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 number λ∗. By using Morse theory, we obtain some new results concerning the existence of nontrivial periodic solution.
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 that H 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 of H(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 replace p 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
gp(n,x)=o(|x|) as |x|→∞ uniformly in n,
gp(n,x)=o(|x|p-1) as |x|→0 uniformly in n.
Case 2 (asymptotically subquadratic case: 1<s<2).
Here we replace q 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
gq(n,x)=o(|x|q-1) as |x|→∞ uniformly in n,
gq(n,x)=o(|x|) as |x|→0 uniformly in n.
Before stating the main results, we introduce space ET={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 numbers a1,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.
If a<λ*(p), then (4) has a nonzero T-periodic solution.
Theorem 3.
If a<λ*(q), then (5) has a nonzero T-periodic solution.
This paper is divided into four sections. Section 2 contains some preliminaries, and the proofs of Theorems 2 and 3 are given in Sections 3 and 4, respectively.
2. Preliminaries2.1. Variational Functional and (PS) Condition
For seeking T-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 a C1-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.
Functional Jp 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 that Jp 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.
Functional Jq satisfies (PS) condition if a<λ*(q).
The proof is similar to that of Lemma 4 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 numbers λ*(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 assumption λ*(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
Let X 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 that φ(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 that C2-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 Propostion 6, one proves the following lemmas with respect to ET=X+⨁X-.
Lemma 7.
If a<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 that Jp(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 exists c4>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).
Case 2(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.
Since e,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 Proposition 6, we obtain Ck(Jp,0)≠0, k=dimW-. The proof is completed.
Lemma 8.
If a<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 Lemma 7 and is omitted.
3. Proof of Theorem 2Lemma 9.
Let a<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.
Set yn(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 Lemma 9, 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.
If a<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 that Jp-K is a strong deformation retraction of Ep+.
By Lemma 9, 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 that E(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 Theorem 2.
Since E(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 Theorem 3
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.
Let a<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.
Let a<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 Theorem 3.
We first prove that Jq0∩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 exists x0∈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, since E(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.
Acknowledgments
This research is supported by the National Natural Science Foundation of China under Grants (11101187) and NCETFJ (JA11144), the Excellent Youth Foundation of Fujian Province (2012J06001), and the Foundation of Education of Fujian Province (JA09152).
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