn0|p=limj→∞∑n=1Tb(n)|yn(j)|p≥0.
Sincee,y0∈W-⨁W0, Λ*(p)=∑n=1Tb(n)|en|p<0 and (W-⨁W0)∖{0} is arcwise connected, then there exists a y-∈(W-⨁W0)∖{0} such that ∑n=1Tb(n)|y-n|p=0. Thus ∥Δx-∥2≥λ*(p)∥x-∥2 by the definition of λ*(p). On the other hand, ∥Δx-∥2≤a∥x-∥2 by the definition of W-⨁W0. This is a contradiction with assumption a<λ*(p). That is to say, the claim is valid.
By Proposition6, we obtain Ck(Jp,0)≠0, k=dimW-. The proof is completed.Lemma 8.
Ifa<λ*(q), then Ck(Jq,∞)≠0 for k=dimX-, where X-=W-⨁W0 as Λ*(q)>0, X-=W- as Λ*(q)<0.
The proof is similar to that of Lemma7 and is omitted.


## 3. Proof of Theorem2

Lemma 9.
Leta<λ*(p). If there exists K1>0 such that for any K>K1, Jp(x)≤-K, then one has ∑n=1Tb(n)|xn|p>0, and (d/dt)Jp(tx)|t=1<0.Proof.
We first claim that∥x∥ is sufficiently large, if x satisfies condition of Lemma 9. Suppose not there exists M>0 such that ∥x∥≤M. So there exists {x(j)}⊂ET, x0∈ET, such that x(j)→x0 as j→∞. Since for any j>K1, we have Jp(x(j))≤-j, thus Jp(x0)=limj→∞Jp(x(j))=-∞. It is a contradiction with Jp(x0)=c.
If∥x∥ is large enough, then we can assume that |xn| is large enough for n∈Z[1,k] and |xn| are bounded for n∈Z[k+1,T]. Therefore, by assumption (A1), for any given ε>0, there exists M1>0 such that
(33)|gp(n,xn)|≤ε|xn|+M1T,|Gp(n,xn)|≤ε|xn|2+M1T,∀(n,xn)∈Z[1,T]×ℝN.
We claim that ∑n=1Tb(n)|xn|p>0. Suppose not that, for j>K1, there exists {x(j)}⊂ET such that
(34)∑n=1T‍b(n)|xn(j)|p≤0.
By Jp(x(j))≤-j≤0, (33) and (34), we have
(35)12∥Δx(j)∥2≤a2∥x(j)∥2+∑n=1T‍Gp(n,xn(j))≤a2∥x(j)∥2+ε∥x(j)∥2+M1.
Set yn(j)=xn(j)/∥x(j)∥ and divided by ∥x(j)∥2 in the previous inequality. Since ε can be small enough, then there exists a subsequence that converges to y0∈ET, such that ∥Δy0∥2≤a, ∥y0∥=1. Moreover, by (33) and (34), we get
(36)0≥1p∑n=1T‍b(n)|xn(j)|p≥j+12∥Δx(j)∥2-a2∥x(j)∥2-∑n=1T‍Gp(n,xn(j))≥-(a2+ε)∥x(j)∥2-M1.
Since p>2 and limj→∞∥x(j)∥=∞, divided by ∥x(j)∥p in the previous inequality, we have ∑n=1Tb(n)|yn0|p=limj→∞∑n=1Tb(n)|yn(j)|p=0, that is, ∥Δy0∥≥λ*(q), which deduce a contradiction. So the claim ∑n=1Tb(n)|xn|p>0 holds.
Next we prove that(d/dt)Jp(tx)|t=1<0 holds. By contradiction, there exists a sequence {x(j)}⊂ET such that, for j>K1,
(37)ddtJp(tx(j))|t=1≥0.
Then, by (7), we get
(38)ddtJp(tx(j))|t=1=∥Δx(j)∥2-a∥x(j)∥2-∑n=1T‍b(n)|xn(j)|p-∑n=1T‍(gp(n,xn(j)),xn(j)),
and by (37) and Jp(x(j))≤-j<0, it follows that
(39)(1-p2)(∥Δx(j)∥2-a∥x(j)∥2)-∑n=1T‍(gp(n,xn(j)),xn(j))+p∑n=1T‍Gp(n,xn(j))=ddtJp(tx(j))|t=1-pJp(x(j))≥0.
Setyn(j)=xn(j)/∥x(j)∥ and divided by ∥x(j)∥2 in the previous formula; since p>2 and ε can be small enough, then there exists a subsequence converges to y0∈ET such that ∥Δy0∥2≤a, ∥y0∥=1. Moreover, by (37) and the first claim, we get
(40)0<∑n=1T‍b(n)|xn(j)|p≤∥Δx(j)∥2-a∥x(j)∥2-∑n=1T‍(gp(n,xn(j)),xn(j)).
Divided by ∥x(j)∥p in the previous formula, and by p>2, it follows that ∑n=1Tb(n)|yn0|p=0. This is a contradiction with the definition of λ*(p) and condition a<λ*(p). So the second claim holds. The proof is completed.Based on Lemma9, we introduce the following notations:
(41)Jp-K={x∈ET:Jp(x)≤-K},Ep+={x∈ET:∑n=1T‍b(n)|xn|p>0},E(Ω+)={x∈ET:xn=0forn∈Z[1,T]∖Ω+}∖{0}.
Clearly, E(Ω+)⊂Ep+. And by Lemma 9, we have Jp-K⊂Ep+. In order to describe the Hq(ET,Jp-K), we need to show the following lemma.Lemma 10.
Ifa<λ*(p), then there exists K1>0, such that for any K>K1, Jp-K is a strong deformation retraction of Ep+. Moreover, E(Ω+) and Ep+ are homotopy equivalent.Proof.
Now we prove thatJp-K is a strong deformation retraction of Ep+.
By Lemma9, we have Jp-K⊂Ep+. Let x∈Ep+. By Lemma 9, there exists a unique tp=tp(x)>0 such that Jp(tpx)=-K. By applying Implicit Function Theorem, tp(x) is a continuous function in Ep+. Let Tp(x)=max{tp(x),1} and define fp(s,x)=(1-s)x+sTp(x)x, then fp:[0,1]×Ep+→Jp-K is a strong deformation retraction. Thus Jp-K is a strong deformation retraction of Ep+.
We next claim thatE(Ω+) is a strong deformation retraction of Ep+. Clearly, in terms of the notations, we have E(Ω+)⊂Ep+.Letξp:Z[1,T]→ℝ be a function such that
(42)ξp(n)=1ifn∈Ω+,ξp(n)=0ifn∈Ω-,ξp(n)∈[0,1]ifn∈Z[1,T]∖(Ω+∪Ω-).
Define
(43)ζp(s,xn)={(1-2s)xn+2sξp(n)xnif0≤s≤12,2(1-s)ξp(n)xn+2(s-12)P(ξp(n)xn)2(1-s)ξp(n)xn+2(s-12)if12≤s≤1,
where P:ET→E(Ω+) is a projection operator. Then ζp:[0,1]×Ep+→E(Ω+) is a deformation retraction. Indeed,
(44)ζp(0,x)=x,ζp(1,x)∈E(Ω+),forx∈Ep+,ζp(s,x)=x,forx∈E(Ω+)ands∈[0,1].
For x∈Ep+, if s∈[0,1/2], then
(45)∑n=1T‍b(n)|ζp(s,xn)|p=∑n∈Ω+‍b(n)|xn|p+∑n∈Ω-‍b(n)(1-2s)p|xn|p≥∑n=1T‍b(n)|xn|p>0,
where 0≤(1-2s)p≤1, that is, ζp(s,x)∈Ep+. If s∈(1/2,1], it follows that
(46)∑n=1T‍b(n)|ζp(s,xn)|p=∑n∈Ω+‍b(n)|2(1-s)ξp(n)xn+2(s-12)P(ξp(n)xn)|p≥0.
We claim that the equality of the previous formula cannot hold. Otherwise, Pxn=-((1-s)/(s-(1/2)))xn, for n∈Ω+, which implies that Pxn=0. Hence xn=0 in Ω+, which contradicts with the fact x∈Ep+. So ∑n=1Tb(n)|ζp(s,xn)|p>0, that is, ζp(s,x)∈Ep+ as s∈(1/2,1]. Therefore, ζp is a deformation retraction from Ep+ onto E(Ω+), and this completes the proof.Proof of Theorem2.
SinceE(Ω+) is well known to be contractile in itself, and by Lemma 10, it follows that Jp-K is homotopically equivalent to E(Ω+) for K large enough, then the Betti numbers (cf. [11, 13]) are(47)βk=dimCk(Jp,∞)=dimHk(ET,Jp-K)=dimHk(ET,E(Ω+))=0,k∈Z[0,NT].
Now we suppose that system (4) has only trivial solution; that is, Jp has only critical point x=0, then we have the Morse-type numbers Mk=dimCk(Jp,0) for k∈Z[0,NT] (cf. [13]). Moreover, by Lemma 7, Ck(Jp,0)≠0 for k=dimW- or k=dim(W-⨁W0). Since Jp satisfies (PS) condition by Lemma 4, then using Morse Relation, we have the following. 							(48)0=∑k=0NT‍(-1)kβk=∑k=0NT‍(-1)kMk≠0,
which is a contradiction. Therefore, Jp has at least one critical point x*≠0 and system (4) has at least a nonzero T-periodic solution.


## 4. Proof of Theorem3

For convenience, we introduce the following notations:(49)Jqc={x∈ET:Jq(x)≤c},c∈ℝ,Eq+={x∈ET:∑n=1T‍b(n)|xn|q>0}.Clearly,Eq+∪{0} is star-shaped with respect to the origin and E(Ω+)⊂Eq+, where E(Ω+) is given in Section 3. Similarly with the proof of Lemmas 9 and 10, we have the following.Lemma 11.
Leta<λ*(q). Then there exists ρ>0 such that (d/dt)Jq(tx)|t=1>0 for any x∈Mρ={x∈Bρ∩Eq+:Jq(x)≥0}, where Bρ stands for the closed ball in ET of radius ρ>0 with the center at zero.Lemma 12.
Leta<λ*(q). Then there exists ρ>0 such that (Jq0∩Bρ)∖{0} is a retract of Eq+∩Bρ, and E(Ω+) is a strong deformation retraction of Eq+.Proof of Theorem3.
We first prove thatJq0∩Bρ is contractible in itself. In fact, it is sufficient to show that Jq0∩Bρ is starshaped with respect to the origin; that is, x∈Jq0∩Bρ implies that tx∈Jq0∩Bρ for all t∈[0,1].
Assume, by a contradiction, that there existsx0∈Jq0∩Bρ and t0∈(0,1), such that Jq(t0x0)>0. It follows from Lemma 11 that (d/dt)Jq(t0x0)>0. By the monotonicity arguments, this implies that
(50)Jq(tx0)>0∀t∈[t0,1].
This contradicts the assumption x0∈Jq0, which implies Jq(x0)≤0.
On the other hand, sinceE(Ω+) is contractible in itself, and Eq+∪{0} is starshaped with respect to the origin, then Eq+∩Bρ is contractible in itself. The retract of the set which is contractible in itself is also contractible (cf. [19]); it follows that the set (Jq0∩Bρ)∖{0} is contractible by Lemma 12.
Combining the previous argument,Jq0∩Bρ and (Jq0∩Bρ)∖{0} are contractible in themselves. 							(51)dimCk(Jq,0)=dimHk(Jq0∩Bρ,(Jq0∩Bρ)∖{0})=0,k∈Z[0,NT].
By Lemma 8, Ck(Jq,∞)≠0 for k=dim(W-⨁W0) or k=dimW-. Therefore, by Morse Relation and the same methods in proof of Theorem 2, it follows that Jq has at least one critical point x*≠0 and system (5) has at least a nonzero T-periodic solution.



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*Source: 517036-2013-04-09.xml*