Solvability of a second-order Hamiltonian system with impulsive effects

R E S E A R C H

Open Access

Solvability of a second-order Hamiltonian

system with impulsive effects

Binxiang Dai

_{and Jia Guo}

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China

Abstract

In this paper, a class of second-order Hamiltonian systems with impulsive effects are considered. By using critical point theory, we obtain some existence theorems of solutions for the nonlinear impulsive problem. We extend and improve some recent results.

MSC: 334B18; 34B37; 58E05

Keywords: Hamiltonian system; impulsive; critical point theory

1 Introduction and main results

Consider the second-order Hamiltonian systems with impulsive effects

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = ,

u˙i_(t

j) =Iij(ui(tj)), i= , , . . . ,N;j= , , . . . ,p,

(.)

where  =t<t<t<· · ·<tp<tp+=T,u(t) = (u(t),u(t), . . . ,uN(t)), Iij:R→R(i=

, , . . . ,N;j= , , . . . ,p) are continuous andF: [,T]×Rn_{→Rsatisfies the following}

as-sumption:

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

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

F(t,x)≤a|x|b(t), ∇F(t,x)≤a|x|b(t)

for allx∈Rn_{and a.e.t∈[,T].}

WhenIij≡ (i= , , . . . ,N;j= , , . . . ,p), (.) is the Hamiltonian system

⎧ ⎨ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = . (.)

In the past years, the existence of solutions for the second-order Hamiltonian systems (.) has been studied extensively via modern variational methods by many authors (see [–]).

When the gradient∇F(t,x) is bounded, that is, there existsg∈L_(,T;R+_{) such that}

_∇F(t,x)≤g(t)

for allx∈RN _{and a.e.t∈[,T], Mawhin-Willem in [] proved the existence of solutions}

for problem (.) under the condition

lim

|x|→+∞ T



F(t,x)dt= +∞

or

lim

|x|→+∞

T



F(t,x)dt= –∞.

When the gradient∇F(t,x) is bounded sublinearly, that is, there existf,g∈L(,T;R+) andα∈[, ) such that

_∇F(t,x)≤f(t)|x|α+g(t)

for allx∈RN _{and a.e.t∈[,T], Tang [] proved the existence of solutions for problem}

(.) under the condition

lim

|x|→+∞|x| –α

T



F(t,x)dt= +∞

or

lim

|x|→+∞|x| –α T



F(t,x)dt= –∞,

which generalizes Mawhin-Willem’s results.

ForIij= , problem (.) gives less results (see [–]). In [], Zhou and Li extended

the results of [] to impulsive problem (.); they proved the following theorems.

Theorem A[] Assume that(A)and the following conditions are satisfied: (h) There existf,g∈L_(,T;R+_{)andα∈[, )such that}

_∇F(t,x)≤f(t)|x|α_+g(t)

for allx∈Rnand a.e.t∈[,T]. (h) lim|x|→+∞|x|–α

T

 F(t,x)dt= +∞.

(h) For anyi= , , . . . ,N;j= , , . . . ,p,

tIij(t)≥, ∀t∈R.

Then problem(.)has at least one weak solution.

(h) For anyi= , , . . . ,N;j= , , . . . ,p,

tIij(t)≤, ∀t∈R.

(h) There existaij,bij> andβij∈(, )such that

Iij(t)≤aij+bij|t|αβij, ∀t∈R,i= , , . . . ,N;j= , , . . . ,p.

Then problem(.)has at least one weak solution.

Let

F(t,x) =f(t)|x|+α+C(x) –λ |x|

_{+h(t),x, (.)}

where C(x) is convex in RN _{(e.g., C(x) =} λ

(|x|+|x| +· · · +|xN|)), λ< π

T, f ∈ L_{[,T] satisfying} T

 f(t)dt>  (e.g., f(t) = T

 –t),  <α< , h∈L(,T;RN) andx=

(x_,x_{, . . . ,x}N_)∈RN_{. It is easy to see thatF(t,x) satisfies the condition (h) but does not}

satisfy the condition (h). The above example shows that it is valuable to further improve Theorem A.

Let

F(t,x) =f(t)|x|+α_{+C(x) –}λ

|x|

_{+h(t),x, (.)}

whereC(x) satisfies that the gradient∇C(x) is Lipschitz continuous and monotone inRN

(e.g., C(x) = λ (|x

_|+|x_|_{+· · ·+|x}N_|_{)),  <λ<} π

T ,f ∈L[,T] satisfies

T

 f(t)dt> 

(e.g.,f(t) =T

 –t),  <α< ,h∈L

_(,T;RN_{) andx= (x}_,x_{, . . . ,x}N_)∈RN_{. It is easy to see}

thatF(t,x) satisfies the condition (h) but does not satisfy the condition (h). The above example shows that it is valuable to further improve Theorem B.

In this paper, we further study the existence of solutions for impulsive problem (.). Our main results are the following theorems.

Theorem . Suppose that F(t,x) =H(t,x) +G(x)satisfies assumption(A)and the follow-ing conditions hold:

(H) There existf,g∈L_(,T;R+_{)andα∈[, )such that}

_∇H(t,x)≤f(t)|x|α_{+g(t) (.)}

for allx∈RN_{and a.e.t∈[,T].}

(H) There exists a positive numberλ<π

T such that

∇G(x) –∇G(y),x–y≥–λ|x–y| (.)

for allx,y∈Rn. (H)

lim

|x|→+∞|x|

–α T



(H) For anyi= , , . . . ,N;j= , , . . . ,p,

tIij(t)≥, ∀t∈R. (.)

Then impulsive problem(.)has at least one weak solution.

Remark . Theorem . generalizes Theorem A, which is a special case of our Theo-rem . corresponding toG(x)≡.

Example . LetN= ,p= . Consider the following impulsive problem: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = ,

u˙i(t) =Ii(ui(t)), i= , , ,

where

F(t,x) =

T

 –t

|x| _–λcosx_+h(t),x,

h∈L,T;R, Ii(t) =t 

 _{(i= , , ).}

Take

G(x) = –λcosx, x=x,x,x∈R,

which is bounded and

H(t,x) =

T

 –t

|x| _+h(t),x,

α=_,f(t) =_(T_ –t),g(t) =|h(t)|. Then all the conditions of Theorem . are satisfied. According to Theorem ., the above problem has at least one weak solution. However,

F does not satisfy the condition (h) in Theorem A. Therefore, our result improves and generalizes the Theorem A.

Theorem . Suppose that F(t,x) =H(t,x) +G(x)satisfies assumption(A)and the condi-tion(H)of Theorem..Furthermore,assume that

(H) There exist≤A<π

T ,B≥such that

_∇G(x) –∇G(y)≤A|x–y|+B (.)

for allx∈Rn_.

(H)

lim

|x|→+∞|x|

–α T



(H) For anyi= , , . . . ,N;j= , , . . . ,p,

tIij(t)≤, ∀t∈R. (.)

(H) There existaij,bij> andβij∈(, )such that

Iij(t)≤aij+bij|t|αβij (.)

for everyt∈R,i= , , . . . ,N;j= , , . . . ,p.

Then impulsive problem(.)has at least one weak solution.

Remark . Theorem . generalizes Theorem B, which is a special case of our Theo-rem . corresponding toG(x)≡.

Example . LetN= ,p= . Consider the following impulsive problem: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = ,

u˙i_(t

) =Ii(ui(t)), i= , , ,

where

F(t,x) =

T

 –t

|x| _–λ

|x|

_+h(t),x,

h∈L,T;R, Ii(t) = –t 

 _{(i= , , ).}

Take

G(x) = –λ |x|

_{, x=x}_,x_,x_∈R_,

which is bounded from above, and

H(t,x) =

T

 –t

|x| _+h(t),x,

α= ,f(t) =

 (

T

–t),g(t) =|h(t)|,ai=bi= ,βi= 

(i= , , ). Then all the conditions of

Theorem . are satisfied. According to Theorem ., the above problem has at least one weak solution. However,Fdoes not satisfy the condition (h) in Theorem B. Therefore, our result improves and generalizes Theorem B.

Theorem . Suppose that F(t,x) =H(t,x) +G(x)and Iij(t)satisfy the assumptions(A),

(H), (H), (H)and(H).Furthermore,assume that

(H)

lim

|x|→+∞|x|

–α

F(t,x) = –∞ (.)

uniformly for allt∈[,T].

Example . LetN= ,p= . Consider the following impulsive problem: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¨

u(t) =∇F(t,u(t)), a.e.t∈[,T],

u() –u(T) =u˙() –u˙(T) = ,

u˙i(t) =Ii(ui(t)), i= , , ,

where

F(t,x) = –|x|+α–λ |x|

_{, I}

i(t) = –t

α

 _{(i= , , ),  <α< ,  <λ<}π 

T .

Take

H(t,x) = –|x|+α_{, G(x) = –}λ

|x|

_{, x=x}_,x_,x_∈R_.

Then all the conditions of Theorem . are satisfied. According to Theorem ., the above problem has at least one weak solution. However,F(t,x) is neither superquadratic inX

nor subquadratic inX.

2 Preliminaries LetH

T ={u: [,T]→RN :uis absolutely continuous,u() =u(T) andu˙ ∈L(,T;RN)}

with the inner product

(u,v) =

T



u(t),v(t)dt+

T



˙

u(t),v˙(t)dt, ∀u,v∈H_T,

inducing the norm

u=

T



u(t)dt+

T



u_˙(t)dt

 

, ∀u∈H_T.

The corresponding functionalφonH

Tgiven by

φ(u) =  

T



u_˙(t)dt+

T



Ft,u(t)dt+

p

j= N

i= ui(tj)



Iij(t)dt (.)

is continuously differentiable and weakly lower semi-continuous onH

T. For the sake of

convenience, we denote

φ(u) =φ(u) +φ(u),

where

φ(u) =  

T



u_˙(t)dt+

T



Ft,u(t)dt

and

φ(u) = p

j= N

i= ui(tj)



For anyu,v∈H

T, we have

φ(u),v=

T



˙

u(t),v˙(t)dt+

p

j= N

i= Iij

ui(tj)

vi(tj)

+

T



∇Ft,u(t),v(t)dt. (.)

Definition . We say that a functionu∈H_T is a weak solution of (.) if the identity

T



˙

u(t),v˙(t)dt+

p

j= N

i= Iij

ui(tj)

vi(tj) + T



∇Ft,u(t),v(t)dt= 

holds for anyv∈H T.

It is well known that the solutions of impulsive problem (.) correspond to the critical point ofφ.

Definition .[] LetXbe a Banach space,φ∈C_{(X,R) andc∈R.}

() φis said to satisfy the(PS)c-condition onXif the existence of a sequence{xn} ⊆X

such thatφ(xn)→candφ(xn)→asn→ ∞implies thatcis a critical value ofφ.

() φis said to satisfy theP.S.condition onXif any sequence{xn} ⊆Xfor whichφ(xn)

is bounded andφ(xn)→asn→ ∞possesses a convergent subsequence inX.

Remark . It is clear that theP.S. condition implies the (PS)c-condition for eachc∈R.

Lemma .[] Ifφ is weakly lower semi-continuous on a reflexive Banach space X(i.e.,

if uku,thenlim infk→∞φ(uk)≥φ(u))and has a bounded minimizing sequence,thenφ

has a minimum on X.

Remark . The existence of a bounded minimizing sequence will be in particular en-sured whenφis coercive,i.e., such that

φ(u)→+∞ ifu →+∞.

Lemma .[] Let X be a Banach space and letφ∈C_{(X,R).Assume that X splits into a} direct sum of closed subspaces X=X–_⊕X+_withdimX–_<∞and

sup

S–_R φ<inf

X+φ,

where

S–_R=u∈X–:u=R.

Let

and

c= inf

h∈Mmaxs∈B–_Rφ

h(s).

Then ifφsatisfies the(PS)c-condition,c is a critical value ofφ.

Lemma .[] If the sequence{uk}converges weakly to u in HT,then{uk}converges uni-formly to u on[,T].

Lemma .[] If u∈H Tand

T

 u(t)dt= ,then

T



u(t)dt≤ T 

π T



u_˙(t)dt (Wirtinger’s inequality)

and

u_∞≤ T



T



u_˙(t)dt (Sobolev’s inequality).

Lemma . There exists C∗> such that if u∈HT,then

u∞≤C∗u.

Moreover,if_Tu(t)dt= ,then

u∞≤C∗˙uL,

whereu∞=maxt∈[,T]|u(t)|anduL= (

T  |u(t)|

_dt)_.

3 Proof of main results Foru∈H_T, letu¯= 

T

T

 u(t)dtandu˜(t) =u(t) –u¯.

Proof of Theorem. It follows from (H) and Sobolev’s inequality that

T



Ht,u(t)–H(t,u¯)dt

=

T

 



∇Ht,u¯+su˜(t),u˜(t)ds dt

≤ T

 



f(t)u¯+su˜(t)αu˜(t)ds dt+

T

 



g(t)u˜(t)ds dt

≤ T



f(t)|¯u|α_+u˜(t)α_u˜(t)dt+ T



g(t)u˜(t)dt

≤|¯u|α+˜uα_∞˜u∞

T



f(t)dt+˜u∞

T

 g(t)dt

≤(π–λT) π_T ˜u



∞+

π_T

(π_–λT₎|¯u| α

T

 f(t)dt

+ ˜uα_∞+ T



f(t)dt+˜u∞

T

 g(t)dt

≤π–λT π

T



u_˙(t)dt+C|¯u|α+C T



u_˙(t)dt

α+ 

+C T



u_˙(t)dt

 

(.)

for allu∈H

Tand some positive constantsC,CandC. By (H) and Wirtinger’s

inequal-ity, we have

T



Gu(t)–G(¯u)dt=

T

 



∇Gu¯+su˜(t)–∇G(u¯),u˜(t)ds dt

=

T

 





s

∇Gu¯+su˜(t)–∇G(u¯),su˜(t)ds dt

≥ T

 



(–λ)su˜(t)ds dt

= –λ 

T



u_˜(t)dt

≥–λT



π T



u_˙(t)dt (.)

for allu∈H T.

From (H), we obtain

φ(u)≥

for allu∈H_T. Therefore we have

φ(u) =  

T



u_˙(t)dt+

T



Ft,u(t)–F(t,u¯)dt+

T



F(t,u¯)dt+φ(u)

=  

T



u_˙(t)dt+

T



Ht,u(t)–H(t,u¯)dt+

T



Gu(t)–G(u¯)dt

+

T



F(t,u¯)dt+φ(u)

≥  

T



u_˙(t)dt+

T



Ht,u(t)–H(t,u¯)dt+

T



Gu(t)–G(u¯)dt

+

T



F(t,u¯)dt

≥  

T



u_˙(t)dt–π

_–λT

π T



u_˙(t)dt–C|¯u|α

–C T



u_˙(t)dt

α+ 

–C T



u_˙(t)dt

 

–λT



π T



u_˙(t)dt

+

T



= π

_–λT

π T



u_˙(t)dt+|¯u|α

|¯u|–α

T



F(t,u¯)dt–C

–C T



u_˙(t)dt

α+ 

–C T



u_˙(t)dt

 

(.)

for allu∈H

T. Asu → ∞if and only if (|¯u|+˙u(t)L) 

 → ∞, (.) and (H) imply

that

φ(u)→+∞ asu → ∞.

By Lemma . and Remark .,φhas a minimum point onH

T, which is a critical point

ofφ. Therefore, we complete the proof of Theorem ..

Lemma . Assume that the conditions of Theorem.hold.Thenφsatisfies the P.S. con-dition.

Proof Let{φ(un)}be bounded andφ(un)→ asn→ ∞. From (H) and Lemma ., we

have T



∇Ht,un(t)

,u˜n(t)

dt

≤ T



_∇H

t,un(t)u˜n(t)dt

≤ T



f(t)u¯n+u˜(t)

α

˜

un(t)dt+ T



g(t)u˜n(t)dt

≤|¯un|α+˜unα∞

˜un∞ T



f(t)dt+˜un∞ T

 g(t)dt

≤(π–AT) π_T ˜un



∞+

π_T

(π_–AT₎|¯un| α T

 f(t)dt



+ ˜unα_∞+ T



f(t)dt+˜un∞

T

 g(t)dt

≤π–AT π

T



u_˙n(t) 

dt+C|¯un|α

+C T



u_˙n(t) 

dt

α+ 

+C T



u_˙n(t) 

dt

 

(.)

for all large n and some positive constants C, C and C. It follows from (H) and

Lemma . that

T



∇Gun(t)

,u˜n(t)

dt=

T



∇Gun(t)

–∇G(u¯n),u˜n(t)

dt

≤ T



_∇G

un(t)

–∇G(u¯n)u˜n(t)dt

≤ T



=A T



u_˜n(t) 

dt+B T



u_˜n(t)dt

≤A T



u_˜n(t) 

dt+B√T T



u_˜n(t) 

dt

 

≤ AT π

T



u_˙n(t) 

dt+BT

√

T

π T



u_˙n(t) 

dt

 

. (.)

By (.), (.), (H) and Young’s inequality, we have

˜un ≥

φ(un),u˜n

=

T



u_˙n(t) 

dt+

T



∇Ft,un(t)

,u˜n(t)

dt+ p j= N i= Iij

ui_n(t)u˜i_n(t)

=

T



_u˙n(t) dt+

T



∇Ht,un(t)

,u˜n(t)

dt+

T



∇Gun(t)

,u˜n(t)

dt + p j= N i= Iij

ui_n(t)u˜i_n(t)

≥π–AT π

T



u_˙n(t)dt–C|¯un|α–C T



u_˙n(t)dt

α+ 

–

C+ BT√T

π

T



u_˙n(t)  dt   – p j= N i=

aij+bijuin(t)

αβij_˜

ui_n(t)

≥π–AT π

T



u_˙n(t) 

dt–C|¯un|α–C T



u_˙n(t)  dt α+  –

C+ BT√T

π

T



u_˙n(t)dt

 

–apN˜un∞

–b p j= N i=

|¯un|αβij+˜un αβij ∞ ˜un∞

≥π–AT π

T



u_˙n(t) 

dt–C|¯un|α–C T



u_˙n(t)  dt α+  –

C+ BT√T

π

T



u_˙n(t)  dt   –apN T  T 

u_˙n(t)  dt   –b p j= N i=

βij|¯un|α– b p j= N i=

 –βij

 ˜un

 –β_ij ∞

– b p j= N i=

˜un αβij+ ∞

≥π–AT π

T



u_˙n(t) 

dt–C|¯un|α–C T



u_˙n(t)  dt α+  –

C+ BT√T

π

T



u_˙n(t)  dt   –apN T  T 

u_˙n(t) 

dt

–b p

j= N

i=

βij|¯un|α–b p

j= N

i=

( –βij)

T



T



u_˙n(t)dt

 –βij

– b p

j= N

i=

T



T



u_˙n(t)dt

αβij+ 

, (.)

where a=max{aij,i= , , . . . ,N;j= , , . . . ,p}, b=max{bij,i= , , . . . ,N;j= , , . . . ,p}.

From Wirtinger’s inequality, we obtain

T



u_˙n(t)dt≤ ˜un≤

 + T



π

T



u_˙n(t)dt. (.)

The inequalities (.) and (.) imply that

C|¯un|α≥ T



u_˙n(t) 

dt

 

–C (.)

for all large n and some positive constants C and C. It follows from (H),

Cauchy-Schwarz’s inequality and Wirtinger’s inequality that

_TGun(t)

–G(¯un)

dt=

T

 



∇Gu¯n+su˜n(t)

–∇G(¯un),u˜n(t)

ds dt

≤ T

 



Asu˜n(t)+Bu˜n(t)ds dt

= A 

T



u_˜n(t) 

dt+B T



u_˜n(t)dt

≤ A 

T



_u˜n(t)

dt+B√T T



_u˜n(t) dt

 

≤ AT π

T



u_˙n(t)dt+ BT√T

π T



u_˙n(t)dt

 

. (.)

Like in the proof of Theorem ., we get

T



Ht,un(t)

–H(t,u¯n)

dt≤π _–AT

π T



u_˙n(t) 

dt+C|¯un|α

+C T



u_˙n(t) 

dt

α+ 

+C T



u_˙n(t)dt

 

. (.)

From (H), we have

for allu∈H

T. Since{φ(un)}is bounded, from (.) and (.), there exists a constantC

such that

C≤φ(un)

=  

T



u_˙n(t)dt+ T



Ft,un(t)

–F(t,u¯n)

dt+

T



F(t,u¯n)dt+φ(un)

≤  

T



u_˙n(t) 

dt+

T



Ht,un(t)

–H(t,u¯n)

dt

+

T



Gun(t)

–G(¯un)

dt+

T



F(t,u¯n)dt

≤ π+AT π

T



u_˙n(t) 

dt+C|¯un|α+C T



u_˙n(t) 

dt

α+ 

+C T



u_˙n(t) 

dt

 

+BT √

T

π u˙n(t) 

dt₊ T



F(t,u¯n)dt

≤ |¯un|α

|¯un|–α

T



F(t,u¯n)dt+C

(.)

for all largenand some constantC. By the above inequality and (H), we know that

{|¯un|}is bounded. In fact, if not, without loss of generality, we may assume that|¯un| → ∞

asn→ ∞. Then, from (.) and the above inequality, we have

lim

n→∞inf|¯un|

–α T



F(t,u¯n)dt> –∞,

which contradicts (H). Hence{|¯un|}is bounded. Furthermore,{un}is bounded from (.)

and (.). Hence, there exists a subsequence ofundefined byu∗nsuch that

u∗_nu inH_T.

By Lemma ., we have

u∗_n→u inC[,T].

On the other hand, we get

φu∗_n–φ(u),u∗_n–u=

T



u_˙∗_n(t) –u˙(t)dt

+

T



∇Ft,u∗_n(t)–∇Ft,u(t),u∗_n(t) –u(t)dt

+

p

j= N

i=

Iij

u*i_n(tj)

–Iij

ui(tj)

u*i_n(tj) –ui(tj)

. (.)

It follows from the above equality, (A) and the continuity ofIijthat

u∗_n→u inH_T.

Now, we give the proof of our Theorem ..

Proof of Theorem. LetWbe the subspace ofH

Tgiven by

W=u∈H_T|¯u= .

ThenH_T =W⊕RN. Firstly, we show that

φ(u)→+∞ asu∈W,u → ∞. (.)

In fact, foru∈W, thenu¯= , by the proof of Theorem ., we have

T



Ht,u(t)–H(t, )dt≤ π _–AT

π T



u_˙(t)dt+C T



u_˙(t)dt

α+ 

+C T



u_˙(t)dt

 

. (.)

Like in the proof of Lemma ., we obtain

T



Gu(t)–G()dt=

T

 



∇G(su) –∇G(),uds dt

≤ A 

T



u(t)dt+B T



u(t)dt

≤ AT π

T



u_˙n(t) 

dt+BT

√

T

π T



u_˙n(t) 

dt

 

. (.)

By (H) and Lemma ., we find

φ(u)=

p

j= N

i= ui(tj)



Iij(t)dt

≤

p

j= N

i= ui(tj)



aij+bij|t|αβij

dt

≤apNu∞+b p

j= N

i=

uαβ∞ij+

≤apNC∗

T



u_˙(t)dt

 

+b p

j= N

i=

C_∗ T



u_˙(t)dt

αβij+ 

(.)

for allu∈W. By (.), (.) and (.), we have

φ(u) =  

T



_u˙(t)dt+

T



Ft,u(t)–F(t, )dt+

T



F(t, )dt+φ(u)

=  

T



u_˙(t)dt+

T



Ht,u(t)–H(t, )dt+

T



+

T



F(t, )dt+φ(u)

≥

π–AT

π

T



u_˙(t)dt–C T



u_˙(t)dt

α+ 

–C T



u_˙(t)dt

 

–BT √

T

π T



u_˙n(t) 

dt

 

–apNC∗

T



u_˙(t)dt

 

–b p

j= N

i=

C_∗ T



u_˙(t)dt

αβ_ij+ 

dt

+

T



F(t, )dt (.)

for allu∈W.

By Lemma ., we have u → ∞ ⇔ ˙u_L → ∞onW. Hence (.) follows from

(.).

On the other hand, by (H), we get

φ(u)≤ (.)

for allu∈H

T. Therefore, from (.) and (H), we obtain

φ(u) =

T



F(t,u)dt+φ(u)≤ |u|α

|u|–α

T



F(t,u)dt

→–∞

as|u| → ∞inRN_{. It follows from Lemma . and Lemma . that problem (.) has at}

least one weak solution.

Proof of Theorem. First we prove thatφsatisfies theP.S. condition. Suppose thatUnis

aP.S. sequence forφ, that is,φ(un)→ asn→ ∞andφ(un) is bounded. In a way similar

to the proof of Theorem ., we have

_T∇Ht,un(t)

,n˜n(t)

dt≤π _–λT

π T



u_˙n(t)dt+C|¯un|α

+C T



u_˙n(t) 

dt

α+ 

+C T



u_˙n(t) 

dt

 

and

T



∇Gun(t)

,u˜n(t)

dt≥–λT



π T



_u˙n(t) dt

for allu∈H

T. Hence we have

˜un ≥

φ(un),u˜n

=

T



u_˙n(t) 

dt+

T



∇Ft,un(t)

,u˜n(t)

dt+

p

j= N

i= Iij

=

T



u_˙n(t) 

dt+

T



∇Ht,un(t)

,u˜n(t)

dt

+

T



∇Gun(t)

,u˜n(t)

dt+

p

j= N

i= Iij

ui_n(t)u˜i_n(t)

≥(π–λT) π

T



u_˙n(t) 

dt–C|¯un|α–C T



u_˙n(t) 

dt

α+ 

–C T



u_˙n(t) 

dt

 

–apN

T



T



u_˙n(t) 

dt

 

–b p

j= N

i=

βij|¯un|α–b p

j= N

i=

( –βij)

T



T



u_˙n(t)dt

 –β_ij

– b p

j= N

i=

T



T



u_˙n(t)dt

αβ_ij+ 

.

From (.) and the above inequalities, we obtain

C|¯un|α≥ T



u_˙n(t)dt

 

–C (.)

for some positive constantsCandC.

By (H) there existsM>  such that

|x|–α_F(t,x)≤

for all|x| ≥Mandt∈[,T], which implies that

F(t,x)≤

for all|x| ≥Mandt∈[,T]. It follows from assumption (A) that

F(t,x)≤ab(t)

for all|x| ≤Mandt∈[,T], wherea=max≤s≤Ma(s).

Letγ(t) =ab(t)∈L(,T), then

F(t,x)≤γ(t)

for allx∈RN_{andt∈[,T].}

By the boundedness ofφ(un), (H) and (.), there exists a constantCsuch that

C≤φ(un) =

 

T



_u˙n(t)dt+

T



Ft,un(t)

dt+φ(u)

≤  

T



u_˙n(t)dt+ T



Ft,un(t)

dt≤C|¯un|α+C



+

T



which implies that|¯un|is bounded. Like in the proof of Lemma ., we know thatφsatisfies

theP.S. condition.

Furthermore, we can prove Theorem . using the same way as in the proof of Theo-rem .. Here, we omit it.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

We express our gratitude to the referees for their valuable criticism of the manuscript and for helpful suggestions. Supported by the National Natural Science Foundation of China (11271371, 10971229).

Received: 24 April 2013 Accepted: 30 May 2013 Published: 25 June 2013

References

1. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

2. Tang, CL: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc.126, 3263-3270 (1998)

3. Ma, J, Tang, CL: Periodic solutions for some nonautonomous second-order systems. J. Math. Anal. Appl.275, 482-494 (2002)

4. Jiang, Q, Tang, CL: Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl.328, 380-389 (2007)

5. Luan, SX, Mao, AM: Periodic solutions of nonautonomous second order Hamiltonian systems. Acta Math. Sin. Engl. Ser.21, 685-690 (2005)

6. Schechter, M: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ.223, 290-302 (2006) 7. Tang, CL: Periodic solutions of nonautonomous second order systems withγ-quasisubadditive potential. J. Math.

Anal. Appl.189, 671-675 (1995)

8. Tang, CL, Wu, XP: Periodic solutions for second order Hamiltonian systems with a change sign potential. J. Math. Anal. Appl.292, 506-516 (2004)

9. Tao, ZL, Tang, CL: Periodic and subharmonic solutions of second-order Hamiltonian systems. J. Math. Anal. Appl.293, 435-445 (2004)

10. Tao, ZL, Yan, SA, Wu, SL: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl.331, 152-158 (2007)

11. Ye, YW, Tang, CL: Periodic solutions for some nonautonomous second order Hamiltonian systems. J. Math. Anal. Appl.

344, 462-471 (2008)

12. Yang, RG: Periodic solutions of some nonautonomous second order Hamiltonian systems. Nonlinear Anal.69, 2333-2338 (2008)

13. Tang, XH, Meng, Q: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal., Real World Appl.11, 3722-3733 (2010)

14. Zhou, JW, Li, YK: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal.72, 1594-1603 (2010)

15. Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.10, 680-690 (2009)

16. Tian, Y, Ge, W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc.51, 509-527 (2008)

doi:10.1186/1687-2770-2013-151