AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/250870 250870 Research Article Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem Bao Longsheng Dai Binxiang Gasinski Leszek School of Mathematics and Statistics Central South University Changsha Hunan 410075 China csu.edu.cn 2014 1072014 2014 19 04 2014 29 06 2014 10 7 2014 2014 Copyright © 2014 Longsheng Bao and Binxiang Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A class of second order impulsive Hamiltonian systems are considered. By applying a local linking theorem, we establish the new criterion to guarantee that this impulsive Hamiltonian system has at least one nontrivial T-periodic solution under local superquadratic condition. This result generalizes and improves some existing results in the known literature.

1. Introduction and Main Results

Consider the second order Hamiltonian systems with impulsive effects (1) u ¨ ( t ) - A u ( t ) + F ( t , u ( t ) ) = 0 , a . e .    t [ 0 , T ] , Δ ( u ˙ i ( t j ) ) = I i j ( u i ( t j ) ) , i = 1,2 , , N , j = 1,2 , , l , u ( 0 ) - u ( T ) = u ˙ ( 0 ) - u ˙ ( T ) = 0 , where u ( t ) = ( u 1 ( t ) , u 2 ( t ) , , u N ( t ) ) , t 0 = 0 < t 1 < t 2 < < t l < t l + 1 = T , T > 0 , Δ ( u ˙ i ( t j ) ) = u ˙ i ( t j + ) - u ˙ i ( t j - ) , where u ˙ i ( t j + ) and u ˙ i ( t j - ) denote the right and left limits of u ˙ i ( t ) at t = t j , respectively, I i j : R R    ( i = 1,2 , , N , j = 1,2 , , l ) are continuous, and F C 1 ( [ 0 , T ] × R N , R ) , F ( t , u ) = F ( t , u ) / u . A = [ a i j ] is a symmetric constant matrix.

Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The theory of impulsive differential systems has been developed by numerous mathematicians (see ). These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see ).

In recent years, many existence results are obtained for impulsive differential systems by critical point theory, such as  and their references. In most superquadratic cases, there is so-called Ambrosetti-Rabinowitz condition (see ): (2) 0 < μ F ( t , u ) ( F ( t , u ) , u ) , t R , u R N { 0 } , where μ > 2 is a constant, which implies that F ( t , u ) is of superquadratic growth as | u | ; that is, (3) lim | u | F ( t , u ) | u | 2 = + , uniformly    for    all    t .

Moreover, Wu and Zhang  study the homoclinic solutions without any periodicity assumption under the local Ambrosetti-Rabinowitz type condition. Two key conditions of the main results of  are listed as follows.

There exist μ > 2 and L 1 > 0 such that (4) μ F ( t , u ) ( F ( t , u ) , u ) , t R , | u | L 1 .

There exists 2 < α < + such that liminf | u | + ( F ( t , u ) / | u | α ) > 0 , uniformly in t R .

In recent paper , Zhang and Tang had obtained some results of the nontrivial T-periodic solutions under much weaker assumptions instead of (A1) and (A2).

There exist constants μ > 2 , 0 < β 0 < 2 , and L 2 > 0 and a function a 0 ( t ) L 1 ( 0 , T ; R + ) such that (5) μ F ( t , u ) ( F ( t , u ) , u ) + a 0 ( t ) | u | β 0 , | u | L 2 , a . e .    t [ 0 , T ] .

There exists a subset E 0 of [ 0 , T ] with meas ( E 0 ) > 0 such that (6) liminf | u | F ( t , u ) | u | 2 > 0 , a . e .    t E 0 .

Remark 1.

Condition (B2) is weaker than (A2) because condition (A2) implies lim | u | ( F ( t , u ) / | u | 2 ) = + .

Recently, applying the local linking theorem (see ), the works in  obtained the existence of periodic solutions or homoclinic solutions with (3) superquadratic condition under different systems. As shown in , condition (B2) is a local superquadratic condition; this situation has been considered only by a few authors.

Motivated by papers [24, 25, 31], in this paper, we aim to consider problem (1) under local superquadratic condition via a version of the local linking theorem (see ). In particular, the impulsive function I i j satisfies a kind of new superquadratic condition which is different from that in the known literature. Our main results are the following theorems.

Theorem 2.

Suppose that F C 1 ( [ 0 , T ] × R N , R ) and I i j C ( R , R ) ,   i = 1,2 , , N , j = 1,2 , , l , satisfies (B2) and consider the following.

There exists a positive constant θ such that (7) ( A u , u ) θ | u | 2 , u R N .

limsup | u | 0 ( | F ( t , u ) | / | u | ) = 0 uniformly for t [ 0 , T ] .

There exist constants d > 1 , c 1 > 0 , and L 3 > 0 such that, for every t [ 0 , T ] and u R N with | u | L 3 , (8) | F ( t , u ) | c 1 ( | u | d + 1 ) .

There exist constants μ > 2 , L 4 > 0 , and b 1 ( 0 , ( μ / 2 - 1 ) θ ) such that, for every t [ 0 , T ] and u R N with | u | L 4 , (9) μ F ( t , u )    ( F ( t , u ) , u ) + b 1 | u | 2 .

There exist constants b i j > 0 and r i j ( 1 , + ) such that (10) | I i j ( u ) | b i j | u | r i j , u R .

There are two constants b 2 > 0 and γ [ 0,2 ) such that (11) I i j ( u ) u μ 0 u I i j ( s ) d s + b 2 | u | γ , u R .

I i j satisfies 0 u I i j ( s ) d s 0 , for all u R .

Then problem (1) has at least one nontrivial T-periodic solution.

Remark 3.

Noting (3), obviously, conditions (B2) and (H4) are weaker than those of (2). From (B2), we only need lim | u | ( F ( t , u ) / | u | 2 ) > 0 to hold in a subset E 0 of [ 0 , T ] . What is more, F in (2) is asked to be positive globally. Here F need not be nonnegative globally; we also generalized Theorems 1.3 and 1.4 in . For example, let (12) F ( t , u ) = 1 8 g ( t ) | u | 4 - 1 , ( t , u ) [ 0 , T ] × R N , where (13) g ( t ) = { sin 2 π t T , t [ 0 , T 2 ] , 0 , t [ T 2 , T ] . Let E 0 = [ T / 8 , T / 4 ] ; then F satisfies our Theorem 2 but does not satisfy (2) and (3) and does not satisfy the corresponding conditions in .

Theorem 4.

Suppose that F C 1 ( [ 0 , T ] × R N , R ) and I i j C ( R , R ) ,    i = 1,2 , , N , j = 1,2 , , l , satisfies (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.

There exist constants μ > 2 and b 3 ( 0 , min { m 1 ( μ - 2 ) , ( μ / 2 - 1 ) θ } ) such that (14) μ F ( t , u ) ( F ( t , u ) , u ) + b 3 | u | 2 , ( t , u ) [ 0 , T ] × R N ,

where (15) m 1 = min { F ( t , u ) t [ 0 , T ] , u R N , | u | = 1 } .

Then problem (1) has at least one nontrivial T-periodic solution.

Theorem 5.

Suppose that F C 1 ( [ 0 , T ] × R N , R ) and I i j C ( R , R ) ,    i = 1,2 , , N , j = 1,2 , , l , satisfies (B2), (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.

There exist constants μ > 2 , β [ 0,2 ) , and L 5 > 0 and a function a ( t ) L 1 ( 0 , T ) and a ( t ) 0 such that, for every t [ 0 , T ] and u R N with | u | L 5 , (16) μ F ( t , u ) ( F ( t , u ) , u ) + a ( t ) | u | β .

Then problem (1) has at least one nontrivial T-periodic solution.

The remaining of this paper is organized as follows. In Section 2, some fundamental facts are given. In Section 3, the main results of this paper are presented.

2. Preliminaries

Let X be a real Banach space with direct sum decomposition X = X 1 X 2 . Consider two sequences of subspaces X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 such that X j = n N X n j ¯ , j = 1,2 . For every multi-index α = ( α 1 , α 2 ) N 2 , let   X α = X α 1 X α 2 ; we define α β α 1 β 1 , α 2 β 2 . A sequence { α n } N 2 is admissible if for every α N 2 there is m N such that n m α n α . For every φ : X R , we define by φ α the function φ restricted to X α .

Definition 6 (see [<xref ref-type="bibr" rid="B25">26</xref>]).

Let φ C 1 ( X , R ) . The function φ satisfies the ( P S ) * condition if every sequence { u α n } , such that { α n } is admissible and (17) u α n X α n , sup n φ ( u α n ) < , φ α n ( u α n ) 0 , as n , possesses a subsequence which converges to a critical point of φ .

Definition 7 (see [<xref ref-type="bibr" rid="B25">26</xref>]).

Let X be a Banach space with direct sum decomposition X = X 1 X 2 . The function φ C 1 ( X , R ) has local linking at 0 with respect to ( X 1 , X 2 ) , if there exists r > 0 such that (18) φ ( u ) 0 , u X 1 with u r , φ ( u ) 0 , u X 2 with u r .

Theorem 8 (see [<xref ref-type="bibr" rid="B25">26</xref>]).

Suppose that φ C 1 ( X , R ) satisfies the following assumptions:

φ has local linking at 0 and X 1 { 0 } ,

φ satisfies ( P S ) * condition,

φ maps bounded sets into bounded sets,

for every m N and u X m 1 X 2 , φ ( u ) - as    u .

Then φ has at least three critical points.

Let us recall some basic notation. In the Sobolev space X : = H 0 1 ( 0 , T ) , consider the inner product (19) ( u , v ) = 0 T ( u ( t ) , v ( t ) ) d t + 0 T ( u ˙ ( t ) , v ˙ ( t ) ) d t , for any u , v X . The corresponding norm is defined by (20) u = ( 0 T | u ( t ) | 2 d t + 0 T | u ˙ ( t ) | 2 d t ) 1 / 2 , for any u X . Moreover, it is well known that X is compactly embedded in C [ 0 , T ] , which implies that (21) u C u , for some constant C > 0 , where u = max t [ 0 , T ] | u ( t ) | .

Define the corresponding functional φ on X by (22) φ ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 1 2 0 T ( A u ( t ) , u ( t ) ) d t + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ( t ) ) d t .

By the conditions of F and I i j , i = 1,2 , , N , j = 1,2 , , l , we get that functional φ is a continuously Gáteaux differential functional whose Gáteaux derivative is the functional φ ( u ) , given by (23) ( φ ( u ) , v ) = 0 T ( u ˙ ( t ) , v ˙ ( t ) ) d t + 0 T ( A u ( t ) , v ( t ) ) d t + j = 1 l i = 1 N I i j ( u i ( t j ) ) v i ( t j ) - 0 T ( F ( t , u ( t ) ) , v ( t ) ) d t .

If u H 0 1 ( 0 , T ) , then u is absolutely continuous and u ˙ L 2 ( 0 , T ) . In this case, Δ u ˙ ( t ) = u ˙ ( t + ) - u ˙ ( t - ) = 0 may not hold for some t ( 0 , T ) ; this leads to impulsive effects.

Following the ideas of [11, 12], we can prove that the critical points of φ are the weak solutions of problem (1).

To prove our main results, we have the following facts (see ).

Letting (24) ϕ ( u ) = 0 T 1 2 [ | u ˙ ( t ) | 2 + ( A u ( t ) , u ( t ) ) ] d t , we see that (25) ϕ ( u ) = 1 2 u 2 - 1 2 0 T ( ( I - A ) u ( t ) , u ( t ) ) d t = 1 2 ( ( I - K ) u , u ) , where K : H T 1 H T 1 is the liner self-adjoint operator defined and I is the N × N identity matrix. By the Riesz representation theorem, we have (26) 0 T ( ( I - A ) u ( t ) , v ( t ) ) d t = ( K u , v ) .

The compact imbedding of H T 1 into C ( [ 0 , T ] , R N ) implies that K is compact. Summing up the above discussion, φ ( u ) can be rewritten as (27) φ ( u ) = 1 2 ( ( I - K ) u , u ) + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s + 0 T F ( t , u ( t ) ) d t .

By classical spectral theory, we can decompose X into the orthogonal sum of invariant subspace for I - K (28) X = H - H 0 H + , where H 0 = ker ( I - K ) and H - and H + are such that, for some δ > 0 , (29) ( ( I - K ) u , u ) - δ u 2 , if u H - , ( ( I - K ) u , u ) δ u 2 , if u H + . Notice that H - is finite dimensional.

In this paper, we set a ¯ : = max 1 i , j N | a i j | .

3. Proof of Main Results 3.1. The Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>

Let X 1 = H + and X 2 = H 0 H - ; then X = X 1 X 2 . Suppose { e n } n 1 is an orthogonal basis of H + . Correspondingly, let (30) X n 1 = span { e 1 , e 2 , , e n } , X n 2 = X 2 , n N ; then X j = n N X n j ¯ , j = 1,2 . We divide our proof into four steps.

Step 1.

φ has local linking at 0 .

In view of (I1), we obtain (31) | 0 u I i j ( s ) d s | b i j r i j + 1 | u | r i j + 1 . Combining this inequality, we have (32) | 0 u I i j ( s ) d s | | u | 2 b i j r i j + 1 | u | r i j - 1 0 , as u 0 . Since r i j > 1 , this implies (33) | 0 u I i j ( s ) d s | | u | 2 0 , as u 0 . Applying (H2) and (33), for any ɛ > 0 , there exists r 1 > 0 such that (34) | F ( t , u ) | 2 ɛ | u | , | 0 u I i j ( s ) d s | ɛ | u | 2 , | u | r 1 , t [ 0 , T ] , which implies that (35) | F ( t , u ) | = | 0 1 ( F ( t , s u ) , u ) d s | 0 1 | F ( t , s u ) | | u | d s 0 1 2 s ɛ | u | 2 d s = ɛ | u | 2 .

On one hand, by (34) and (35), for all u X 1 = H + with u r 2 : = r 1 / C . Choose ɛ = δ / 4 ( l N + T ) C 2 ; then one has (36) φ ( u ) = 1 2 ( ( I - K ) u , u ) + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ( t ) ) d t δ 2 u 2 - j = 1 l i = 1 N ( ɛ | u i ( t j ) | 2 ) - ɛ 0 T | u | 2 d t δ 2 u 2 - l N C 2 ɛ u 2 - T C 2 ɛ u 2 = δ 4 u 2 0 .

On the other hand, since dim X 2 is finite, there exists a constant K 1 > 0 such that (37) u K 1 u - , u X 2 . For all u X 2 = H - H 0 with u r 3 : = r 1 / C . Choose ɛ = δ / 4 ( l N + T ) C 2 K 1 2 ; by (34), (35), and (37), we obtain (38) φ ( u ) = 1 2 ( ( I - K ) u - , u - ) + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ( t ) ) d t - δ 2 u - 2 + j = 1 l i = 1 N ( ɛ | u i ( t j ) | 2 ) + ɛ 0 T | u | 2 d t - δ 2 u - 2 + ( l N + T ) C 2 ɛ u 2 - δ 2 u - 2 + ( l N + T ) C 2 K 1 2 ɛ u - 2 = - δ 4 u - 2 0 . Let r = min { r 2 , r 3 } ; one has (39) φ ( u ) 0 , u X 1 , u r , φ ( u ) 0 , u X 2 , u r .

Step 2.

φ maps bounded sets into bounded sets.

By (H3) and F C 1 ( [ 0 , T ] × R N , R ) , there exists c 2 > 0 such that (40) | F ( t , u ) | c 2 + c 1 | u | d , ( t , u ) [ 0 , T ] × R N , which implies that (41) | F ( t , u ) | 0 1 | F ( t , s u ) | | u | d s 0 1 ( c 2 | u | + c 1 | u | d + 1 ) d s = c 2 | u | + c 1 | u | d + 1 , ( t , u ) [ 0 , T ] × R N . Note that (42) | φ ( u ) | = | j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s 1 2 0 T | u ˙ ( t ) | 2 d t + 1 2 0 T ( A u ( t ) , u ( t ) ) d t + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ( t ) ) d t | 1 2 0 T | u ˙ ( t ) | 2 d t + 1 2 N a ¯ 0 T | u ( t ) | 2 d t + j = 1 l i = 1 N ( b i j r i j + 1 | u i ( t j ) | r i j + 1 ) + c 2 0 T | u | d t + c 1 0 T | u | d + 1 d t ( 1 2 + 1 2 N a ¯ ) u 2 + j = 1 l i = 1 N ( b i j r i j + 1 C r i j + 1 u r i j + 1 ) + c 2 C T u + c 1 C d 1 + 1 T u d + 1 . It implies that φ maps bounded sets into bounded sets.

Step 3.

φ satisfies the ( P S ) * condition.

Consider a ( P S ) * sequence { u α n } such that { α n } is admissible. Then there exists a constant M 1 > 0 such that (43) | φ ( u α n ) | M 1 , φ α n ( u α n ) M 1 . By (41), for | u | L 4 , one has (44) | F ( t , u ) | c 2 | u | + c 1 | u | d + 1 c 2 L 4 + c 1 L 4 d + 1 ; together with (H4), one has (45) μ F ( t , u ) ( F ( t , u ) , u ) + b 1 | u | 2 + μ ( c 2 L 4 + c 1 L 4 d + 1 ) , for all ( t , u ) [ 0 , T ] × R N .

It follows from (21), (43), (45), (H1), and (I2) that (46) M 1 + M 1 μ u α n φ ( u α n ) - 1 μ ( φ ( u α n ) , u α n ) = ( 1 2 - 1 μ ) 0 T | u ˙ α n | 2 d t + ( 1 2 - 1 μ ) 0 T ( A u α n , u α n ) d t + 1 μ 0 T ( ( F ( t , u α n ) , u α n ) - μ F ( t , u α n ) ) d t + 1 μ j = 1 l i = 1 N ( μ 0 u α n i ( t j ) I i j ( s ) d s - I i j ( u α n i ( t j ) ) u α n i ( t j ) ) ( 1 2 - 1 μ ) 0 T | u ˙ α n | 2 d t + ( θ 2 - θ μ - b 1 μ ) 0 T | u α n | 2 d t - μ T ( c 2 L 4 + c 1 L 4 d + 1 ) - b 2 μ j = 1 l i = 1 N u α n γ min { ( 1 2 - 1 μ ) , ( 1 2 - 1 μ ) θ - b 1 μ } u α n 2 - b 2 l N μ C γ u α n γ - μ T ( c 2 L 4 + c 1 L 4 d + 1 ) .

Since μ > 2 , γ [ 0,2 ) , and b 1 ( 0 , ( μ / 2 - 1 ) θ ) , (46) shows that { u α n } is bounded in H T 1 . By a method similar to that of , we can prove that { u α n } has a convergent subsequence. Thus φ satisfies ( P S ) * condition.

Step 4.

For every m N and u X m 1 X 2 , φ ( u ) - as u .

Since dim ( X m 1 X 2 ) is finite, there exists M 2 > 0 such that (47) u M 2 ( E 0 | u | 2 d t ) 1 / 2 , u X m 1 X 2 . In fact, for M 3 = M 2 2 ( max { 1 / 2 , ( 1 / 2 ) N a ¯ } + δ / 2 ) > 0 , by (B2), there exists M 4 > 0 such that (48) F ( t , u ) M 3 | u | 2 - M 4 , u R N , a . e .    t E 0 . Hence, it follows from (47), (48), and (I3) that (49) φ ( u ) = 1 2 ( ( I - K ) u , u ) + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ) d t = 1 2 ( ( I - K ) u - , u - ) + 1 2 ( ( I - K ) u + , u + ) + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ) d t - δ 2 u - 2 + 1 2 0 T | u ˙ + | 2 d t + 1 2 0 T ( A u + , u + ) d t - E 0 F ( t , u ) d t - δ 2 u - 2 + max { 1 2 , 1 2 N a ¯ } u + 2 - M 3 E 0 | u | 2 d t + M 4 T - δ 2 u - 2 + max { 1 2 , 1 2 N a ¯ } u + 2 - ( max { 1 2 , 1 2 N a ¯ } + δ 2 ) u 2 + M 4 T - δ 2 u - 2 + max { 1 2 , 1 2 N a ¯ } u + 2 - ( max { 1 2 , 1 2 N a ¯ } + δ 2 ) ( u + 2 + u 0 2 ) + M 4 T - δ 2 u 2 + M 4 T , for u X m 1 X 2 and a.e. t E 0 , which implies that (50) φ ( u ) - , as    u    on    u X m 1 X 2 . Therefore, all the assumptions of Theorem 8 are verified. Then, the proof of Theorem 2 is completed.

3.2. The Proof of Theorem <xref ref-type="statement" rid="thm1.2">4</xref>

Following the same procedures in the proof of Theorem 2, we can prove that φ satisfies (A1), (A2), and (A3) in Theorem 8.

To prove (A4), set ϕ ( ξ ) = F ( t , ξ u ) ξ - μ + ( b 3 / ( μ - 2 ) ) ( 1 - ξ 2 - μ ) | u | 2 , ξ ( 0 , + ) ; then by (H5), we have (51) ϕ ( ξ ) = ( F ( t , ξ u ) , u ) ξ - μ - μ ξ - μ - 1 F ( t , ξ u ) + b 3 ξ 1 - μ | u | 2 = ( ( F ( t , ξ u ) , ξ u ) - μ F ( t , ξ u ) ) ξ - μ - 1 + b 3 ξ 1 - μ | u | 2 0 . When 0 < ξ 1 , it follows from (51) that (52) ϕ ( 1 ) = F ( t , u ) ϕ ( ξ ) = F ( t , ξ u ) ξ - μ + b 3 μ - 2 ( 1 - ξ 2 - μ ) | u | 2 ; this implies (53) F ( t , u ) ( F ( t , u | u | ) - b 3 μ - 2 ) | u | μ + b 3 μ - 2 | u | 2 , if | u | 1 . Set m 2 = max { F ( t , u ) t [ 0 , T ] , u R N , | u | 1 } ; then by (53), we have (54) F ( t , u ) ( m 1 - b 3 μ - 2 ) | u | μ + b 3 μ - 2 | u | 2 - m 2 , ( t , u ) [ 0 , T ] × R N ; since   dim ( X m 1 X 2 ) is finite, there exists M 5 > 0 such that (55) u μ M 5 0 T | u | μ d t , u X m 1 X 2 . By (54), (55), and (I3), we have (56) φ ( u ) = 1 2 0 T | u ˙ | 2 d t + 1 2 0 T ( A u , u ) d t + j = 1 l i = 1 N 0 u i ( t j ) I i j ( s ) d s - 0 T F ( t , u ) d t 1 2 0 T | u ˙ | 2 d t + 1 2 N a ¯ 0 T | u | 2 d t - ( m 1 - b 3 μ - 2 ) 0 T | u | μ d t + b 3 μ - 2 0 T | u | 2 d t - m 2 T max { 1 2 , 1 2 N a ¯ } u 2 - 1 M 5 ( m 1 - b 3 μ - 2 ) u μ + b 3 M 5 ( μ - 2 ) u 2 - m 2 T . Since μ > 2 and m 1 - ( b 3 / ( μ - 2 ) ) > 0 , (56) implies (57) φ ( u ) - , as    u    on    u X m 1 X 2 . Consequently, the conclusion follows from Theorem 8. This completes the proof.

3.3. The Proof of Theorem <xref ref-type="statement" rid="thm1.3">5</xref>

Similar to the proof of Theorem 2, φ satisfies all conditions of Theorem 8. Thus, problem (1) has at least one nontrivial T-periodic solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (no. 11271371).

Bainov D. D. Covachev V. C. Periodic solutions of impulsive systems with a small delay Journal of Physics A: Mathematical and General 1994 27 16 5551 5563 10.1088/0305-4470/27/16/020 MR1295379 2-s2.0-21844483310 Nieto J. J. Impulsive resonance periodic problems of first order Applied Mathematics Letters 2002 15 4 489 493 10.1016/S0893-9659(01)00163-X MR1902284 2-s2.0-31244432616 Benchohra M. Henderson J. Ntouyas S. Impulsive Differential Equations and Inclusions 2006 New York, NY, USA Hindawi Publishing Corporation 10.1155/9789775945501 MR2322133 Dai B. Li Y. Luo Z. Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses Applied Mathematics and Computation 2011 217 18 7478 7487 10.1016/j.amc.2011.02.049 MR2784594 ZBL1221.92077 2-s2.0-79953214540 Nenov S. I. Impulsive controllability and optimization problems in population dynamics Nonlinear Analysis: Theory, Methods & Applications 1999 36 7 881 890 10.1016/S0362-546X(99)00627-6 MR1682836 2-s2.0-0033148258 Carter T. E. Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion Dynamics and Control 2000 10 3 219 227 10.1023/A:1008376427023 MR1809484 2-s2.0-0034217074 Jiang G. Lu Q. Qian L. Complex dynamics of a Holling type II prey-predator system with state feedback control Chaos, Solitons & Fractals 2007 31 2 448 461 10.1016/j.chaos.2005.09.077 MR2259769 2-s2.0-33746005301 Luo Z. Nieto J. J. New results for the periodic boundary value problem for impulsive integro-differential equations Nonlinear Analysis 2009 70 6 2248 2260 10.1016/j.na.2008.03.004 MR2498308 2-s2.0-59049102714 Dai B. Su H. Hu D. Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse Nonlinear Analysis: Theory, Methods & Applications 2009 70 1 126 134 10.1016/j.na.2007.11.036 MR2468223 2-s2.0-55549130813 Bai L. Dai B. Three solutions for a p -Laplacian boundary value problem with impulsive effects Applied Mathematics and Computation 2011 217 24 9895 9904 10.1016/j.amc.2011.03.097 MR2806376 2-s2.0-79959235699 Ding W. Qian D. Periodic solutions for sublinear systems via variational approach Nonlinear Analysis: Real World Applications 2010 11 4 2603 2609 10.1016/j.nonrwa.2009.09.007 MR2661927 2-s2.0-77955770889 Zhang H. Li Z. Variational approach to impulsive differential equations with periodic boundary conditions Nonlinear Analysis: Real World Applications 2010 11 1 67 78 10.1016/j.nonrwa.2008.10.016 MR2570525 2-s2.0-70350706450 Nieto J. J. Variational formulation of a damped Dirichlet impulsive problem Applied Mathematics Letters 2010 23 8 940 942 10.1016/j.aml.2010.04.015 MR2651478 2-s2.0-77955518624 Zhou J. Li Y. Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2010 72 3-4 1594 1603 10.1016/j.na.2009.08.041 MR2577560 2-s2.0-71549115120 Sun J. Chen H. Nieto J. J. Otero-Novoa M. The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2010 72 12 4575 4586 10.1016/j.na.2010.02.034 MR2639205 2-s2.0-77950370388 Sun J. Chen H. Nieto J. J. Infinitely many solutions for second-order Hamiltonian system with impulsive effects Mathematical and Computer Modelling 2011 54 1-2 544 555 10.1016/j.mcm.2011.02.044 MR2801909 ZBL1225.37070 2-s2.0-79955482702 Zhang H. Li Z. Periodic and homoclinic solutions generated by impulses Nonlinear Analysis: Real World Applications 2011 12 1 39 51 10.1016/j.nonrwa.2010.05.034 MR2728662 2-s2.0-77958004466 Zhou J. Li Y. Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects Nonlinear Analysis: Theory, Methods & Applications 2009 71 7-8 2856 2865 10.1016/j.na.2009.01.140 MR2532812 2-s2.0-67349168447 Zhang Z. Yuan R. An application of variational methods to Dirichlet boundary value problem with impulses Nonlinear Analysis, Real World Applications 2010 11 1 155 162 10.1016/j.nonrwa.2008.10.044 MR2570535 2-s2.0-70350716916 Sun J. Chen H. Yang L. The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method Nonlinear Analysis: Theory, Methods & Applications 2010 73 2 440 449 10.1016/j.na.2010.03.035 MR2650827 2-s2.0-77955414425 Tian Y. Ge W. Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations Nonlinear Analysis. Theory, Methods & Applications 2010 72 1 277 287 10.1016/j.na.2009.06.051 MR2574937 2-s2.0-71749107090 Bai L. Dai B. Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory Mathematical and Computer Modelling 2011 53 9-10 1844 1855 10.1016/j.mcm.2011.01.006 MR2782888 ZBL1219.34039 2-s2.0-79951943819 Zhang D. Dai B. Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions Computers & Mathematics with Applications 2011 61 10 3153 3160 10.1016/j.camwa.2011.04.003 MR2799840 2-s2.0-79955868117 Wu X. Zhang W. Existence and multiplicity of homoclinic solutions for a class of damped vibration problems Nonlinear Analysis: Theory, Methods & Applications 2011 74 13 4392 4398 10.1016/j.na.2011.03.059 MR2810736 2-s2.0-79957570815 Zhang Q. Tang X. H. New existence of periodic solutions for second order non-autonomous Hamiltonian systems Journal of Mathematical Analysis and Applications 2010 369 1 357 367 10.1016/j.jmaa.2010.03.033 MR2643874 ZBL1201.37094 2-s2.0-77952581930 Li S. J. Willem M. Applications of local linking to critical point theory Journal of Mathematical Analysis and Applications 1995 189 1 6 32 10.1006/jmaa.1995.1002 MR1312028 ZBL0820.58012 2-s2.0-0003361352 Duan S. Wu X. The local linking theorem with an application to a class of second-order systems Nonlinear Analysis 2010 72 5 2488 2498 10.1016/j.na.2009.10.045 MR2577814 2-s2.0-72649097239 Gasiński L. Papageorgiou N. S. Nontrivial solutions for Neumann problems with an indefinite linear part Applied Mathematics and Computation 2010 217 6 2666 2675 10.1016/j.amc.2010.08.004 MR2733711 2-s2.0-77958007830 Li C. Ou Z. Tang C. Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems Nonlinear Analysis: Theory, Methods & Applications 2012 75 4 2262 2272 10.1016/j.na.2011.10.026 MR2870916 2-s2.0-84855202655 Wan L. Tang C. Existence of homoclin ic for second order Hamiltonian systems without (AR) condition Nonlinear Analysis: Theory, Methods & Applications 2011 74 16 5303 5313 10.1016/j.na.2011.05.011 MR2819275 2-s2.0-79959741709 Wang Z. Zhang J. Zhang Z. Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential Nonlinear Analysis: Theory, Methods & Applications 2009 70 10 3672 3681 10.1016/j.na.2008.07.023 MR2504454 2-s2.0-61749098533 Mawhin J. Willem M. Critical Point Theory and Hamiltonian Systems 1989 74 New York, NY. USA Springer 10.1007/978-1-4757-2061-7 MR982267