# Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

Authors: Jingli Xie; Zhiguo Luo; Guoping Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the existence of  homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a2kT-periodic approximation solution is a homoclinic solution of our problem.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the second-order impulsive differential equation(1)q¨(t)+Vq(t,q)=f(t),t≠tj,t∈ℝ,(2)-Δq˙(tj)=gj(q(tj)),j∈ℤ,
where q∈ℝN, f∈(ℝ,ℝN), Vq(t,q)=gradqV(t,q), gj(q)=gradqGj(q), Gj∈(ℝN,ℝN) for each j∈ℤ, and the operator Δ is defined as Δq˙(tj)=q˙(tj+)-q˙(tj-), where q˙(tj+)(q˙(tj-)) denotes the right-hand (left-hand) limit of q˙ at tj. There exist an m∈ℕ and a T>0 such that 0=t0<t1<⋯<tm=T, tj+m=tj+T, and gj+m=gj, j∈ℤ. V:ℝ×ℝN→ℝ satisfies(V1)
V(t,q)=-K(t,q)+W(t,q), K,W∈C1(ℝ×ℝN,ℝ), and is T-periodic in its first variable.We are mainly concerned with the existence of homoclinic solutions of system (1) and (2). A function q(t)∈C(ℝ,ℝN) is said to be a (classical) solution of (1) and (2) if q(t) satisfies (1) and (2). A (classical) solution q(t) of (1) and (2) is a homoclinic solution and if q(t)→0 as t→±∞ and q˙(t±)→0 as t→±∞.WhenΔq˙(tj)≡0, f(t)=0, and V(t,q)=(1/2)(L(t)q,q)+W(t,q), system (1) and (2) reduces to Hamiltonian system
(3)q¨(t)+Vq(t,q)=0,t∈ℝ.
Rabinowitz [1] studied the existence of nontrivial homoclinic solutions of it.WhenΔq˙(tj)≡0 and V(t,q) satisfied (V1), system (1), (2) reduces to Hamiltonian system
(4)q¨(t)+Vq(t,q)=f(t),t∈ℝ.
Izydorek and Janczewska [2] studied the existence of homoclinic solutions of it.Some classical tools such as some fixed point theorems in cones, topological degree theory, the upper and lower solutions method combined with monotone iterative technique, and variational methods [3–20] have been widely used to get solutions of impulsive differential equations. However, the existence of homoclinic solutions for the impulsive systems is paid little attention. It is well known that the homoclinic orbit rupture phenomenon can lead to chaos, which has been interesting to the mathematicians in recent years [21–26]. In the literature, Coti-Zelati et al. [27] used dual variational methods, and Lions [28] and Hofer and Wysocki [29] employed concentration compactness method, Ekeland's variational principle, that they established the existence of homoclinic solutions of the first-order Hamiltonian systems. Rabinowitz [1] and Izydorek and Janczewska [2] obtained homoclinic solutions of a class of second order Hamiltonian systems as a limit of its periodic solutions.In recent paper [18], Zhang and Li studied the existence of homoclinic solutions of an impulsive Hamiltonian system q′′(t)+Vq(t,q)=f(t), t≠tj, t∈ℝ, Δq′(tj)=gj(q(tj)), j∈ℤ as a limit of its periodic solutions. In detail, they obtained the following theorem.Theorem A (see [18]).
Assume thatf∈C(ℝ,ℝℕ)⋂‍L2(ℝ,ℝℕ), gj is continuous and m-periodic in j, and V, gj satisfy the following conditions:  (H1)
V:ℝ×ℝN→ℝ is continuous differentiable T-periodic, and there exist positive constants b1,b2>0 such that
(5)b1|q|2≤-V(t,q)≤b2|q|2,∀(t,q)∈[0,T]×ℝℕ;(H2)
-V(t,q)≤-Vq(t,q)q≤-2V(t,q) for all (t,q)∈[0,T]×ℝℕ;(H3)
lim|q|→0(gj(q)/|q|)=0 for j=1,2,…,m;(H4)
there exists aμ>2 such that
(6)gj(q)q≤μGj(q)<0,forq∈ℝℕ∖{0},j=1,2,…,m,then the Hamiltonianq¨(t)+Vq(t,q)=f(t), t≠tj, t∈ℝ, Δq′(tj)=gj(q(tj)), j∈ℤ, possesses at least one nonzero homoclinic solution.Motivated by papers [1, 2, 18], in this paper, we synthesize their methods to study the existence of homoclinic solutions of systems (1), (2). In detail, firstly, we introduce the following sequence equations:
(7)q′′(t)+Vq(t,q)=fk(t),t≠tj,t∈ℝ,
where for each k∈ℕ, fk:ℝ→ℝN is a 2kT-periodic extension of the restriction of f to the interval [-kT,kT]. Secondly, we study periodic solutions of (2) and (7) by converting the problem to the existence of critical points of some variational structure. finally, we find the homoclinic solutions of (1) and (2) as the limit of the periodic solutions of (2) and (7).Part of the difficulty in treating (1) and (2) is subjected to the impulsive perturbation which destroys continuities of the velocity and when we apply the Mountain Theorem to prove our main result, we need the constant ρ, α appearing in the theorem to be independent of k.Our result is the following theorem.Theorem 1.
Assume thatV satisfies (V1), and K, W, and f satisfy the following:(K1)
there exist constantsa1>0 and γ∈(1,2] such that for all (t,q)∈[0,T]×ℝN(8)K(t,0)=0,K(t,q)≥a1|q|γ;(K2)
there existsθ∈(1,2] such that
(9)K(t,q)≤Kq(t,q)q≤θK(t,q),∀(t,q)∈[0,T]×ℝN;(W1)
W(t,0)=0, Wq(t,q)=o(|q|), as |q|→0 uniformly for t∈[0,T];(W2)
there exist constantsr>2 and b1>0 such that
(10)W(t,q)≤b1|q|r,∀(t,q)∈[0,T]×ℝN;(W3)
there existμ>1, a2>0, and β(t)∈L1(ℝ,ℝ+) such that μ>r-γ and
(11)Wq(t,q)q-2W(t,q)≥a2|q|μ-β(t),∀(t,q)∈[0,T]×ℝN;(W4)
W(t,q)/|q|2→+∞ as |q|→+∞ uniformly in t∈[0,T];(G1)
Gj(0)=0, gj(q)=o(|q|), as |q|→0, j=1,2,…,m;(G2)
there existsb2>0 such that Gj(q)≤b2|q|r, q∈ℝN, j=1,2,…,m;(G3)
gj(q)q-2Gj(q)≥0, q∈ℝN∖{0}, j=1,2,…,m;(F1)
f∈C(ℝ,ℝN)∩L2(ℝ,ℝN)∩Lμ/(μ-1)(ℝ,ℝN), (∫-∞+∞‍|f(t)|2dt)1/2<min{1/2,a1-b1-M}(1/C), where M=sup{Gj(q):j=1,2,…,m,|q|=1}, a1>b1+M, and C is a constant of (17). Then the system (1) and (2) possesses at least one nonzero homoclinic solution.The rest of this paper is organized as follows. In Section2 we present some preliminary results. Our main result's proofs are given in Section 3.


## 2. Preliminaries

Let(12)H2kT={q:ℝ→ℝN∣q,q˙∈L2(([-kT,kT]),ℝN),u(t)=u(t+2kT),t∈ℝ:ℝ→ℝNq,q˙∈L2(([-kT,kT]),ℝN)}.
Then H2kT is a Hilbert space with the norm defined by
(13)∥q∥H2kT=(∫-kTkT‍(|q˙(t)|2+|q|2)dt)1/2,q∈H2kT.
For the norm in L2kT∞(ℝ,ℝ), which denotes a space of 2kT periodic essentially bounded measurable functions from ℝ into ℝN, ∥q∥L2kT∞=esssup{|q(t)|:t∈[-kT,kT]}. Next we set Ωk={-km+1,-km+2,…,0,1,2,…,km-1,km} and define a functional φk as
(14)φk(q)=12ηk2(q)-∫-kTkT‍Wdt-∑j∈Ωk‍Gj(q(tj))+∫-kTkT‍fkqdt,q∈H2kT,
where
(15)ηk(q)=(∫-kTkT‍|q˙(t)|2+2K(t,q(t))dt)1/2.
Note that φk is Fréchet differentiable at any q∈H2kT, and for any p∈H2kT, we have
(16)φk′(q)(p)=limh→0φk(q+hp)-φk(q)h=∫-kTkT‍(q˙(t)p˙(t)-Vq(t,q)p+fkp)dt-∑j∈Ωk‍gj(q(tj))p(tj).
It is clear that critical points of the functional φk are classical 2kT-periodic solutions of system (2) and (7).Lemma 2 (see [2]).
There is a positive constantC such that for each k∈N and q∈H2kT the following inequality holds:
(17)∥q∥L2kT∞≤C∥q∥H2kT,∀q∈H2kT.Lemma 3.
Setm1=inf{Gj(q):|q|=1,j=1,2,…,m}; then for every ζ∈ℝ∖{0} and q∈H2kT∖{0}, we have
(18)∑j∈Ωk‍Gj(ζq(tj))≥m1|ζ|2∑j∈Ωk‍|q(tj)|2-2kmm1.Proof.
SetAk={j∈Ωk:|ζq(tj)|≤1}, Bk={j∈Ωk:|ζq(tj)|≥1}, and φ(s)=s2Gj(q/s), j∈Ωk, s>0. By (G3), we have
(19)φ′(s)=s[2Gj(qs)-gj(qs)qs]≤0.
So we have Gj(q)≥|q|2Gj(q/|q|), |q|≥1. If Bk is empty, we have q(tj)=0, j∈Ωk, which implies ∑j∈ΩkGj(ζq(tj))=0≥-2kmm1. Therefore, Without loss of generality, we can assume that Bk is nonempty, and we have
(20)∑j∈Ωk‍Gj(ζq(tj))≥∑j∈Bk‍Gj(ζq(tj))≥∑j∈Bk‍Gj(ζq(tj)|ζq(tj)|)|ζq(tj)|2≥m1∑j∈Bk‍|ζq(tj)|2≥(∑j∈Ωk‍|ζq(tj)|2-∑j∈Ak‍|ζq(tj)|2)m1≥m1|ζ|2∑j∈Ωk‍|q(tj)|2-2kmm1.Lemma 4 (see [30]).
LetE be a real Banach space and let φ:E→ℝ be a C1-smooth functional satisfying the Palais-Smale condition and φ(0)=0. If φ satisfies the following conditions:(i)
there exist constantsρ,α>0 such that φ∣∂Bρ≥α,(ii)
there existse∈E∖B¯ρ, such that φ(e)≤0, then φ possesses a critical value c≥α given by
(21)c=infg∈Γmaxs∈[0,1]φ(g(s)),where(22)Γ={g∈C([0,1],E):g(0)=0,g(1)=e}.Lemma 5 (see [2]).
Letq:ℝ→ℝN be a continuous mapping such that q˙∈Lloc2(ℝ,ℝN). For every t∈ℝ the following inequality holds:
(23)|q(t)|≤2(∫t-1/2t+1/2(|q(s)|2+|q˙(s)|2)ds)1/2.


## 3. Proof of Theorem1

We have divided the proof of Theorem1 into a sequence of lemmas.Lemma 6.
Assume that (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1) are satisfied; system (2), (7) possesses a 2kT-periodic solution.Proof.
It is clear thatφk(0)=0. It is well known that Lemma 4 holds true with the (C) condition replacing the usual (PS) condition. We say the functional φk satisfies the (C) condition; that is, for every sequence {qn}⊂H2kT, {qn} has a convergent subsequence if {φk(qn)} is bounded and limn→∞(1+∥q∥H2kT)φk′(qn)=0.
Step  1. Pick {qn}⊂H2kT such that {φk(qn)} is bounded and limn→∞(1+∥q∥H2kT)φk′(qn)=0 then there exists a constant C1>0 such that
(24)C1≥2φk(qn)-φk′(qn)(qn)=∫-kTkT‍[2K(t,qn)-Kq(t,q)]dt-∫-kTkT‍[2W(t,qn)-Wq(t,qn)qn]dt-∑j∈Ωk‍[2Gj(qn(tj))-gj(qn(tj))qn(tj)]+∫-kTkT‍fkqndt≥α1∫-kTkT‍|qn|μdt-∫-∞+∞‍β(t)dt-(∫-∞+∞‍|f|μ′dt)1/μ′(∫-kTkT‍|qn|μdt)1/μ,
where μ′>1 and 1/μ′+1/μ=1. (24) implies (∫-kTkT‍|qn|μdt)1/μ is bounded; that is, there exists a constant C2>0 such that
(25)(∫-kTkT‍|qn|μdt)1/μ<C2.
From (W2), (G2), (17), and (25), we have
(26)12ηk2(qn)=φk(qn)+∫-kTkT‍W(t,qn)dt-∫-kTkT‍fkqndt+∑j∈Ωk‍Gj(qn(tj))≤C3+b1(C∥qn∥H2kT)r-μ∫-kTkT‍|qn|μdt+(∫-∞+∞‍|f(t)|μ′dt)1/μ′C2+b2(C∥qn∥H2kT)r-μ∑j∈Ωk‍|qn(tj)|μ≤C3+(b1+b2)C2μCr-μ(∥qn∥H2kT)r-μ+(∫-∞+∞‍|f(t)|μ′dt)1/μ′C2.
On the other hand, it follows from (K1) and (17) that
(27)12ηk2(qn)≥12∫-kTkT‍[|q˙n|2+2a1|qn|γ]dt≥12∫-kTkT‍|q˙n|2dt+a1(C∥qn∥H2kT)γ-2∫-kTkT‍|qn|2dt≥min{12∥qn∥H2kT2,a1Cγ-2∥qn∥H2kTγ}.
Combining (26) and (27), we obtain
(28)min{12∥qn∥H2kT2,a1Cγ-2∥qn∥H2kTγ}≤C3+(b1+b2)C2μCr-μ(∥qn∥H2kT)r-μ+(∫-∞+∞‍|f(t)|μ′)1/μ′C2.
Since r-μ<γ≤2, it follows that ∥qn∥H2kT is bounded. In a similar way to [21, Proposition B35], we can prove that {qn} has a convergent subsequence. So, the functional φk satisfies the (C) condition.
Step  2. We show that the functional φk satisfies the assumption (i) of Lemma 4. Set φ(s)=s2Gj(q/s), j=1,2,…,m, s>0. By (G3), we have
(29)φ′(s)=s[2Gj(qs)-gj(qs)qs]≤0.
Hence when 0<∥q∥L2kT∞≤1, we have
(30)∑j∈Ωk‍Gj(q(tj))≤∑j∈Ωk‍Gj(q(tj)|q(tj)|)|q(tj)|2≤M∑j∈Ωk‍|q(tj)|2≤M∥q∥H2kT2.
From (K1), (W2), and (30), we have
(31)φk(q)=12ηk2(q)-∫-kTkT‍W(t,q)dt-∑j∈Ωk‍Gj(q(tj))+∫-kTkT‍fkqdt≥min{12,(a1-b1-M)}∥q∥H2kT2-(∫-∞+∞‍|f(t)|2dt)1/2∥q∥H2kT.
Set
(32)ρ=1C,α=min{12,(a1-b1-M)}1C2-(∫-∞+∞‍|f(t)|2dt)1/21C>0.
Let ∥q∥H2kT=ρ; then 0<∥q∥L2kT∞≤1 and (31) gives φk(q)≥α>0.
Step  3. We show that the functional φk satisfies assumption (ii) of Lemma 4.
In order to verify (ii), we chooseζ∈ℝ, Q∈H2T∖{0} such that Q(±T)=0 and M2=max|t|≤T,|q|≤1K(t,q). Set h(s)=s-θK(t,sq). By (K2), we have h′(s)=(Kq(t,sq)sq-θK(t,sq))/sθ+1≤0. So we have
(33)K(t,q){≥|q|θK(t,q|q|),|q|≤1,t∈[0,T],≤|q|θK(t,q|q|),|q|≥1,t∈[0,T].
Define(34)Q~={Q,t∈[-T,T],0,t∈[-kT,kT][-T,T].
TakeA>(1+2M2∥Q∥H2T2)/2∫-TT‍Q(t)dt. By (W4), there exists B>0 such that
(35)W(t,q)≥A|q|2-B,t∈[0,T]×ℝN.
By (33), (34), (35), and Lemma 3, we have
(36)φk(ζQ~)=12η12(ζQ)-∫-TT‍W(t,ζQ)dt-∑j∈Ω1‍Gj(ζQ(tj))+ζ∫-TT‍f1Qdt≤12(1+2M2)|ζ|2∥Q∥H2T2-A|ζ|2∫-TT‍|Q(t)|2dt+2(M2T+TB+mm2)-m1|ζ|2∑j∈Ω1‍|Q(tj)|2+|ζ|(∫-TT‍|f1|2dt)1/2(∫-TT‍|Q|2dt)1/2≤|ζ|2(1+2M22∥Q∥H2T2-A∫-TT‍|Q(t)|2dt)+2(M2T+TB+mm1)+|ζ|(∫-TT‍|f1|2dt)1/2(∫-TT‍|Q|2dt)1/2.
Clearly, φk(ζq)→-∞ as |ζ|→+∞, so (ii) holds. By Lemma 4, φk possesses a critical value ck≥α>0. Let qk denote the corresponding critical point of φk on H2kT; that is,
(37)φk(qk)=ck,φk′(qk)=0.
Hence the system (2), (7) possesses a nontrivial 2kT-periodic solution qk.Lemma 7.
Let{qk} be the sequence given by (37). Then there exist a subsequence {qj,j} of {qk} and a function q0∈Wloc1,2∩Lloc∞(ℝ,ℝℕ) such that {qj,j} converges to q0 weakly in Wloc1,2 and strongly in Lloc∞(ℝ,ℝℕ).Proof.
We assert that there is a constantM3>0 independent of k such that
(38)∥qk∥H2kT≤M3.
Let e1∈H2T∖{0} such that e1(±T)=0, e1(tk)≠0 for some tk∈(-T,T), and φ1(e1)≤0. Define
(39)ek={e1,|t|≤T,0,T≤|t|≤kT.
We extend ek in the way of 2kT-periodic to ℝ. For simplicity, we also note it again by ek. It is clear that ek∈H2kT and φk(ek)=φ1(e1)≤0. Define gk:[0,1]→H2kT by gk(s)=sek for s∈[0,1]. Then, we have
(40)ck≤maxs∈[0,1]φk(gk(s))=maxs∈[0,1]φ1(g1(s))≡c0,
independently of k. The rest detailed argument is similar to the proof of Step 1 in Lemma 6 and we thus omit it here.
Hence,{qk} is a bounded sequence in W1,2((-T,T),ℝN) and we may pick a subsequence {q1,k} such that {q1,k} converges weakly in W1,2((-T,T),ℝN) and strongly in L∞((-T,T),ℝN). Next {q1,k} is a bounded sequence in W1,2((-2T,2T),ℝN), so we may pick a subsequence {q2,k} such that {q2,k} converges weakly in W1,2((-2T,2T),ℝN) and strongly in L∞((-2T,2T),ℝN). We can repeat this process and obtain, for any positive integer n, a sequence {qn,k} which converges weakly in W1,2((-nT,nT),ℝN) and strongly in L∞((-nT,nT),ℝN), and
(41){qk}⊃{q1,k}⊃{q2,k}⊃⋯⊃{qn,k}⊃⋯.
Therefore, for any positive integer n, the sequence {qk,k} converges weakly in W1,2((-nT,nT),ℝN) and strongly in L∞((-nT,nT),ℝN). Hence there exists a function q∈Wloc1,2(ℝ,ℝN)∩Lloc∞(ℝ,ℝN) such that the sequence {qk,k} converges weakly in q∈Wloc1,2(ℝ,ℝN) and strongly in Lloc∞(ℝ,ℝN).Lemma 8.
The functionq0 determined by Lemma 7 is a non-zero homoclinic solution of the system (1), (2).Proof.
The proof will be divided into four steps.
Firstly, we show thatq0 is a solution of the system (1), (2). Here, for simplicity, we denote {qk,k} by {qk}. For any given interval (a,b)⊂(-kT,kT) and any p∈W01,2((a,b),ℝN), define
(42)p1={p,t∈(a,b),0,t∈(-kT,kT)∖(a,b),
so for any p∈W01,2((a,b),ℝN), we have
(43)0=φk′(qk)p1=∫ab‍(q˙kp˙-Vq(t,qk)p+fkp)dt+∑tj∈(a,b)‍gj(qk(tj))p(tj).
Therefore, one has
(44)∫ab‍(q˙0p˙-Vq(t,q0)p+fp)dt+∑tj∈(a,b)‍gj(q0(tj))p(tj)=limk→+∞(∑tj∈(a,b)‍∫ab‍(q˙kp˙-Vq(t,qk)p+fkp)dt+∑tj∈(a,b)‍gj(qk(tj))p(tj))=0.
The remained detailer argument is similar to the proof of Lemma 2.5 in [13] and we thus omit it here, so q0(t) is a solution of system (1) and (2).
Secondly we show thatq0(t)→0, as t→±∞. {qk} is weak continuity, so it is weak lower semicontinuity. One has
(45)∫-∞+∞‍(|q0|2+|q˙0|2)dt=limk→+∞∫-kTkT‍(|q0|2+|q˙0|2)dt≤limk→+∞limj→+∞inf∫-kT+kT‍(|qj|2+|q˙j|2)dt≤M32,
and so
(46)∫|t|≥r‍(|q0|2+|q˙0|2)dt→0,asr⟶+∞.
By (23) and (46), we obtain q0(t)→0, as t→±∞.
Thirdly, we prove thatq˙0(t±)→0 as t→±∞. We have proved q0(t) is a solution of system (1) and (2), so we have
(47)∫tj-1tj‍|q¨0(s)|2ds=∫tj-1tj‍(-Vq(s,q0(s))+f(s))2ds≤2∫tj-1tj‍(|Vq(s,q0(s))|2+|f(s)|2)ds.
By (V1), (K1), and (W1), one has Vq(t,0)=-Kq(t,0)+Wq(t,0)=0. Hence ∫tj-1tj‍|q¨0(s)|2ds→0 as j→±∞. By (23), one has
(48)|q˙0(t)|2≤2∫tj-1tj‍(|q˙0(s)|2+|q¨0(s)|2)ds≤2∫tj-1tj‍(|q0(s)|2+|q˙0(s)|2)ds+2∫tj-1tj‍|q¨0(s)|2ds,t∈(tj-1,tj).
Therefore one has q˙0(t±)→0 as t→±∞.
Finally, we showq0≠0 when f≡0. Since 0=t0<t1<⋯<tm=T, tj+m=tj+T, j∈ℤ, we can let δ_=minj∈ℤ{tj-tj-1} and δ¯=maxj∈ℤ{tj-tj-1}. By Hölder inequality and qk(tj)=qk(τ)+∫τtj‍q˙k(s)ds, τ∈[tj-1,tj], j∈Ωk, we have
(49)∑j∈Ωk‍|qk(tj)|2≤1δ_∑j∈Ωk‍∫tj-1tj‍|qk(tj)|2dτ=1δ_∑j∈Ωk‍∫tj-1tj‍(qk(τ)+∫τtj‍q˙k(s)ds)2dτ≤2δ_∑j∈Ωk‍∫tj-1tj‍(|qk(τ)|2+|∫τtj‍q˙k(s)ds|2)dτ≤2δ_∫-kTkT‍|qk(τ)|2dτ+2δ_∑j∈Ωk‍∫tj-1tj‍((tj-τ)∫τtj‍|q˙k(s)|2ds)dτ≤2δ_∫-kTkT‍|qk(τ)|2dτ+2δ¯2δ_∫-kTkT‍|q˙k(τ)|2dτ≤max{1,δ¯2}2δ_∥qk∥H2kT2.
Let M4=max{1,δ¯2}(2/δ_)>0, which is a constant independent of k.
It is clearly thatqk(t+jT), j∈ℤ, is a 2kT periodic solution of (2), (7). So we can assume the maximum of qk occurs in [-T,T]. Now we assume q0=0, so there is ∥qk∥L2kT∞=maxt∈[-T,T]|qk|→0 as k→+∞; therefore there exists integer N1>0 such that ∥qk∥L2kT∞<1. Combining (G1), there exists an integer N2>0 such that when k>N2, one has
(50)|gj(qk(tj))||qk(tj)|≤12M4min{1,a1},j∈ℤ.
By (49) and (50), when k>N2, one has
(51)∑j∈Ωk‍gj(qk(tj))qk(tj)≤∑j∈Ωk‍|gj(qk(tj))||qk(tj)||qk(tj)|2≤12min{1,a1}∥qk∥H2kT2.
Define a function Y:ℝ+→ℝ+ by
(52)Y(s)={0,s=0,maxt∈[0,T],|ξ|≤s|ξWq(t,ξ)||ξ|2,s>0.
It is clear that Y∈C(ℝ+,ℝ+) and is monotone nondecreasing, so we have
(53)|qkWq(t,qk(t))||qk(t)|2≤Y(∥qk∥L[-kT,kT]∞),t∈[-kT,kT].
Hence we have
(54)∫-kTkT‍|qkWq(t,qk(t))|dt≤Y(∥qk∥L[-kT,kT]∞)∫-kTkT‍|qk(t)|2dt≤Y(∥qk∥L[-kT,kT]∞)∥qk∥H2kT2.
Sincef≡0, qk is a solution of the system (1) and (2), so when k>max{N1,N2}, we have
(55)∫-kTkT‍qkWq(t,qk(t))dt+∑j∈Ωk‍gj(qk(tj))qk(tj)≥min{1,a1}∥qk∥H2kT2.
Combining (51), (54), and (55) we have
(56)Y(∥qk∥L[-kT,kT]∞)∥qk∥H2kT2+12min{1,a1}∥qk∥H2kT2≥min{1,a1}∥qk∥H2kT2.
Hence, we have
(57)Y(∥qk∥L[-kT,kT]∞)≥12min{1,a1}>0.
By the property of the function Y, there exists a constant M5>0 such that ||qk||L[-kT,kT]∞≥M5>0. This is a contradiction. Hence the system (1), (2) has a nontrivial homoclinic solution even if f≡0.Next, we give an example to illustrate our main result.Example 9.
Let(58)K(t,q)=(4+sint)|q|4/3,W(t,q)=(2+cost)|q|2ln(1+|q|),Gj(q(tj))=|sin(tj2)|qr(tj),gj(q(tj))=r|sin(tj2)|qr-1(tj),f(t)=0,
where r>2, tj=2πj/m, j∈ℕ. It is easy to verify that K, W, Gj, gj, and f satisfy conditions (V1), (K1), (K2), (W1), (W2), (W3), (W4), (G1), (G2), (G3), and (F1). So, system (1), (2) with K, W, Gj, gj, and f as in (58) has a nontrivial homoclinic solution.



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*Source: 583107-2013-08-26.xml*