Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term

R E S E A R C H

Open Access

Subharmonic solutions for a class of

second-order impulsive Lagrangian systems

with damped term

Xingyong Zhang

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P.R. China

Abstract

In this paper, by using the mountain pass theorem, we investigate the existence of subharmonic weak solutions for a class of second-order impulsive Lagrangian systems with damped term under asymptotically quadratic conditions. Some new existence criteria are established. Finally, an example is presented to verify our results.

MSC: 37J45; 34C25; 70H05

Keywords: impulsive Lagrangian systems; damped term; subharmonic weak solutions; mountain pass theorem

1 Introduction and main results

In this paper, we investigate the existence of subharmonic weak solutions for the following second-order impulsive Lagrangian system with damped term:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d(P(t)u(t))˙

dt + (q(t)P(t) +B)u˙(t) + ( 

q(t)B–A(t))u(t) +∇F(t,u(t))

= , a.e.t∈R,

(P(tj)u˙(tj)) =P(tj)u˙(t+j) –P(tj)u˙(t–j) =∇Ij(u(tj)), j= , . . . ,p,

(.)

whereT> ,p∈N,t=  <t<t<· · ·<tp<tp+=T,u(t) = (u(t), . . . ,uN(t)),Ij:RN→R,

q∈C(R,R) satisfyingq(t+T) =q(t) and_Tq(t)dt= ,Bis a skew-symmetricN×N

constant matrix,P(t) andA(t) are symmetric and continuousN×Nmatrix-value func-tions onRsatisfyingP(t+T) =P(t) andA(t+T) =A(t), andF:R×RN _{→Rsatisfies}

F(t,x) = –K(t,x) +W(t,x), whereK,W areT-periodic in their first variable, and the fol-lowing assumption:

(A) F(t,x)is measurable intfor everyx∈RN _{and continuously differentiable inxfor}

a.e.t∈[,T], and there exista∈C(R+_,R+_)andb∈L_(,T;R+_{)withb(t+T) =b(t)}

such that

F(t,x)≤a|x| b(t), ∇F(t,x)≤a|x| b(t),

Ij(x)≤a

|x| , ∇Ij(x)≤a

|x|

for allx∈RN _{and a.e.t∈[,T].}

Lagrangian systems are applied extensively to study the fluid mechanics, nuclear physics and relativistic mechanics. Especially, as a special case of Lagrangian systems, the follow-ing second-order Hamiltonian systems are considered by many authors:

¨

u(t) =∇Ft,u(t) , a.e.t∈R, (.)

whereF:R×RN _{→RsatisfiesF(t+T,x) =F(t,x), and the existence and multiplicity of}

periodic solutions, subharmonic solutions and homoclinic solutions are obtained via vari-ational methods. We refer readers to [–]. Especially, in , under the asymptotically quadratic conditions, Tang and Jiang [] obtained the following interesting result.

Theorem A(see [], Theorem .) Assume that F satisfies

(F) F(t,x) = –K(t,x) +W(t,x)andK,W∈C(R×RN,R)areT-periodic in their first variable withT> ,and thatKandWsatisfy the following assumptions:

(H) There exist constantsb> andγ ∈(, ]such that

K(t, ) = , K(t,x)≥b|x|γ for(t,x)∈[,T]×RN;

(H) (∇K(t,x),x)≤K(t,x)for(t,x)∈[,T]×RN_;

(H) lim sup_|x|→W(t,x)_|

x| <buniformly fort∈[,T]; (H) There exists a functiong∈L_{([,T],R)such that}

∇W(t,x),x – W(t,x)≥g(t) for(t,x)∈[,T]×RN

and

lim

|x|→∞

∇W(t,x),x – W(t,x)= +∞ for a.e.t∈[,T];

(H) There exist constantsa> andd> such that

W(t,x)≤a|x|+d for(t,x)∈[,T]×RN;

(H) There existsx∈RN such that T



K(t,x) –W(t,x) –

g(t) 

dt< .

Then system(.)has a nontrivial T -periodic solution.

In recent years, variational methods have been applied to study the existence and multi-plicity of solutions for impulsive differential equations and lots of interesting results have been obtained, see [–].

In [], Nieto introduced a variational formulation for the following one-dimensional damped nonlinear Dirichlet problem with impulses:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u(t) +g(t)u(t) +λu(t) =f(t,u(t)), t∈[,T],

u(tj) =Ij(u(t–j)), j= , , . . . ,p,

u() =u(T) = ,

(.)

and gave the concept of a weak solution for such a problem. They obtained that the weak solutions of problem (.) are indeed the critical points of the functional:

ϕ(v) =  

T



eG(t)v(t) dt+λ 

T



eG(t)v(t)dt

+

p

j=

eG(tj)

v(tj)



Ij(t)dt– T



eG(t)Ft,v(t) dt, (.)

whereG(t) =_tg(t)dtandF(t,v) =_vf(t,s)ds. In [] and [], the authors also dealt with some one-dimensional impulsive problems with damped term by variational methods.

For higher dimensional dynamical systems, some interesting results have also been ob-tained (see [–]). In [], Zhou and Li investigated the second-order Hamiltonian sys-tem with impulsive effects:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = ,

u˙i_(t

j) =u˙i(tj+) –u˙i(t–j) =Iij(ui(tj)), i= , , . . . ,N,j= , , . . . ,p.

(.)

By using the least action principle and the saddle point theorem, they obtained some ex-istence results of solutions under sublinear condition and some reasonable conditions. In [], system (.) withF(t,u) = _A(t)u·u–λW(t,u) –μG(t,u), whereλ,μ∈R, was also investigated. By using variational methods, the authors obtained that system (.) has at least three weak solutions. In [], the authors investigated system (.) with

F(t,u) =_A(t)u·u–W(t,u). They obtained that system (.) has infinitely many solutions under the assumptions that nonlinear term is superquadratic, asymptotically quadratic and subquadratic, respectively.

In recent years, via variational methods, some authors have been interested in study-ing the existence and multiplicity of periodic solutions and homoclinic solutions for the following Lagrangian systems with damped term:

d(P(t)u˙(t))

dt +Bu˙(t) +∇F

t,u(t) = , (.)

whereP(t) is a symmetric and continuousN×N matrix-valued function,Bis a skew-symmetricN×Nconstant matrix andF:R×RN_{→R. They obtained some interesting}

In , Liet al.[] investigated the following system, more general than system (.), withP(t)≡IN:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) + (q(t)IN×N+B)u˙(t) + (_q(t)B–A(t))u(t) +∇F(t,u(t))

= , a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = .

(.)

Motivated by [], in [], we investigated the following system, more general than sys-tem (.):

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d(P(t)u(t))˙

dt + (q(t)P(t) +B)u˙(t) + ( 

q(t)B–A(t))u(t) +∇F(t,u(t))

= , a.e.t∈[,T],

u() –u(T) =P()u˙() –P(T)u˙(T) = .

(.)

By variational methods, under superquadratic or subquadratic conditions, we obtained that system (.) has infinitely many solutions. One can see more details of our results and more research background of system (.) in [].

In [], Luoet al.investigated the existence of subharmonic solutions with prescribed minimal period for the following one-dimensional second-order impulsive differential equation:

⎧ ⎨ ⎩

u(t) +f(t,u(t)) = , a.e.t∈J,

u(tm) =Im(u(tm)), m∈Z,

(.)

wheref ∈C(R×R,R),Z=Z+∪Z–,J=R\{tm|m∈Z},Im∈C(R,R+∪ {}),  <t<

t<· · ·<tp<T,Im+p=Imandtm=tm+p–Tifm∈Z+, whiletm=tm+p+–Tifm∈Z–.

In this paper, motivated by [, , , , , ] and [], we focus on the existence of subharmonic weak solutions for system (.), which is of impulsive conditions, and we study the problem under asymptotically quadratic conditions. To the best of our knowl-edge, there are few papers that consider such a problem for system (.). We call a solution

usubharmonic ifuiskT-periodic for somek∈N. Let

A= sup

t∈[,T]

max

|x|=,x∈RN A(t)x

= sup

t∈[,T]

maxλ(t) :λ(t) is the eigenvalue ofAτ(t)A(t)

and

B= max

|x|=,x∈RN|Bx|

=max√λ:λis the eigenvalue ofBτB.

(P) There exists a constantm>_ such that the matrixP(t)satisfies

P() =P(T), P(t)x,x >m(x,x) for all(t,x)∈R×RN/{};

(K) There exist constantsa> andγ ∈(, ]such that

K(t, ) = , K(t,x)≥

B+ A

 +a

|x|γ for allx∈RNand a.e.t∈[,T];

(K) (∇K(t,x),x)≤K(t,x)for allx∈RNand a.e.t∈[,T]; (K) There existsD> such that

K(t,x)≤D|x| for allx∈RN and a.e.t∈[,T];

(W) lim|x|→W(t,x)_|x| <auniformly for a.e.t∈[,T]; (W) There exist constantsb> andd> such that

W(t,x)≤b|x|+d for allx∈RN and a.e.t∈[,T]; (W) There exists a functionh∈L_{(,T;R)such that}

eQ(t)∇W(t,x),x – W(t,x)≥h(t) for allx∈RN _{and a.e.t∈[,T]}

and

lim

|x|→∞e

Q(t)_{∇W(t,x),x – W(t,x)= +∞ for a.e.t∈[,T],}

whereQ(t) =_tq(s)ds; (W) There existsx∈RN such that

T



eQ(t)

 

A(t)x,x +K(t,x) –W(t,x) –

h(t) 

dt< ;

(W) There exists a constantD>A_+Dsuch that

W(t,x)≥D|x| for allx∈RNand a.e.t∈[,T];

(I) There exist constantslj> (j= , . . . ,p) such that

Ij(x)≤lj (j= , . . . ,p)for allx∈RN;

(I) p_j=eQ(tj)Ij() = and p

j=eQ(tj)Ij(x)≥for allx∈RN;

(I) There exists a constantCsuch that

p

j=

eQ(tj)Ij(x) –

This paper is organized as follows. In Section , we present the definition of a subhar-monic classical solution, a subharsubhar-monic weak solution and the variational structure for system (.) and make some preliminaries. In Section , we present our main theorems and their proofs. In Section , an example is given to verify our main theorems.

2 Preliminaries

In this section, we present the variational structure of system (.), which is motivated by [–, ] and [].

Let

H_kT =u:R→RN|uis absolutely continuous,u(t+kT) =u(t) andu˙∈L(,kT). Define

u,v=

kT



eQ(t)u(t),v(t) dt+

kT



eQ(t)P(t)u˙(t),v˙(t) dt

and

u=

kT



eQ(t)u(t)dt+

kT



eQ(t)P(t)u˙(t),u˙(t) dt

/

for eachu,v∈H_kT . Then (H_kT ,·,·) is a Hilbert space. It is well known that

u_H

kT= kT



u(t)dt+

kT



u_˙(t)dt

/

is also a norm onH_kT . Obviously, if the condition (P) holds,u_H

kTanduare equivalent.

Moreover, there existsCk>  such that u∞= max

t∈[,kT]

u(t)≤Cku_H

kT

(see Proposition . in []). Hence, there exist positive constantsCk,Cksuch that u∞≤Cku, u_H

kT≥Cku. (.)

For anya,b∈R, define

H(a,b) =u:R→RN|bothuandu˙are absolutely continuous on (a,b), andu¨∈L(a,b),

H[a,b) =u:R→RN|bothuandu˙are absolutely continuous ona,b), andu¨∈L(a,b),

H(a,b] =u:R→RN|bothuandu˙are absolutely continuous on (a,b, andu¨∈L(a,b).

Definition . Assume thatu∈H

kT∩H[,t)∩( p–

j= H(tj,tj+))∩H(tp,T]∩H[T,kT]

and the limitsu˙(t+

j) andu˙(tj–) (j= , , . . . ,p) exist. Ifusatisfies system (.), then we say that

uis a subharmonic classical solution of system (.).

Remark . In [], impulsive effects may occur periodically intj,j∈ {, . . . ,p}. In order

to obtain a sequence of distinct subharmonic weak solutions (see Theorem . below), different from [], in Definition ., we assume that the impulsive effects only occur intj,

j= , . . . ,p, which belong to (,T). In other words,uis absolutely continuous onRandu˙

is absolutely continuous on [,t)∪( p–

j=(tj,tj+))∪(tp,T]∪[T,kT]. Moreover, note that

u(t) =u(t+kT). Then it is easy to see thatu˙() =u˙(kT).

Note thatQ(t) =_tq(s)ds. Then, byT-periodicity ofq, we haveQ(kT) =k_Tq(t)dt= . Moreover, obviously,Q(t) is continuous onR. We transform system (.) into the following system:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d(eQ(t)_P(t)u(t))˙

dt +eQ(t)Bu˙(t) +eQ(t)( 

q(t)B–A(t))u(t) +eQ(t)∇F(t,u(t))

= , a.e.t∈R,

(P(tj)u˙(tj)) =P(tj)u˙(t+j) –P(tj)u˙(tj–) =∇Ij(u(tj)), j= , . . . ,p.

(.)

Then system (.) is equivalent to system (.) and its solutions are the solutions of sys-tem (.).

By the idea in [], we takev∈H

kTand multiply the two sides of the equality

d(eQ(t)_P(t)u˙(t))

dt +e

Q(t)_{Bu˙(t) +e}Q(t)



q(t)B–A(t)

u(t) +eQ(t)∇Ft,u(t) = 

byvand integrate it from  tokT. Then we obtain

kT



d(eQ(t)_P(t)u˙(t))

dt +e

Q(t)_{Bu˙(t) +e}Q(t)



q(t)B–A(t)

u(t)

+eQ(t)∇Ft,u(t) ,v(t)

dt= . (.)

Note thatP(t+T) =P(t),P(t) is continuous onRandu˙() =u˙(kT). By integration by parts and the continuity ofv, we obtain

kT



d(eQ(t)_P(t)u˙(t))

dt ,v(t)

dt

=

p

j= tj+

tj

d(eQ(t)_P(t)u˙(t))

dt ,v(t)

dt+

kT

T

d(eQ(t)_P(t)u˙(t))

dt ,v(t)

dt

=

p

j=

eQ(tj+)_P(t

j+)u˙

t_j+– ,v(tj+) –

eQ(tj)P(tj)u˙

t+_j ,v(tj)

–

tj+

tj

eQ(t)P(t)u˙(t),v˙(t) dt

–

kT

T

eQ(t)P(t)u˙(t),v˙(t) dt

=eQ(T)P(T)u˙(T),v(T) –eQ()P()u˙(),v()

–

p

j=

eQ(tj)P(tj)u˙

t_j+ ,v(tj) –

eQ(tj)P(tj)u˙

t–_j ,v(tj)

–

T



eQ(t)P(t)u˙(t),v˙(t) dt

+eQ(kT)P(kT)u˙(kT),v(kT) –eQ(T)P(T)u˙(T),v(T) –

kT

T

eQ(t)P(t)u˙(t),v˙(t) dt

=eQ(kT)P(kT)u˙(kT),v(kT) –eQ()P()u˙(),v() –

p

j=

eQ(tj)_P(t j)u˙

t_j+ ,v(tj) –

eQ(tj)_P(t j)u˙

t–_j ,v(tj)

–

kT



eQ(t)P(t)u˙(t),v˙(t) dt

= –

p

j=

eQ(tj)P(tj)u˙(tj) ,v(tj) – kT



eQ(t)P(t)u˙(t),v˙(t) dt. (.)

Definition . u∈H

kT is called a subharmonic weak solution of system (.) if

kT



eQ(t)P(t)u˙(t),v˙(t) dt+

p

j=

eQ(tj)∇Ij

u(tj) ,v(tj)

=

kT



eQ(t)

 q(t)

Bu(t),v(t) +Bu˙(t),v(t)

–A(t)u(t),v(t) +∇Ft,u(t),v(t) dt

holds for anyv∈H_kT .

Lemma . If u∈H

kT is a subharmonic weak solution of system(.),then u is a

subhar-monic classical solution of system(.).

Proof Motivated by [], forj∈ {, , , . . . ,p}, choosev∈H_kT withv(t) =  for everyt∈

[,tj]∪[tj+,kT]. Then, by Definition ., we obtain tj+

tj

eQ(t)P(t)u˙(t),v˙(t) dt

=

tj+

tj

eQ(t)

 q(t)

Bu(t),v(t) +Bu˙(t),v(t)

Choosev∈H

kTwithv(t) =  for everyt∈[,T]. Then we obtain kT

T

eQ(t)P(t)u˙(t),v˙(t) dt

=

kT

T

eQ(t)

 q(t)

Bu(t),v(t) +Bu˙(t),v(t)

–A(t)u(t),v(t) +∇Ft,u(t),v(t) dt. (.)

Equations (.) and (.) imply thatu∈H

kT∩H[,t)∩( p–

j= H(tj,tj+))∩H(tp,T]∩

H_{[T,kT] andusatisfies}

d(eQ(t)_P(t)u˙(t))

dt +e

Q(t)_{Bu˙(t) +e}Q(t)



q(t)B–A(t)

u(t) +eQ(t)∇Ft,u(t) = , a.e.t∈[,kT].

Multiplying the above equality byvand integrating between  andkT, combining the argument of (.) and Definition ., we obtain that

p

j=

eQ(tj)P(tj)u˙(tj) ,v(tj) = p

j=

eQ(tj)∇Ij

u(tj) ,v(tj).

Hence,(P(tj)u˙(tj)) =∇Ij(u(tj)) for everyj= , , . . . ,p. This completes the proof.

For everyk∈N, defineϕk:HkT →Rby

ϕk(u) = kT



eQ(t)

 

P(t)u˙(t),u˙(t) + 

Bu(t),u˙(t)

+ 

A(t)u(t),u(t) –Ft,u(t) dt+

p

j=

eQ(tj)Ij

u(tj) .

It follows from assumption (A) and Theorem . in [] that the functionalϕkis

continu-ously differentiable and

ϕ_k(u),v=

kT



eQ(t)P(t)u˙(t),˙v(t) – q(t)

Bu(t),v(t) –Bu˙(t),v(t)

+A(t)u(t),v(t) –∇Ft,u(t) ,v(t) dt+

p

j=

eQ(tj)∇Ij

u(tj) ,v(tj) (.)

foru,v∈H

kT. Obviously, ifu∈HkT is a critical point ofϕk,i.e.,ϕk(u) = , thenuis a

subharmonic weak solution of system (.).

We will use the following mountain pass theorem to prove our results.

Lemma .(see []) Let E be a real Banach space,and letφ∈C_{(E,R)satisfy the(PS)}

condition.Ifφsatisfies the following conditions: (i) φ() = ;

(iii) There existse∈E/B¯ρ()such thatφ(e)≤,thenφpossesses a critical valuec≥α

given by

c=inf

g∈smax∈[,]φ

g(s),

whereBρ()is an open ball inEof radiusρcentered at,and

=g∈C[, ],E :g() = ,g() =e.

Remark . As shown in [], a deformation lemma can be proved by replacing the usual

(PS)-condition with the condition (C), and it turns out that Lemma . holds true under the condition (C). We say thatφsatisfies the condition (C),i.e., for every sequence{un} ⊂ E,{un}has a convergent subsequence ifφ(un) is bounded and ( +un)φ(un) → as

n→ ∞.

3 Main results

Theorem . Assume that(P), (K), (K), (W)-(W)and(I)-(I)hold.Then,for every k∈N,system(.)has at least one kT -periodic weak solution in H

kT.

Proof We use Lemma . to prove the theorem. LetE=H_kT .

Step . We prove thatϕksatisfies assumption (ii) of Lemma .. It follows from (W) and

(W) that there exist  <ε<m  –



,θ>  andC>  such that

W(t,x)≤(a–ε)|x|+C|x|θ. (.)

Choose  <δ<  such thatε(_Cδ k)

_–C Ckθ(

δ

Ck)

θkT

 e

Q(t)_{dt> . Then it follows from (K),}

(I), (.) and (.) that for allu∈H

kT withu=δ/Ck:=ρk,

ϕk(u) = kT



eQ(t)

 

P(t)u˙(t),u˙(t) + 

Bu(t),u˙(t) + 

A(t)u(t),u(t) dt

+

kT



eQ(t)Kt,u(t) –Wt,u(t) dt+

p

j=

eQ(tj)Ij

u(tj)

≥ kT



eQ(t)

m

u˙(t)



– 

Bu(t)+u˙(t) –A|u(t)|





dt

+

B+ A

 +a

kT



eQ(t)u(t)γdt– (a–ε)

kT



eQ(t)u(t)dt

–C kT



eQ(t)u(t)θdt

≥

m

 –

 

kT



eQ(t)u˙(t)dt+a

kT



eQ(t)u(t)γdt

– (a–ε)

kT



eQ(t)u(t)dt–C kT



eQ(t)u(t)θdt

≥

m

 –

 

kT



eQ(t)u˙(t)dt+ε kT



–Cuθ_∞

kT



eQ(t)dt

≥εu–CCkθu

θ

kT



eQ(t)dt

=ε

δ

Ck 

–CCkθ

δ

Ck θ kT



eQ(t)dt

=:αk> .

Step . We prove thatϕksatisfies assumption (iii) of Lemma .. Setϕ(s) =s–W(t,sx)

fors> . By the argument in [], we know that (W) implies that

W(t,sx)≥sW(t,x) +

h(t) 

s–  for a.e.t∈[,T],s> , (.)

and (K) implies that

K(t,sx)≤sK(t,x) for a.e.t∈[,T],s> . (.)

It follows from (.), (.), (W) and (I) that for sufficiently larges,

ϕk(sx) =

s



kT



eQ(t)A(t)x,x dt+ kT



eQ(t)K(t,sx) –W(t,sx)

dt

+

p

j=

eQ(tj)Ij(sx)

= s



k

T



eQ(t)A(t)x,x dt+k T



eQ(t)K(t,sx) –W(t,sx)

dt

+

p

j=

eQ(tj)_I j(sx)

≤sk

T



eQ(t)

 

A(t)x,x +K(t,x) –W(t,x) –

h(t) 

dt

+ 

kT



eQ(t)h(t)dt+

p

j=

eQ(tj)lj.

By (W), we can choose sufficiently largesksuch thatskx>ρkandϕk(skx)≤. Let

ek=skx. Thenϕksatisfies assumption (iii) of Lemma ..

Step . We prove thatϕsatisfies the (C)-condition onH

kT. The proof is motivated by [].

For every{un} ⊂H_kT , assume that there exists a constantCk>  such that ϕk(un)≤Ck,

 +un ϕ_k(un)≤Ck for alln∈N. (.)

Then it follows from antisymmetry ofB, (K) and (I) that Ck≥ϕk(un) –

ϕ_k(un),un

=  

kT



+

kT



eQ(t)∇Ft,un(t),un(t) – F

t,un(t) dt

+

p

j=

eQ(tj)Ij

un(tj) –

∇Ij

un(tj) ,un(tj)

=

kT



eQ(t)Kt,un(t) –

∇Kt,un(t) ,un(t) dt

+

kT



eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt

+

p

j=

eQ(tj)Ij

un(tj) –

∇Ij

un(tj) ,un(tj)

≥ kT



eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt+C. (.)

Next we prove that{un}is bounded. Assume thatun → ∞asn→ ∞. Letzn=un_un.

Thenzn= , and so there exists a subsequence, still denoted by{zn}, such thatznz

onH

kT. Then, by Proposition . in [], we getzn–z∞→. Hence, we havekT|zn(t) –

z(t)|_{dt→ andz}

n(t)→z(t) for a.e.t∈[,kT]. Thus, by conditions (P), (W) and (I),

we have

ϕk(un) = kT  eQ(t)  

P(t)u˙n(t),u˙n(t) +

 

Bun(t),u˙n(t)

+ 

A(t)un(t),un(t) +K

t,un(t) –W

t,un(t) dt

+

p

j=

eQ(tj)Ij

un(tj)

≥ kT  eQ(t)  

P(t)u˙n(t),u˙n(t) –

B_|u

n(t)|+|˙un(t)|

 –

A|un(t)|

 dt –b kT 

eQ(t)un(t)dt–d kT



eQ(t)dt

≥   –  m kT 

eQ(t)P(t)u˙n(t),u˙n(t) dt

–B

_{+ A+ b}



kT



eQ(t)un(t) 

dt–d

kT



eQ(t)dt

=   –  m un–

 –  m kT 

eQ(t)un(t) 

dt

–B

_{+ A+ b}



kT



eQ(t)un(t) 

dt–d

kT



eQ(t)dt

=   –  m un–

B_{+ A+ b}

 +  –  m kT 

eQ(t)un(t) 

dt

–d

kT



Hence, we have

ϕk(un) un ≥

 –

 m

–

B_{+ A+ b}

 +

 –

 m

kT



eQ(t)|un(t)|



un dt

–d

kT  e

Q(t)_dt un .

Letn→ ∞. Then, by (.), we get 

– 

m≤

B+ A+ b

 +

 –

 m

kT



eQ(t)zn(t) 

dt. (.)

Then it follows fromm>_ and (.) that_kTeQ(t)_|z(t)|_{dt>  and soz= . LetS={t∈}

[,kT] :lim_|x|→∞eQ(t)_{[W(t,x) – (∇W(t,x),x)] = +∞} andS}

 ={t∈S:z(t)= }. Then

measS>  and

lim

n→∞un(t)= +∞ for everyt∈S. (.)

Let fn(t) = eQ(t)[(∇W(t,un(t)),un(t)) – W(t,un(t))]. Then (.) and T-periodicity of

W(t,x) intimply that

lim

n→∞fn(t) = +∞ for everyt∈S. (.)

It follows from (.) and Lemma  in [] that there exists a subsetSofSwithmeasS> 

such that

lim

n→∞fn(t) = +∞ uniformly fort∈S. (.)

By (W), we have

kT



eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt

=

S

eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt

+

[,kT]/S

eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt ≥

S

eQ(t)∇Wt,un(t),un(t) – W

t,un(t) dt+

[,kT]/S

h(t)dt.

Letn→ ∞. Then by Fatou’s lemma and (.), we have

kT



eQ(t)∇Wt,un(t) ,un(t) – W

t,un(t) dt→+∞,

which contradicts (.). Hence{un}is bounded. Going if necessary to a subsequence, as-sume thatunuinHkT . Then, by Proposition . in [], we haveun–u∞→ and so kT

easy to obtain that_kTeQ(t)_(P(t)(u˙

n–u˙),u˙n–u˙)dt→. Hence,un–u → asn→ ∞.

Hence,ϕksatisfies the (C)-condition.

Finally, (K), (W) and (I) imply thatϕk() = . Hence, combining Step -Step  with

Lemma . and Remark ., we obtain thatϕkhas at least a critical pointukinHkT and ϕk(uk) =ck≥αk> . Then system (.) has at least onekT-periodic solutionuk inHkT .

This completes the proof.

Remark . It is easy to see that Theorem . generalizes Theorem A. To be precise, when

k= ,P(t)≡IN,q(t)≡,B= ,A(t)≡ andIj(x)≡, Theorem . reduces to Theorem A.

Theorem . Assume(P), (K)-(K), (W)-(W)and(I)-(I)hold.Then system(.)has a sequence of distinct subharmonic weak solutions with period kjT satisfying kj∈Nand

kj→ ∞as j→ ∞.

Proof By Theorem ., we know that for everyk∈N, system (.) has at least onekT -periodic solutionukinHkT andϕk(uk) =ck≥αk> . By Lemma ., we have

ϕk(uk) =ck=inf g∈smax∈[,]ϕk

g(s) ,

where

=g∈C[, ],E :g() = ,g() =ek

.

Letg(s) =sek=sskx,s∈[, ]. Obviously,g∈. Hence, by (K), (W) and (I), we have ϕk(uk)≤max

s∈[,]ϕk(ssx)

= max

s∈[,] kT



eQ(t)

 

A(t)ssx,ssx +K(t,ssx) –W(t,ssx)

dt

+

p

j=

eQ(tj)Ij(ssx)

= max

s∈[,]

k

T



eQ(t)

 

A(t)ssx,ssx +K(t,ssx) –W(t,ssx)

dt

+

p

j=

eQ(tj)Ij(ssx)

≤max

s∈[,]

k

T



eQ(t)

A

 |ssx|

_+D

|ssx|–D|ssx|

dt+

p

j=

eQ(tj)lj

= max

s∈[,]

A

 +D–D

ss_|x|k

T



eQ(t)dt+

p

j=

eQ(tj)lj

≤ p

j=

eQ(tj)lj:=M. (.)

Obviously, we can findk∈N/{}such thatk> Mα, then we claim thatukis distinct to

ufor allk≥k. In fact, ifuk=ufor somek≥k, it is easy to check that

ϕk(uk) =kϕ(u)≥kα.

Then, by (.), we havek≤k≤Mα, a contradiction. We can also findk>max{k,

kM αk} such that ukk =uk for all k≥

k

k. Otherwise, ifukk =uk for some k≥k, we have

ϕkk(ukk) =kϕk(uk)≥kαk. Then by (.), we have

k

k ≤k≤

M

αk_, a contradiction. In

the same way, we can obtain that system (.) has a sequence of distinct periodic solutions with periodkjTsatisfyingkj∈Nandkj→ ∞asj→ ∞. This completes the proof.

4 Example

The following example is inspired partially by Example . in []. Let T = π, N= . Consider the following impulsive Lagrangian system with damped term:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

d(P(t)u(t))˙

dt + (q(t)P(t) +B)u˙(t) + ( 

q(t)B–A(t))u(t) +∇F(t,u(t))

= , a.e.t∈R,

(P(π_)u˙(π_)) =P(π_)u˙(π_+) –P(π_)u˙(π_–) =∇I(u(π_)), (P(π)u˙(π)) =P(π)u˙(π+) –P(π)u˙(π–) =∇I(u(π)),

(.)

where

P(t) =

⎛ ⎜ ⎝

sint+   

 sint+  

  cost+ 

⎞ ⎟

⎠, A(t) =

⎛ ⎜ ⎝

cost  

 cost 

  cost

⎞ ⎟ ⎠,

B=

⎛ ⎜ ⎝

  

–  

 – 

⎞ ⎟

⎠, q(t) =sint, K(t,x) = sint



+ .

|x|,

W(t,x) = cost



+ 

|x|

 – 

ln(e+|x|₎

,

I(x) = |

x|

 +|x|, I(x) =

ln( +|x|)

|x| .

Obviously, the condition (P) holds andA= ,B=√ and (K), (K), (W) and (W) hold witha=  andγ = .

eQ(t)∇W(t,x),x – W(t,x)=(|cos

t |+ )e–

cost_|x|

(e+|x|_)[ln(e+|_x|_)] .

Then (W) holds withh(t)≡. Moreover,

π



eQ(t)

 

A(t)x,x +K(t,x) –W(t,x) –h(t) 

dt

=

π



e–cost

 cost|x|

_{+ sin}t



+ .

– cost



+ 

|x|

 – 

ln(e+|x|₎

dt

≤e

π



 cost|x|

_{+ sin}t



+ .

|x|

– cost



+ 

|x|

 – 

ln(e+|x|₎

dt

=e

( + π)|x|– ( + π)|x|

 – 

ln(e+|x|₎

.

Hence, it is easy to see that there existsx∈RN such that (W) holds by the above

in-equality. Obviously, (I) and (I) hold. Note that



j=

eQ(tj)Ij(x) –

∇Ij(x),x

=eQ(π)

|x|

 +|x| –

|x|

( +|x|₎

+eQ(π)

ln( +|x|₎ |x| –

|x|

+|x| – ln( +|x|)

|x|

= |x|



 +|x|e Q(π_)

 – 

 +|x|

+eQ(π)

ln( +|x|₎ |x| –

  +|x|

≥–eQ(π) 

 +|x| ≥–eQ(π).

So (I) holds. Hence, by Theorem ., we obtain that system (.) has at least onekT -periodic solution for everyk∈N.

Moreover, it is easy to see that (K) holds withD= . Since

W(t,x) = cost



+ 

|x|

 – 

ln(e+|x|₎

≥|x|

 – 

ln(e+|x|₎

≥|x|.

ChooseD= . Then (W) holds. Hence, by Theorem ., we obtain that system (.) has

a sequence of distinct subharmonic weak solutions with periodkjT satisfyingkj∈Nand

kj→ ∞asj→ ∞.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Acknowledgements

This work is supported by Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).

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