Periodic solutions for a class of nonautomous second order systems

Vol. 1, 2013-14, 66-75

ISSN: 2349-0632 (P), 2349-0640 (online) Published on 22 May 2014

www.researchmathsci.org

66

Journal of

Periodic solutions for a class of nonautomous second

order systems

Er Ye and Yabin Shao

College of Mathematics and Computer Science

Northwest University for Nationalities, Lanzhou – 730030, Lanzhou, China E-mail: yb-shao@163.com

Received 14 April 2014; accepted 20 may 2014

Abstract. In this paper, some multiplicity theorem is obtained for periodic solutions of nonautomous second order systems with partially periodic potential by the minimax methods.

Keywords: Nonautomous second order systems; periodic solutions; minimax methods AMS Mathematics Subject Classification (2010): 34A34, 34A12

1. Introduction and main results

Consider the second-order systems

{

() (, ()) 0 a.e.t [0, ], )

( ) ( ) 0 ( ) ( ) 0 (

T t

u t F t u

t e T u u T u u

∈ =

∇ −

= − = − ɺ ɺ

ɺ

ɺ (1) where T>0, and F [ ]T _×RN_→R

, 0

: satisfies the following assumption: )

, ( )

(H Ft x is measurable in t for each x∈RN and continuously differentiable in x for

[ ]T t e

a.. ∈0, and there exist a∈C(R+,R+), _b∈L1(0,_T;_R+) _{such that} )

( ) ( ) , ( ), ( ) ( ) ,

(t x a x bt F t x a x bt

F ≤ ∇ ≤

for all _x∈×RN _{and a.e.t}∈

[ ]

_{0,T .}

The existence of periodic solutions for problem (1) has been studied extensively, a lot of existence and multiplicity results have been obtained, we refer the readers to [1-17] and the reference therein.

Suppose that F( xt, ) is T_i −periodic in

x

_i, 1≤i≤r, that is

_F

(

)

₍

_,

₎

1

x

t

F

x

t,

r kTe

i i i i

=

∑

+

₌ , (2) for alla.e.t∈

[ ]

0.T , _x∈×_RN _{and all integers}

i

k

,1≤i≤r , where

{ }

ei (1≤i≤r) is the

canonical basis of N R .

With periodic potentials, that is, (2) holding with r=N, the existence and multiplicity theorems are obtained for the nonautonomous second-order system (1) in [1] and [2] respectively. When the nonlinearity is bounded, In [3-4], Chang and Liu have studied the nonautonomous second order system (1) with partially periodic (that is,(2)holding with1≤r≤N)a- nd partially uniformly coercive potentials (F( xt, )→+∞for every ( x,

r

r R

x

x2⋅⋅⋅, )∈ as(xr+1,xr+2⋅ ⋅⋅xN) tends to infinity inR r N−_).

In [5], Wu has obtained the following result.

Theorem A. Suppose that F satisfies conditions (H) and (2). Assume that there exist )

; , 0 (

1 +

∈L T R

) ( ) ,

(t x g t

F ≤

∇ and

_∫

TF t xdt→+∞

0 (, )

,

for every

(

x

₁

,

x

₂

⋅

⋅⋅

,

x

_r

)

∈

R

r as

(

x

r+1

,

x

r+2

⋅

⋅⋅

,

x

N

)

tends to infinity in

R

N−r .Then problem (1) has at least

r

+

1

geometrically distinct solutions in

H

T

1

.

In 2003, Tang [6] generalized Theorem A and obtained the following results.

Theorem B. Assume that there exist , ∈ 1(0, ; +)

R T L g

f and 0≤α<1 such that ∇F(t,x)≤f(t)

x

+g(t)

α

(3)

for all N

R

x∈ and a.e.t∈

[ ]

0,T . Suppose that F satisfies conditions (H) and (2). and x−

_∫

TF t xdt_→+∞

0 2

) , (

α _{, (4)}

for every

(

x

1

,

x

2

...,

x

r

)

∈

R

r as

(

x

r+1

,

x

r+2

...,

x

N

)

tends to infinity in RN+r .Then

problem (1) has at least r+1 geometrically distinct solutions in _HT 1_.

In 2011, Zhang and Tang [7] investigated the existence of periodic solutions for problem (1) and obtained the following results.

Theorem C. Assume that there exist f, 1(0, ; )

R T L

g∈ + and 0≤α<1 such that

) g( ) ( ) ,

(t x f t x t

F ≤ +

∇ α ,

for all N R

x∈ and a.e.t∈

[ ]

0,T . Suppose that F satisfies conditions (H) and (2).and

2

0 0

2

)

)

(

(

8

)

,

(

inf

lim

x

T

F

t

x

dt

T

T

f

t

dt

x

∫

>

∫

α − ∞

→ ,

for every (x1,x2⋅⋅⋅,xr)∈Rras

(

x

r+1

,

x

r+2

⋅

⋅⋅

,

x

N

)

tends to infinity in r N

R

−.Then problem (1) has at least r+1 _{geometrically distinct solutions in H} 1T .

In this paper, we consider the following second-order systems:

{

ɺ_u(uɺ(t₀)₎+₋A_u(uɺ_T(t₎)₌+_uɺ∇₍₀F₎₋(t_uɺ,u(_(Tt₎))₌=₀0, a.e.t∈

[ ]

0,T , (5)

where T>0, A is antisymmetry constant matric with A

<

2_Tπ . The Hilbert space 1

T

H is defined by HT1 =

{

u:

[ ]

0,T ֏RN| u is a absolutely continuous u(0)=u(T) and uɺ∈L2(0,T;RN)

}

_and 1

T

H is endowed with the norm

(

)

2

1

0 0

2 2

∫

+

∫

= T T

dt u dt u

u ɺ ,

By sobolev embeded theorems, there exist C>0, for all u _HT

1

∈ , such that _{u ≤Cu}

∞ . (6)

where u ∞=_t∈max_{[ ]0,T} u(t) .Motivated by [1-7], The following main results are obtained by the minimax methods.

Theorem 1.1. Suppose that F satisfies conditions (H), and there exist

,

1

(

0

,

;

)

R

T

L

g

f

∈

+

such that

∇_F(t,x) ≤ _f(t)x +_g(t), (7)

68

   

 

−

  

  

=

=

∫

T A

T dt

t f M

T

π π δ

2 1 2 , 2 1 min , )

( ₂

2 0

1 .

δ

<

2 1 2M C , Assume that (2) holds and

=

_∫

T _→+∞ dt x t F x

0 2

) ,

( , (8) as x tends to infinity in ∈₀×_RN−r.Then problem (5) has at least _r+₁geometrically distinct

solutions inHT

1_.

Remark 1.1. System (5) generalizes system (1) obviously, for if A= 0, then system (5) becomes system (1). We consider the system (5) with partially periodic potentials and linear nonlinearity.

Theorem 1.1 is a new result, which completes the theorem B and theorem C with α_{=1 in (3). Theorem 1.1 generalized Theorem A , which is the special case of Theorem} 1.1 corresponding to f(t)≡_0.

Example 1.1. There are functions F satisfying our Theorem 1.1 and not satisfying the results in [1]-[7]. For example, let

2

1 2

1 sin sin

sin 1 ) ,

(

∑

+ =

+ + ⋅⋅ ⋅ + + + +

= N

r j

j

r x

x x

x r

x t

F ,

where

{

}

N

N R

x x x

x= 1, 2,⋅ ⋅⋅, ∈ .

Theorem 1.2. Suppose that F satisfies conditions (H), (2),(7) and

_∫

T _{→ −∞} dt x t F

x2 ₀ ( , ) , (9) as x tends to infinity in ∈0×RN−r.Then problem (5) has at least

r

+

1

geometrically

distinct solutions in_HT1.

Example 1.2. There are functions F satisfying our Theorem 1.2 and not satisfying the results in [1]-[7]. For example, let

) sin

sin sin 1 ( ) , (

2

1 2

1

∑

+ =

+ + ⋅⋅ ⋅ + + + + −

= N

r j

j

r x

x x

x r

x t

F ,

where

{

}

N

N R

x x x

x= 1, 2,⋅ ⋅⋅, ∈ . 2. Preliminaries

Foru∈H_T1,let 1 0 ( )

T T

u = ∫ u t dt ,and ~u(t)=u(t)−u.Then we have Sobolev’s inequality _u T T_ut 2_dt

0 2

) ( 12

~ _≤

_∫

ɺ , (10)

and Wirtinger’s inequality

_∫

Tu t dt_≤ T

_∫

Tut dt

0 2 2 2 2

0 () ₄ ()

~ ɺ

π (11)

for allu∈H_T1 (see Proposition 1.3 in [8]).

[

]

i r

i

i i i i

N

r i

i e Qu u e lT e

e u u

P

∑

∑

= +

=

− =

=

1 1

) , ( ,

) , (

and li(1≤i≤r) is the unique integer such that 0≤(u,ei)−liTi <Ti

Define the functional

ϕ

on HT1 by

dt t u t F dt t u t Au dt

t u

u =

∫

T +

∫

T −

∫

T

ϕ 0 0

2

0 ( ( ), ( )) ( , ( ))

2 1 ) ( 2 1 )

( ɺ ɺ

Then

ϕ

is continuously differentiable by Lemma 1 in [9] and the solutions of systems (5) correspond to the critical points of ϕ. Moreover, one has

dt t v t u t F dt

t v t Au dt

t v t u v

u =

∫

T +

∫

T −

∫

T ∇

ϕ 0 0( (), ( )) 0( (, ( )), ( ))

2 1 )) ( ), ( ( ),

(

Let

1

| ,1

r

i i i i i

G k T e k Z i r

=

 

= ∈ ≤ ≤ 



∑



be a discrete subgroup of H and letT1

π

:

1 1

T T

H → H G be the canonical surjection. It is obvious that 1

T

H G = X×V ,

whereX =Y⊕Z, = ~1=

{

∈ 1| =0

}

u H u

Y H_{T T}

Z

=

span

{

e_r+1,⋅⋅⋅,e_N

}

,andV span e=

{

1, ,⋅⋅⋅er

}

G is

isomorphic to the torus r

T .Define ψ ;X×Y→R by

) ( )) (

(π u ϕ u

ψ = .

It follows from (2) that ψ is well-defined. Moreover,

ψ

is continuously differentiable.

3. Proof of theorem

Now we begin to prove our main result, for the sake of convenience, c denote some constant.

Proof of Theorem 1.1. the proof relies on the generalized saddle point theorem due to Liu [4]. Assume that π((u_n))→0 is a (PS) sequence for

ψ

,that is, ψ(π(u_n)) is bounded and ψ′(π(un))→0.Then

ϕ

(un) is bounded and

φ

′(un)→0 .

Let

{ }

1

T

n H

u ∈ be such that

ϕ

(

u

_n

)

≤

c

,

ϕ′

(

u

_n

)

→

0

,

(

n

→

∞

)

(12) First,we shall prove that

{ }

u_n is bounded in H_T1.By contradiction, we assume

∞ →

n

u as n→∞. For

{ }

1

T

n H

u ∈ ,let utdt T u=

∫

T

0 ()

1 _{,and u}~_{(t)=u(t)−u.By (12), for} 1

T H

∈ φ

∀ ,we have

) ( 0 ) ), , ( ( )

, ( )

,

( _{0 0}

0 φ +

∫

φ −

∫

∇ φ = φ

∫

u dt Au dt T F t un dt

T n T

n

ɺ ɺ

ɺ .

Taking φ=u~_n,we have

∫

₀Tuɺ_n2dt+

∫

₀T(Au_n,uɺ_n)dt−

∫

₀T(∇F(t,u_n),u~_n)dt≤ u~_n . (13) By the Wirtinger inequality (11) and 2

T

A≤ π_{, we have}

dt

u

Au

dt

u

T _{n n}

T

n

∫

∫

+

0

2

0

(

,

)

70 dt u A

T T 2

0 ) 2 1 (

∫

π − ≥ ɺ dt u A T T dt u A T T n T n 2 0 2 2 2 0 ~ ) 2 1 ( 4 2 1 ) 2 1 ( 2 1

∫

∫

π − π + π − ≥ ɺ 2 2 2 2 ~ ~ ) 2 1 ( 2 , 2 1

min A un un

T

T  − π =δ

 



 π

≥ (14)

where ) 2 1 ( 2 , 2 1 min ₂ 2 A T T  − π

    π = δ .

By (7), (11) and the Hoɺɺlder inequality, one has

dt u u t F T n n

∫

0 ∇ )

~ ), , ( (

dt

u

t

g

dt

u

u

t

f

_n T _n

T n

~

)

(

~

)

(

0 0

∫

∫

+

≤

∞ ∞ ∞

∫

∫

+ ≤ n T n T u dt t g u u dt t

f( ) ~ ₀ ( ) ~

0

∫

∫

+

≤

T n n n T

u

dt

t

g

C

u

u

dt

t

f

C

0 0 2

~

)

(

~

)

(

n T n

n

u

C

g

t

dt

u

u

C

M

~

(

)

~

0 2

1

+

∫

=

(15) where =∫T

dt t f

M1 0 () .

By (13),(14) and (15), we have

dt u u t F dt u Au dt u

u_n ≥

∫

T _n +

∫

T _{n n} −

∫

T ∇ _{n n}

0 0 2 0 ) ~ ), , ( ( ) , ( ɺ ɺ n T o n n

n M C u u C g t dtu

u~ 2 ~ ( ) ~

1 2

∫

− − δ ≥ , Therefore

_u MC _{u C}

n

n ≤ δ +

2 1

~ (16)

It follows from (15) and (16) that

∫

T ∇

n

n u dt

u t F 0 ) ~ ), , ( ( n T n

n u C g t dt u

u C

M ~ () ~

0 2

1 +

∫

≤ C u C u C M n

n + +

δ

≤ ₁2 4 2 _{, (17)}

By (16), 2<δ 1

2MC and the boundedness of Qu_{n , we have}

2 1 ~ 2 1 < δ ≤ →m MC u

u

n

δ − ≥ − → − −

≥ 1 2

2 1 1 ~

1 m MC

u u Q u u u u P n n n n n

n , (n→∞) (18)

By (18), for ∀ε>0,such that ₀ 2

1 2

1 −ε>

δ

−MC ,we have

n

n

P

u

u

C M ε − δ −

≤

2 1 2 1

1

_{. (19)}

It follows from (13),(16), (17) and (19) that

dt

u

Au

dt

u

T _{n n}

T

n

∫

∫

+

0

2

0

(

,

)

ɺ

ɺ

n T

n

n

u

dt

u

u

t

F

(

,

),

~

)

~

(

0

∇

+

≤

∫

M C un +cun +c

δ ≤ 2 4 2 1 c u P c u P C M n n C

M + +

δ ≤ ε − δ − 2 4 2 1 2 1 2 1

1 ₍₂₀₎

By (16) and (19), we have

n n

n Pu

C M c u C M u C M ε − δ − δ ≤ + δ ≤ 2 1 2 1 1

~ 1 2

2

1 . (21)

By (6),(7),(21), the Hoɺɺlder _{inequality and the boundedness of} Qu_{n ,we have}

∫

T −

∫

T

n

n dt F t Pu dt

u t F

0 (,ˆ ) 0 ( , )

≤

∫ ∫

∇

+

+

T

n n

n

s

Q

u

u

u

P

t

F

0 1 0

))

~

(

,

(

dt u u Q u u Q u P t

f n n

T

n n

n ~ ) ~

)( (

0 + + +

≤

_∫

g t Qun undt

T ~ ) ( 0 + +

_∫

∞ ∞ ∞+ + +

≤

_∫

n n n n n

T u u Q u u Q u P dt t

f( ) ( ~ ) ~

0 +

∫

+ ∞

T n n u u Q dt t g 0 ~ ) ( n n n n n T u u Q u u Q u P dt t f

C () ( ~ ) ~

0

2 _{+ + +}

≤

_∫

n n T u u Q dt t g

C ( ) ~

0 +

∫

2 2

1 2

1C Pun Qun ~un MC Qun u~n

M + + +

= n n

T u u Q dt t g

C () ~

0 + +

_∫

c u P c u P C M C M n

n + +

− −

≤ ₁2 4 2

2 1 2 1 1 ε δ δ 2 2 1 2 2 1 2 4 2 1 ) ( 1 2

1C n

M M C Pu

C M

ε

δ

− _δ − + 2 4 2 1 2 2 1 2 1 2 1 ) 2 1 ( 2 1 1 n u P C M C M C M C M           − − − − − = ε δ δ ε δ

72

c u P c u P C

M

n n

C

M + +

δ =

ε − δ −

ε

− ₂

4 2 1

2 2 1

) 2

1 (

2 1

(22)

By (20) and (22), we have

ϕ

(

u

_n

)

=

ϕ

(

u

ˆ

_n

)

_{Pu cPu c}

C M C

M

n

n + +

− −

≤ ₁2 4 2

2 2 1

) 2

1 (

1

ε δ δ

[

]

∫

∫

− −

− T

n T

n

n F t Pu dt F t Pu dt

u t F

0

0 (,ˆ ) ( , ) (, )

t u P C

M C

M

n 2 4

2 1

2 2 1 ₎

2 1 (

1 ε δ

δ _{− −}

≤

2

2 2

1 2 1 4 2 1

)

( M1C2 n

u P C

M

ε

ε

δ

_{− δ −}

− +

dt u P t F c u P

c n + −

∫

T n

+

0 (, )

  

 

  

 

− − −

−

=

_∫

T

n n

dt u P t F u P C

M C

M

0 2 4

2

1 1 _{(, )}

) 2

1 (

1

2 2

1 ε

δ ε

δ

Pu_n 2+c Pu_n +c (23)

Since u n → ∞ as n→∞, by (19), we have _→∞

n

u

P as n→∞.

Let

ε

small enough, by (8), we get that φ(u_n)→−∞ as n→∞,this contradicts (12) and hence

{ }

u_{n is bounded in H}1

T .Arguing then as in Proposition 4.1 in [9], we conclude that the (PS) condition is satisfied.

Now we check the link condition that (a) inf

{

ψ(π(u))|π(u)∈Y×X

}

>−∞.

(b)

ψ

(

π

(

x

))

→

−∞

uniformly for π_(Qx) as Px→∞,where x∈RN.

For π(u)∈Y×V,

u

=

u

~

+

Q

u

. In a similar way to (22), we can get that

∫

T

+

−

∫

T

dt

t

F

dt

u

Q

t

u

t

F

0

(

)

)

0

(

,

0

)

~

,

(

c

u

C

T

M

u

C

M

+

+

≤

~

2

~

1

2 2 2

1 , （24）

It follows from (24), A <2_Tπ

and the boundedness of Qun that

)

~

(

))

~

(

(

))

(

(

π

u

=

ψ

π

u

+

Q

u

=

ϕ

u

+

Q

u

ψ

Tu dt T(Au(t),u(t))dt

2 1 2

1

0 2 0

ɺ

ɺ

∫

∫

+

=

∫

∫

∫

_−



 _{+ −}

− T T T

dt t F dt

t F dt u Q t u t F

0

0 ( ) ) 0 ( ,0) ( ,0)

2 2 1 2 2

2

~ ~

) 2 1 ( 2 , 2 1 min 2 1

u C M u A T

T  − −



   ≥

π

π _{− + −}

_∫

T

dt t F c u C T M

0

2 ~ (,0)

2 1

By 2<δ 1

2MC , we have ψ(π(x))→+∞, as u →∞, for all π(u)∈Y×V,which

implies (a).

It follows from (7) and the boundedness of Q u n that

)

ˆ

(

)

(

))

(

(

π

x

=

ϕ

x

=

ϕ

x

ψ

=−

∫

₀TF(t,Px+Qx)dt

∫ ∫

∇

+

−

∫

−

=

T T

dt

Px

t

F

dsdt

Qx

sQx

Px

t

F

0

1

0

(

(

,

),

)

0

(

,

)

dt

Qx

M

dt

Qx

Qx

Px

M

dt

Px

t

F

T T

T

∫

∫

∫

+

+

+

−

≤

0

(

,

)

0 1 ₀ 2

c Px c dt Px t F

T

+ + −

≤

∫

0 ( , )

   

   

+ + −

=

∫

T

Px c Px

c dt Px t F Px

Px 2 ₀ 2

2

) , (

1 ，

for all N

R

x∈ .Hence we have, by (8),(b) is satisfied. It follows from the generalized saddle point theorem (Theorem 1.7 in [4]) that has at least

r

+

1

critical points .Hence ϕ _{has at least} r+1 _{geometrically distinct critical points. Therefore, problem (5) has at} least r+1 _{geometrically distinct solutions in} 1

T

H .

Proof of Theorem 1.2 the proof relies on Theorem 4.12 in [8]. In a similar way to (22), we can get that

_∫

T ₋

_∫

T

dt u P t F dt u t F

0 (,ˆ) 0 (, )

≤M C Pu u~ +c Pu +M C2 u~ 2 +cu~ +c

1 2

1 (25)

It follows from (14) and (25) that ϕ(u)=ϕ(uˆ)

_≥δ ₋

[

_∫

T ₋

_∫

T

]

₋

_∫

T

dt u P t F dt u P t F dt u t F

u~2 ₀ ( ,ˆ) ₀ ( , ) ₀ ( , ) 2

_{MC )u}~ _{MC Pu u}~ 2

( 1 2

2 2

1 −

−

≥ δ +cPu +cu +c−

∫

TF t Pu

0 ( , )

~

2 2

1 2

1

) 2 ( 2 ~ ) 2 (

2

1 

 

 

  

 

− − −

= Pu

C M C M u C

M _δ

δ +c Pu +cu~ +c

2

0 2 2

1 2

4 2

1 1 _{(, )}

) (

4 MC _Pu F t Pu dt Pu

C

M T

   

   

+ −

− _δ

_∫

74

c u c u P c u P C M u

C M

C M

+ + +

 

 

 

 

− −

δ =

−

δ ~

) (

2 ~ ) 2

(

2 2

1 2

1

2 1

2

2

0 2 4

2

1 1 _{( , )}

) (

4 2

1

2

u P dt u P t F u P C

M T

C

M 



 

 

 

+

−

∫

−

δ (26)

By 2<δ 1

2MC and (9), we get that

ϕ

is bounded from below. Moreover, the functional

ϕ

satisfies the

G

PS)

( condition; that is, for every sequence

(

u

_n

)

in HT1 such that ϕ(u_n)→0 is bounded and ϕ(un)→0,the sequence

π

(

u

n

)

has a convergent subsequence (see Definition 4.2 in [8]). In fact, the boundedness of

ϕ

(

u

_n

)

, (9) and (26) imply that (u~n) and (Pu are bounded. Hence n) (uˆn) bounded. As in the proof of

Proposition 4.1 in [8], _{(ˆ )}

n

u has a convergent subsequence, so we have π(un)=π(uˆn). Now the Theorem in this case follows from Theorem 4.12 in [8].

Acknowledgments

This work is supported by the National Natural Scientific Fund of China(11161041).

REFERENCES

1. M.Willem, Oscillations forces de systemes hamiltoniens, Public Semin.Analyse nonlineaire Univ. Besancon, 1981.

2. P.H.Rabinowitz, On a class of functionals invariant under a Zn action, Trans. Amer.Math. Soc, 310(1) (1988) 303-311.

3. K.C. Chang, on the periodic nonlinearity and the multiplicity of solutions, Nonlinear Analysis TMA, 13(5) (1989) 527-537.

4. J.Q.Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989) 372-385.

5. X.P.Wu, Periodic solutions for nonautonomous second-order systems with bounded nonlinearity, J. Math. Anal, Appl., 230(1) (1999) 135-141.

6. C.L.Tang, A note on periodic solutions of nonautonomous second order systems, Porc. Amer. Math. Soc., 132 (5) (2003) 1295-1393.

7. X.Y.Zhang and X.H.Tang, Periodic solutions for an ordinary p-Laplacian system, Taiwanese Journal of Mathematic, 3 (2011) 1369-1396.

8. J.Mawhin and M.Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, New York 1989.

9. F.J.Meng and J.Yang, Periodic Solutions for a Class of Non-autonomous Second Order Systems, International Journal of Nonlinear Science, 10(3) (2010) 342-348. 10. Z.Y.Wang and J.H.Zhang, Periodic solutions of nonautomous second orderr systems

11. J.X.Feng and Z.Q.Han, Periodic solutions to differential systems with unbounded or periodic nonlinearities, J. Math. Anal. Appl, 323 (2006) 1264-1278.

12. Y.W.Ye and C.L.Tang, Periodic solutions for some nonautonomous second order Hamiltonian systems, J. Math. Anal .Appl., 344(1) (2008) 462-471.

13. Q.Meng and X.H.Tang, Solutions of a second order Hamiltonian systems, Nonlinear Anal, 9(2010) 1053-1067.

14. Y.Liu and H.b.Chen, Existence and multiplicity of periodic solutions generated by impulses, Abstract and Applied Analysis, 2 (2011) 1-15.

15. X. H. Tang and X.Y.Zhang, Periodic solutions for second-order Hamiltonian systems with a p-Laplacian, Annales Universitatis Mariae, 1 (2010) 93-113.

16. P.Zhang and C.Tang, Infinitely many periodic solutions for nonautonomous sublinear second order Hamiltonian systems, Abstract and Applied Analysis, 12 (2010) 1-10. 17. Y.H.Su and Z.S.Feng, A nonautonomous Hamiltonia n system on time scales,