Homoclinic solutions for Hamiltonian system with impulsive effects

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R E S E A R C H

Open Access

Homoclinic solutions for Hamiltonian

system with impulsive effects

Jian Liu

1

_{, Lizhao Yan}

2*

_{, Fei Xu}

3

_{and Mingyong Lai}

1

*_{Correspondence:}

yanbine@126.com

2_{School of Business, Hunan Normal}

University, Changsha, P.R. China Full list of author information is available at the end of the article

Abstract

In this article, we investigate a class of impulsive Hamiltonian systems with a p-Laplacian operator. By establishing a series of new suﬃcient conditions, the existence of homoclinic solutions to such type of systems is revealed. We show the existence of homoclinic orbit induced by impulses by introducing some conditions. To illustrate the applications of the main results in this article, we create an example.

Keywords: Impulsive Hamiltonian system;p-Laplacian operator; Homoclinic solution

1 Introduction

The aim of the article is to investigate the impulsive Hamiltonian systems withp-Laplacian operator of the form

p

u(t)–∇Ft,u(t)= 0, t=ti,t∈R, (1)

–p

u(ti)

=gi

u(ti)

, i∈Z. (2)

Here, we are particularly interested in the existence of homoclinic orbits for such sys-tems. In the system,u∈R,(p(u(ti))) =|u(t+i)|p–2u(t+i) –|u(t–i)|p–2u(ti–) withu(ti±) = lim_t_→_t±

i u

₍_t_),_∇_F₍_t_,_u_{) =}_grad

uF(t,u), gi(u) =graduGi(u),Gi∈C1(Rn,Rn) for eachi∈Z.

There exist anm∈N and aT > 0 such that 0 =t0<t1<· · ·<tm=T,ti+m=ti+T and gi+m=gifor alli∈Z.

The existence and multiplicity of homoclinic orbits attracted the attention of researchers from all over the world and as such have been extensively investigated in the literature [1–8]. Mathematical techniques such as the dual variational method [9], concentration compactness method and Ekeland variational principle [10,11], and the approximation method [12] have been used in evaluating the existence of homoclinic orbits for Hamilto-nian systems.

Real-world systems display a variety of abrupt changes, and such changes can be mod-eled using impulsive differential equations. Impulsive effects have been integrated into different types of differential equations to describe the consequences of abrupt changes. Such systems have been investigated in the literature [13–24].

A special case of system (1)–(2) wherep= 2 has been considered by Zhang and Li [20]. The authors get the following result.

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Theoerem 1([20]) Assume that,for j= 1, 2, . . . ,m,gjis continuous and m-periodic in j, and F and gjsatisfy the following conditions:

(H1) F:R×RN→Ris continuously diﬀerentiable andT-periodic,and there exist positive constantsr1,r2> 0such that

r1|u|2≤F(t,u)≤r2|u|2, ∀(t,u)∈[0,T]×Rn;

(H2) F(t,u)≤(∇F(t,u),u)≤2F(t,u),∀(t,u)∈[0,T]×Rn; (H3) there existsμ> 2such that

0 < –μGj(u)≤–ugj(u), foru∈Rn\ {0},j= 1, 2, . . . ,m;

(H4) lim|u|→0

gj(u)

|u| = 0forj= 1, 2, . . . ,m.

Then the second order impulsive Hamiltonian system

u(t) –∇Ft,u(t)= 0, t=ti,t∈R,

–u(ti) =gi

u(ti)

, i∈Z,

has at least one nonzero homoclinic solution generated by impulses.

For the general case ofp= 2, due to the complex structure of problem (1) and (2), it is challenging to construct an appropriate functional such that the existence of its criti-cal point implies a homoclinic orbit of the system. Since the domain under consideration is unbounded, the Sobolev embedding might not be compact. In order to complete the proof, we show that the homoclinic orbituis obtained as the limit of 2kT-periodic solu-tionsukof (1)–(2) ask→ ∞. Due to the impulsive perturbation, the velocity is no longer

continuous. Besides, ifp= 2, the Sobolev spaceW₂1,_kTp is not a Hilbert space. When we use the mountain pass theorem to prove the main results of the paper, it is necessary to guar-antee that constantsρandαare required to be independent ofk. We also need to show that the approximating solution sequence{uk}has a bound, which is independent ofk.

Before introducing the main results, we make the following assumptions.

Theoerem 2 Assume that F,gisatisﬁes the following conditions:

(A1) F∈C1₍_R_×_RN_,_R₎_,_F₍_t_{, 0)}_≡₀_and_F_is_T_{-periodic in its ﬁrst variable}_;

(A2) F(t,u) = 1

p|u|

p₊_H₍_t_,_u₎_,_where_H_∈_C1₍_R_×_RN_,_R₎_;

(A3) for everyt∈Randu∈Rn\{0},there existsμ>psuch that

pH(t,u)≥u∇H(t,u) > 0;

(A4) there existsμ>p,such thatgi(u)u≤μGi(u)≤0,u∈Rn\ {0},i= 1, 2, . . . ,m. Then the system(1)–(2)possesses at least one nonzero homoclinic solution generated by impulses.

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The rest of the paper is organized as follows. In Sect.2, we present some preliminaries and statements, which will be used in proving our main results. The proof of the main results of this article is given in Sect.3. We then present an example to illustrate the ap-plicability of our results in Sect.4.

2 Preliminaries and statements

For eachk∈N, set

Ek=

u:R→Rn|u,u∈Lp[–kT,kT],Rn,u(t) =u(t+ 2kT),t∈R. Thus,Ekis a Hilbert space with the norm deﬁned by

uE_k=

kT

–kT

u(t)p+u(t)pdt 1

p

.

LetLp₂_kT(R,Rn_{) denote the Hilbert space of 2}_kT_{-periodic functions on}_R_{with values in}_Rn

under the norm

u_Lp

2kT(R,Rn)=

kT

–kT

u(t)pdt 1

p

,

andL∞₂_kT(R,Rn_{) be a space of 2}_kT_{-periodic essentially bounded measurable functions from} RtoRn_{under the norm}

uL∞

2kT(R,Rn)=esssupu(t):t∈[–kT,kT]

.

Setk={–km+ 1, –km+ 2, . . . , 0, 1, 2, . . . ,km– 1,km}and deﬁne

Ik(u) = kT

–kT

1

pu

₍_t₎p

+Ft,u(t)dt+

i∈_k

Gi

u(ti)

= 1

pu p Ek+

kT

–kT

Ht,u(t)dt+

i∈k

Gi

u(ti)

. (3)

It follows thatIkis Frechet diﬀerentiable at anyu∈Ek. For anyu∈Ek, we thus have

I_k(u)v=

kT

–kT

u(t)p–2u(t)v(t) +∇Ft,u(t)v(t)dt

+

i∈k

gi

u(ti)

v(ti). (4)

The above analysis shows that the critical points of the functionalIk are classical 2kT

-periodic solutions of system (1)–(2).

Lemma 1([25]) There exists a positive constant C independent of k such that,for each k∈N and u∈Ek,one has

uL∞

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Lemma 2([26]) Let u:R→Rn_{be a continuous mapping such that u}_∈_Lp

loc(R,Rn).For

every t∈R,the following inequality holds:

u(t)≤2

p–1

p

t+1₂

t–1₂

_u₍_s₎p

+u(s)pds 1

p

. (6)

Lemma 3([27]) Let X be a real Banach space and I∈C(X,R)satisfying the Palais–Smale

(PS)-condition.Suppose that I satisﬁes the following conditions: (i) I(0) = 0;

(ii) there exist constantsρ,α> 0such thatI|∂Bρ(0)≥α; (iii) there existse∈X\ ¯Bρ(0)such thatI(e)≤0.

Then I possesses a critical value c≥αgiven by

c=inf g∈smax∈[0,1]I

g(s),

where Bρ(0)is an open ball in X of radiusρcentered at0,and

=g∈C[0, 1],X:g(0) = 0,g(1) =e.

3 Proof of Theorem2

Before starting the proof of Theorem2, we recall some properties of the functionH(t,u).

Remark1 If (A3) holds, then, for everyt∈[0,T], there exista1,a2> 0 such that:

H(t,u)≥a1|u|p, if 0 <|u|< 1, (7)

H(t,u)≤a2|u|p, if|u| ≥1, (8)

wherea1=mint∈[0,T],|u|=1H(t,u),a2=maxt∈[0,T],|u|=1H(t,u).

Proof To prove this fact, it suﬃces to show that, for everyu= 0 andt∈[0,T] the function ζ →H(t,ζ–1u)ζp (where ζ ∈(0, +∞)) is nonincreasing, which is an immediate

conse-quence of (A3).

Remark2 Assume that (A4) hold, then there existb1,b2> 0, such that

Gi(u)≥–b1|u|μ, if 0 <|u|< 1, (9)

Gi(u)≤–b2|u|μ, if|u| ≥1. (10)

Proof Since the proof is similar to that of Fact 2.2 of [11], here it is omitted.

We divide the proof of Theorem2into several lemmas.

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Proof By (A1) and (A4), we getIk(0) = 0.

we can show that{un}has a convergent subsequence. It thus follows thatIksatisﬁes the

(PS)-condition.

We now show that the functionalIksatisﬁes Assumption (ii) in Lemma3. Choose 0 <

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≥ kT

–kT

1

pu

j(t) p

+1

puj(t) p

+a1u(t)pdt–b1

i∈k

|u|μ

≥min

1

p,

1

p+a1

up_E

k–b1u

μ

Ek

=min

1

p,

1

p+a1

δp

Cp – b1δμ

Cμ

:=α> 0. (15)

It follows from (15) thatuEk=

δ

C=ρ, which implies thatIk(u)≥α.

It remains to prove that the functionalIksatisﬁes Assumption (iii) of Lemma3. From

(3), (8) and (10) we have

Ik(ζu) = |

ζ|p

p u p Ek+

kT

–kT

Ht,ζu(t)dt+

i∈k

Gi

ζu(ti)

≤|ζ|p p u

p

Ek+a2|ζ|

p_up

Ek–b2|ζ|

μ

i∈k

u(ti)

μ

. (16)

TakeU∈E1such thatU(±T) = 0. Sinceμ>pandb1> 0, then by (16) there existsξ ∈

R\{0}such thatξUE1>ρandI1(ξU) < 0. Fork> 1, sete1(t) =ξU(t) and

ek(t) = ⎧ ⎨ ⎩

e1(t), |t| ≤T,

0, T<|t| ≤kT. (17)

Thenek∈Ek,ekEk=e1E1>ρandIk(ek) =I1(e1) < 0 for everyk∈N.

By Lemma3,Ikpossesses a critical valueck≥α> 0. Then, for everyk∈N, there exists uk∈Eksatisfying

Ik(uk) =ck, Ik(uk) = 0. (18)

Hence, system (1)–(2) has a nontrivial 2kT-periodic solutionuk.

Lemma 5 Let{uk}be the sequence given by(18).There exist a subsequence{uj,j}of{uk}

and a function u0∈Wloc1,p∩L∞loc(R,Rn)such that{uj,j}converges to u0weakly in Wloc1,pand

strongly in L∞_loc(R,Rn_).

Proof Our ﬁrst step is to show that the sequence{ck}k∈N is bounded. For eachk∈N, let gk: [0, 1]→Ek be a curve given bygk(s) =sek, whereek is deﬁned by (17). Thengk∈k

andIk(gk(s)) =I1(g1(s)) for allk∈Nands∈[0, 1]. Therefore, it follows from the mountain

pass theorem that

ck≤max s∈[0,1]I1

g1(s)

≡C0, (19)

which is independent ofk∈N. SinceI_k(uk) = 0, from (A3), (A4) and (3) we obtain

ck =Ik(uj) –

1 μI

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=

Sinceμ>pand all the constants in (20) are independent ofk, then there exists a constant

L1> 0 independent ofksuch that

ukEk≤L1. (21)

The boundedness ofukE_k implies the boundedness of the set

ukW1,p_((–_kT_,_kT_),_Rn₎

for each positive integerk. In particular, whenk= 1, since{uk}is a bounded sequence in

W1,p(–T,T),Rn,

that, for any positive integerm, there is a sequence{um,k}which converges weakly in

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Lemma 6 The function u0determined by Lemma5is a nonzero homoclinic solution of the

system(1)–(2).

Proof The proof is divided into four steps.

First we show thatu0is a solution of system (1)–(2). For convenience, we denote{uk,k}

by{uk}. For any given interval (a,b)⊂(–kT,kT) and anyv∈W₀1,p((a,b),Rn_{), deﬁne}

v1(t) =

⎧ ⎨ ⎩

v(t), t∈(a,b),

0, t∈(–kT,kT)\(a,b).

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Then, for anyv∈W₀1,p((a,b),Rn_{), one has}

b

a

u₀v–∇F(t,u0)v

dt+

ti∈(a,b)

gi

u0(ti)

v(ti)

= lim k→+∞

b

a

u_kv–∇F(t,uk)v

dt+

ti∈(a,b)

gi

uk(ti)

v(ti)

= 0. (23)

Letk1be a positive integer and seta= –k1T andb=k1T. By a similar method to the one

proposed in [20], using (23), we can show thatu0|(ti,ti+1)satisﬁes (1) in classical sense for i= –k1m+ 1, . . . ,k1m, andu0satisﬁes

u₀t+_ip–2u₀t_i+=u₀t–_ip–2u₀t_i–+gi

u0(ti)

, i= –k1m+ 1, . . . ,k1m– 1.

Sincev∈W₀1,p((–k1T,k1T),Rn), we cannot show that

u₀t+_k1mp–2u₀t+_k1m=u₀t_k1m– p–2u₀t_k1m– +gk1m

u0(tk1m)

from (23). Therefore,u0is a solution of (1)–(2) in (–k1T,k1T). Sincek1is arbitrary,u0is

a solution of system (1)–(2) inR.

Next, we prove thatu0(t)→0, ast→ ±∞. Since{uk}is weakly continuous, it is weakly

lower semicontinuous, and thus we have

+∞ –∞

u0(t)

p

+u₀(t)pdt = lim k→+∞

kT

–kT u0(t)

p

+u₀(t)pdt

≤ lim

k→+∞j→lim+∞inf

kT

–kT uj(t)

p

+u_j(t)pdt ≤Lp₁.

Therefore,

|t|≥r u0(t)

p

+u₀(t)pdt→0, asr→+∞. (24)

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Suppose thatukL∞→0, then, whenksuﬃciently large, by (A4), (9), (25) and (26), we have

b1μ

i∈k

uk(ti)

μ

≥–

i∈k

gi

uk(ti)

=

kT

–kT

u_k(t)p+∇Ft,uk(t)

uk(t)

dt

=

kT

–kT

u_k(t)p+uk(t) p

dt+

kT

–kT

∇Ht,uk(t)

uk(t)dt

≥ ukp_E_k

≥ θ

2p–1_max{_1,_θp}

i∈k

_u_k₍_t_i₎p

.

This is impossible whenukL∞→0. Hence, the system (1)–(2) has a nontrivial

homo-clinic solution.

Lemma 7 The u0given in Lemma6is generated by impulses.

Proof In order to complete the proof, we only need to show that under the conditions of Theorem2, the system (p(u(t)))–∇F(t,u(t)) = 0 has no nontrivial homoclinic solution.

Ifuis a homoclinic solution of (p(u(t)))–∇F(t,u(t)) = 0, then

lim

t→→±∞u(t) = 0,

lim t→→±∞u

₍_t_{) = 0.}

By

0 = –

b

a

_u₍_t₎p–2

u(t)u(t) –∇Ft,u(t)u(t)dt

=

b

a

u(t)p+∇Ft,u(t)u(t)dt–u(t)p–2u(t)u(t)

b

a

=

b

a

u(t)p+u(t)pdt+

b

a

Ht,u(t)u(t)dt–u(t)p–2u(t)u(t)

b

a

,

we have

b

a

_u₍_t₎p

+u(t)pdt+

b

a

Ht,u(t)u(t)dt

≤ lim

a→–∞blim→∞u ₍_t₎p–2

u(t)u(t)b

a= 0.

Moreover, by (A3), we know that

b

a

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Then

b

a

u(t)p+u(t)pdt+

b

a

Ht,u(t)u(t)dt≥0.

Therefore,u= 0. It thus follows that the system (p(u(t)))–∇F(t,u(t)) = 0 has only a

trivial homoclinic solution.

4 Example

In this section, we present an example to illustrate the application of the main results obtained in precious sections.

Example1 Let

p= 6, H(t,u) = (2 +sint)u2,

Gi

u(ti)

= –costi

2

u10(ti), gi

u(ti)

= –10costi

2

u9(ti),

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whereti=2_mπi,i∈Z.

Forμ= 7 andp= 6, we haveF(t,u) =1₆|u|6_{+ (2 +}_sin_t₎_u2_{. It is easy to see that}_F₍_t_{, 0) = 0.}

We notice that (A1), (A2) are satisﬁed.

ByH(t,u) = (2 +sint)u2_{, one has}_∇_H₍_t_,_u_{) = 2(2 +}_sin_t₎_u_{. Therefore,}

0 <u∇H(t,u) = 2(2 +sint)u2< 6(2 +sint)u2=pH(t,u),

which implies that condition (A3) is satisﬁed.

ByGi(u(ti)) = –|cost₂i|u10(ti),gi(u(ti)) = –10|cost₂i|u9(ti), we have

gi

u(ti)

u(ti) = –10 costi

2

u10(ti) < –7 costi

2

u10(ti) =μGi

u(ti)

< 0.

Thus, condition (A4) is satisﬁed.

By Theorem2, system (1)–(2) withF,H,Giandgideﬁned in (27) has a nontrivial

ho-moclinic solution.

Acknowledgements

The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.

Funding

This work is partially supported by the National Natural Science Foundation of China (No. 71501069, No. 71420107027, No. 71871030), the Hunan Provincial Natural Science Foundation of China (No. 2015JJ3090, No. 2017JJ3330), the China Postdoctoral Science Foundation Funded Project (No. 2016M590742).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript, and read and approved the ﬁnal manuscript.

Author details

1_{School of Economics and Management, Changsha University of Science and Technology, Changsha, P.R. China.}2_School

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