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Variational approach to a class of impulsive differential equations

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Abstract

In this article, the author discusses the existence of solutions for a class of impulsive differential equations by means of a variational approach different from earlier approaches.

MSC:34B37, 45G10, 47H30, 47J30.

1 Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years [13]. There is a vast literature on the existence of solutions by using topological methods, including fixed point theorems, Leray-Schauder degree theory, and fixed point index theory [415]. But it is quite difficult to apply the variational approach to an impulsive differential equation; therefore, there was no result in this area for a long time. Only in the recent five years, there appeared a few articles which dealt with some impulsive differential equations by using variational methods [1620]. Motivated by [17], in this article we shall use a different variational approach to discuss the existence of solutions for a class of impulsive differential equations and we only deal with classical solutions.

Consider the boundary value problem (BVP) for the second-order nonlinear impulsive differential equation:

{ u ( t ) = f ( t , u ( t ) ) , t J , Δ u | t = t k = c k ( k = 1 , 2 , 3 , , m ) , Δ u | t = t k = d k ( k = 1 , 2 , 3 , , m ) , u ( 0 ) = u ( 1 ) = 0 ,
(1)

where J=[0,1], 0< t 1 << t k << t m <1, J =J{ t 1 ,, t k ,, t m }, c k and d k (k=1,2,,m) are any real numbers, f(t,u) is a real function defined on J×R, where R denotes the set of all real numbers, and f(t,u) is continuous on J ×R, left continuous at t= t k , i.e.

lim t t k 0 , w u f(t,w)=f( t k ,u)

for any uR (k=1,2,,m), and the right limit at t= t k exists, i.e.

lim t t k + 0 , w u f(t,w)

(denoted by f( t k + ,u)) exists for any uR (k=1,2,,m). Δu | t = t k denotes the jump of u(t) at t= t k , i.e.

Δu | t = t k =u ( t k + ) u ( t k ) ,

where u( t k + ) and u( t k ) represent the right and left limits of u(t) at t= t k , respectively. Similarly,

Δ u | t = t k = u ( t k + ) u ( t k ) ,

where u ( t k + ) and u ( t k ) represent the right and left limits of u (t) at t= t k , respectively. Let PC[J,R] = {u:u is a real function on J such that u(t) is continuous at t t k , left continuous at t= t k , and u( t k + ) exists, k=1,2,,m} and P C 1 [J,R] = {uPC[J,R]: u (t) is continuous at t t k and u ( t k + ), u ( t k ) exist, k=1,2,,m}. A function uP C 1 [J,R] C 2 [ J ,R] is called a solution of BVP (1) if u(t) satisfies (1).

Let us list some conditions.

(H1) There exist p>2, a>0 and b>0 such that

|f(t,u)|a+b | u | p 1 ,tJ,uR.

(H2) There exist 0<c< π 2 4 and d>0 such that

0 u f(t,v)dvc u 2 +d,tJ,uR.

Lemma 1 uP C 1 [J,R] C 2 [ J ,R] is a solution of BVP (1) if and only if vC[J,R] is a solution of the integral equation

v(t)= 0 1 G(t,s)g ( s , v ( s ) ) ds,tJ,
(2)

where

G(t,s)= { s ( 1 t ) , 0 s t 1 ; t ( 1 s ) , 0 t < s 1 ,
(3)
g(t,v)=f ( t , v + a ( t ) a ( 1 ) t ) ,tJ,vR
(4)

and

v(t)=u(t)a(t)+a(1)t,a(t)= 0 < t k < t [ c k + ( t t k ) d k ] ,tJ.
(5)

Proof For uP C 1 [J,R] C 2 [ J ,R], we have the formula (see [21], Lemma 1(b))

u ( t ) = u ( 0 ) + t u ( 0 ) + 0 t ( t s ) u ( s ) d s + 0 < t k < t { [ u ( t k + ) u ( t k ) ] + ( t t k ) [ u ( t k + ) u ( t k ) ] } , t J .
(6)

So, if uP C 1 [J,R] C 2 [ J ,R] is a solution of BVP (1), then, by (1) and (6), we have

u ( t ) = t u ( 0 ) 0 t ( t s ) f ( s , u ( s ) ) d s + 0 < t k < t [ c k + ( t t k ) d k ] = t u ( 0 ) 0 t ( t s ) f ( s , u ( s ) ) d s + a ( t ) , t J .
(7)

It is clear, by (5), that

a(t)=0,0t t 1 ;a(1)= k = 1 m [ c k + ( 1 t k ) d k ] ,
(8)

so

v (0)= u (0)+a(1).
(9)

Substituting (9) into (7), we get

v ( t ) = t v ( 0 ) 0 t ( t s ) f ( s , u ( s ) ) d s = t v ( 0 ) 0 t ( t s ) f ( s , v ( s ) + a ( s ) a ( 1 ) s ) d s = t v ( 0 ) 0 t ( t s ) g ( s , v ( s ) ) d s , t J .
(10)

By virtue of (5), we see that vC[J,R] (in fact, v C 1 [J,R]) and

v(1)=u(1)a(1)+a(1)=u(1)=0,

so, letting t=1 in (10), we find

v (0)= 0 1 (1s)g ( s , v ( s ) ) ds.
(11)

Substituting (11) into (10), we get

v ( t ) = t 1 t ( 1 s ) g ( s , v ( s ) ) d s + 0 t s ( 1 t ) g ( s , v ( s ) ) d s = 0 1 G ( t , s ) g ( s , v ( s ) ) d s , t J ,

so v(t) is a solution of the integral equation (2).

Conversely, suppose that vC[J,R] is a solution of (2), i.e.

v(t)=(1t) 0 t sg ( s , v ( s ) ) ds+t t 1 (1s)g ( s , v ( s ) ) ds,tJ.
(12)

By (4), it is clear that g(t,v(t)) is continuous on J , so differentiation of (12) gives

v ( t ) = 0 t s g ( s , v ( s ) ) d s + ( 1 t ) t g ( t , v ( t ) ) + t 1 ( 1 s ) g ( s , v ( s ) ) d s t ( 1 t ) g ( t , v ( t ) ) = 0 t s g ( s , v ( s ) ) d s + t 1 ( 1 s ) g ( s , v ( s ) ) d s , t J .
(13)

Differentiating again, we get

v (t)=tg ( t , v ( t ) ) (1t)g ( t , v ( t ) ) =g ( t , v ( t ) ) ,t J .
(14)

From (13) we see that v ( t k + ) and v ( t k ) (k=1,2,,m) exist and

v ( t k + ) = v ( t k ) = 0 t k sg ( s , v ( s ) ) ds+ t k 1 (1s)g ( s , v ( s ) ) ds.
(15)

It follows from (4), (5), (12), (14), and (15) that uP C 1 [J,R] C 2 [ J ,R] and u(t) satisfies (1). □

Lemma 2 Let condition (H1) be satisfied. If v L p [J,R] is a solution of the integral equation (2), then vC[J,R].

Proof It is clear, for function a(t) defined by (5),

|a(t)| a 0 ,tJ; a 0 = k = 1 m ( | c k | + ( 1 t k ) | d k | ) .
(16)

By (4), (5), (16), and condition (H1), we have

| g ( t , v ) | a + b | v + a ( t ) a ( 1 ) t | p 1 a + b ( | v | + 2 a 0 ) p 1 a + b ( 2 max { | v | , 2 a 0 } ) p 1 a + b 2 p 1 ( | v | p 1 + ( 2 a 0 ) p 1 ) , t J , v R ,

so,

|g(t,v)| a 1 + b 1 | v | p 1 ,tJ,vR,
(17)

where

a 1 =a+b 2 2 ( p 1 ) a 0 p 1 , b 1 =b 2 p 1 .

It is clear that g(t,v) satisfies the Caratheodory condition, i.e. g(t,v) is measurable with respect to t on J for every vR and is continuous with respect to v on R for almost tJ (in fact, g(t,v) is discontinuous only at t= t k (k=1,2,,m)), so (17) implies [22, 23] that the operator g defined by

(gv)(t)=g ( t , v ( t ) ) ,tJ
(18)

is bounded and continuous from L p [J,R] into L q [J,R], where 1 p + 1 q =1 (q>1).

Let v L p [J,R] be a solution of the integral equation (2). Then by the Hölder inequality,

|v( t 1 )v( t 2 )| ( 0 1 | G ( t 1 , s ) G ( t 2 , s ) | p d s ) 1 p ( 0 1 | g ( s , v ( s ) ) | q d s ) 1 q , t 1 , t 2 J,

which implies by virtue of the uniform continuity of G(t,s) on J×J that vC[J,R]. □

2 Variational approach

Theorem 1 If conditions (H1) and (H2) are satisfied, then BVP (1) has at least one solution uP C 1 [J,R] C 2 [ J ,R].

Proof By Lemma 1 and Lemma 2, we need only to show that the integral equation (2) has a solution v L p [J,R]. The integral equation (2) can be written in the form

v=Ggv,
(19)

where G is the linear integral operator defined by

(Gv)(t)= 0 1 G(t,s)v(s)ds,tJ,
(20)

and the nonlinear operator g is defined by (18), which is bounded and continuous from L p [J,R] into L q [J,R] ( 1 p + 1 q =1). It is well known that G(t,s) is a L 2 positive-definite kernel with eigenvalues { 1 n 2 π 2 } (n=1,2,3,) and, by the continuity of G(t,s), we have

0 1 0 1 [ G ( t , s ) ] p dsdt<,
(21)

so [22, 23] the linear operator G defined by (20) is completely continuous from L 2 [J,R] into L 2 [J,R] and also from L q [J,R] into L p [J,R], and G=H H , where H= G 1 2 (the positive square-root operator of G) is completely continuous from L 2 [J,R] into L p [J,R] and H denotes the adjoint operator of H, which is completely continuous from L q [J,R] into L 2 [J,R]. We now show that (19) has a solution v L p [J,R] is equivalent to the equation

u= H gHu
(22)

has a solution u L 2 [J,R]. In fact, if v L p [J,R] is a solution of (19), i.e. v=H H gv, then H gv= H gH H gv, so, u= H gv L 2 [J,R] and u is a solution of (22). Conversely, if u L 2 [J,R] is a solution of (22), then Hu=H H gHu=GgHu, so, v=Hu L p [J,R] and v is a solution of (19). Consequently, we need only to show that (22) has a solution u L 2 [J,R]. It is well known [22, 23] that the functional Φ defined by

Φ(u)= 1 2 (u,u) 0 1 dt 0 ( H u ) ( t ) g(t,v)dv,u L 2 [J,R]
(23)

is a C 1 functional on L 2 [J,R] and its Fréchet derivative is

Φ (u)=u H gHu,u L 2 [J,R].
(24)

Hence we need only to show that there exists a u L 2 [J,R] such that Φ (u)=θ (θ denotes the zero element of L 2 [J,R]), i.e. u is a critical point of functional Φ.

By (4), (5), (16), and condition (H1), we have

0 u g(t,v)dv= 0 u + a ( t ) a ( 1 ) t f(t,w)dw 0 a ( t ) a ( 1 ) t f(t,w)dw,tJ,uR
(25)

and

| 0 a ( t ) a ( 1 ) t f ( t , w ) d w | | a ( t ) a ( 1 ) t | ( a + b | a ( t ) a ( 1 ) t | p 1 ) 2 a 0 ( a + b 2 p 1 a 0 p 1 ) = a 2 , t J .
(26)

So, (25), (26), and condition (H2) imply

0 ( H u ) ( t ) g ( t , v ) d v 0 ( H u ) ( t ) + a ( t ) a ( 1 ) t f ( t , w ) d w + a 2 c { ( H u ) ( t ) + a ( t ) a ( 1 ) t } 2 + d + a 2 2 c { [ ( H u ) ( t ) ] 2 + [ a ( t ) a ( 1 ) t ] 2 } + d + a 2 2 c [ ( H u ) ( t ) ] 2 + 8 c a 0 2 + d + a 2 , u L 2 [ J , R ] , t J .
(27)

It is well known [24],

G= λ 1 = 1 π 2 ,
(28)

where G is defined by (20) and is regarded as a positive-definite operator from L 2 [J,R] into L 2 [J,R], and λ 1 denotes the largest eigenvalue of G. It follows from (23), (27), and (28) that

Φ ( u ) 1 2 ( u , u ) 2 c ( H u , H u ) 8 c a 0 2 d a 2 = 1 2 ( u , u ) 2 c ( G u , u ) 8 c a 0 2 d a 2 1 2 ( u , u ) 2 c π 2 ( u , u ) 8 c a 0 2 d a 2 = ( 1 2 2 c π 2 ) u 2 8 c a 0 2 d a 2 , u L 2 [ J , R ] ,
(29)

which implies by virtue of 0<c< π 2 4 (see condition (H2)) that

lim u Φ(u)=.
(30)

So, there exists a r>0 such that

Φ(u)>Φ(θ)=0,u L 2 [J,R],u>r.
(31)

It is well known [22, 23] that the ball T(θ,r)={u L 2 [J,R]:ur} is weakly closed and weakly compact and the functional Φ(u) is weakly lower semicontinuous, so, there exists u T(θ,r) such that

Φ ( u ) = inf u T ( θ , r ) Φ(u)Φ(θ).
(32)

It follows from (31) and (32) that

Φ ( u ) = inf u L 2 [ J , R ] Φ(u).

Hence Φ ( u )=θ and the theorem is proved. □

Example 1 Consider the BVP

{ u ( t ) = 9 2 u ( t ) sin ( t u ( t ) ) t 3 , t J , Δ u | t = t k = c k ( k = 1 , 2 , , m ) , Δ u | t = t k = d k ( k = 1 , 2 , , m ) , u ( 0 ) = u ( 1 ) = 0 ,
(33)

where J=[0,1], 0< t 1 << t k << t m <1, J =J{ t 1 ,, t k ,, t m }, c k and d k (k=1,2,,m) are any real numbers.

Conclusion BVP (33) has at least one solution uP C 1 [J,R] C 2 [ J ,R].

Proof Evidently, (33) is a BVP of the form (1) with

f(t,u)= 9 2 usin(tu) t 3 .
(34)

It is clear that fC[J×R,R]. By (34), we have

|f(t,u)| 9 2 |u|+1,tJ,uR.
(35)

Moreover, it is well known that

|u| 1 2 ( 1 + u 2 ) ,uR.
(36)

So, (35) and (36) imply that

|f(t,u)| 9 4 u 2 + 13 4 ,tJ,uR,

and consequently, condition (H1) is satisfied for p=3, a= 13 4 and b= 9 4 . On the other hand, choose ϵ 0 such that

0< ϵ 0 < 1 4 ( π 2 9 ) .
(37)

For |u| 1 ϵ 0 , we have |u| ϵ 0 u 2 , so,

|u| ϵ 0 u 2 + 1 ϵ 0 ,uR.
(38)

By (35), we have

0 u f(t,v)dv 9 4 u 2 +|u|,tJ,uR.
(39)

It follows from (38) and (39) that

0 u f(t,v)dv ( 9 4 + ϵ 0 ) u 2 + 1 ϵ 0 ,tJ,uR.
(40)

Since, by virtue of (37),

0< 9 4 + ϵ 0 < π 2 4 ,

we see that (40) implies that condition (H2) is satisfied for c= 9 4 + ϵ 0 and d= 1 ϵ 0 . Hence, our conclusion follows from Theorem 1. □

By using the Mountain Pass Lemma and the Minimax Principle established by Ambrosetti and Rabinowitz [25, 26], we have obtained in [23] the existence of a nontrivial solution and the existence of infinitely many nontrivial solutions for a class of nonlinear integral equations. Since (2) is a special case of such nonlinear integral equations, we get the following result for (2).

Lemma 3 (Special case of Theorem 1 and Theorem 2 in [23])

Suppose the following.

  1. (a)

    There exist p>2 and a>0, b>0 such that

    |g(t,v)|a+b | v | p 1 ,tJ,vR.
  2. (b)

    There exist 0τ< 1 2 and M>0 such that

    0 v g(t,w)dwτvg(t,v),tJ,|v|M.
  3. (c)

    g ( t , v ) v 0 as v0 uniformly for tJ and g ( t , v ) v as |v| uniformly for tJ.

Then the integral equation (2) has at least one nontrivial solution in L p [J,R]. If, in addition,

  1. (d)

    g(t,v)=g(t,v), tJ, vR.

Then the integral equation (2) has infinite many nontrivial solutions in L p [J,R].

Let us list more conditions for the function f(t,u).

(H3) There exist 0τ< 1 2 and M>0 such that

0 u f ( t , v + a ( t ) a ( 1 ) t ) dvτuf ( t , u + a ( t ) a ( 1 ) t ) ,tJ,|u|M.

(H4) f ( t , u + a ( t ) a ( 1 ) t ) u 0 as u0 uniformly for tJ, and f ( t , u + a ( t ) a ( 1 ) t ) u as |u| uniformly for tJ.

(H5) f(t,u+a(t)a(1)t)=f(t,u+a(t)a(1)t), tJ, uR.

Theorem 2 Suppose that conditions (H1), (H3), and (H4) are satisfied. Then BVP (1) has at least one solution uP C 1 [J,R] C 2 [ J ,R]. If, in addition, condition (H5) is satisfied, then BVP (1) has infinitely many solutions u n P C 1 [J,R] C 2 [ J ,R] (n=1,2,3,).

Proof In the proof of Lemma 2, we see that condition (H1) implies condition (a) of Lemma 3 (see (17)). On the other hand, it is clear that conditions (H3), (H4), (H5) are the same as conditions (b), (c), (d) in Lemma 3, respectively. Hence the conclusion of Theorem 2 follows from Lemma 3, Lemma 2, and Lemma 1. □

Example 2 Consider the BVP

{ u ( t ) = { [ u ( t ) t ] 3 , 0 t < 1 2 ; [ u ( t ) + 3 t 3 ] 3 , 1 2 < t 1 , Δ u | t = 1 2 = 1 , Δ u | t = 1 2 = 4 , u ( 0 ) = u ( 1 ) = 0 .
(41)

Conclusion BVP (41) has infinite many solutions u n P C 1 [J,R] C 2 [ J ,R] (n=1,2,3,).

Proof Obviously, (41) is a BVP of form (1). In this situation, J=[0,1], m=1, t 1 = 1 2 , J =[0,1]{ 1 2 }, c 1 =1, d 1 =4, and

f(t,u)= { ( u t ) 3 , 0 t 1 2 ; ( u + 3 t 3 ) 3 , 1 2 < t 1 .
(42)

It is clear that f(t,u) is continuous on J ×R, left continuous at t= t 1 , and the right limit f( t 1 + ,u) exists. By (42), we have

| f ( t , u ) | ( | u | + 3 2 ) 3 ( 2 max { | u | , 3 2 } ) 3 2 3 ( | u | 3 + ( 3 2 ) 3 ) = 8 | u | 3 + 27 , t J , u R ,

so, condition (H1) is satisfied for p=4, a=27 and b=8. By (5), we have

a(t)= { 0 , 0 t 1 2 ; 3 4 t , 1 2 < t 1 ,
(43)

so, a(1)=1 and (42) and (43) imply

f ( t , u + a ( t ) a ( 1 ) t ) = u 3 ,tJ,uR,
(44)

and, consequently, (H3) is satisfied for τ= 1 4 and any M>0. On the other hand, from (44) we see that conditions (H4) and (H5) are all satisfied. Hence, our conclusion follows from Theorem 2. □

References

  1. 1.

    Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.

  2. 2.

    Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.

  3. 3.

    Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.

  4. 4.

    Agarwal RP, O’Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 2000, 114: 51-59. 10.1016/S0096-3003(99)00074-0

  5. 5.

    Yan B: Boundary value problems on the half line with impulses and infinite delay. J. Math. Anal. Appl. 2001, 259: 94-114. 10.1006/jmaa.2000.7392

  6. 6.

    Agarwal RP, O’Regan D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput. 2005, 161: 433-439. 10.1016/j.amc.2003.12.096

  7. 7.

    Kaufmann ER, Kosmatov N, Raffoul YN: A second-order boundary value problem with impulsive effects on an unbounded domain. Nonlinear Anal. 2008, 69: 2924-2929. 10.1016/j.na.2007.08.061

  8. 8.

    Guo D: Positive solutions of an infinite boundary value problem for n th-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 2009, 70: 2078-2090. 10.1016/j.na.2008.02.110

  9. 9.

    Guo D: Multiple positive solutions for first order impulsive superlinear integro-differential equations on the half line. Acta Math. Sci. Ser. B 2011, 31(3):1167-1178. 10.1016/S0252-9602(11)60307-X

  10. 10.

    Guo D, Liu X: Multiple positive solutions of boundary value problems for impulsive differential equations. Nonlinear Anal. 1995, 25: 327-337. 10.1016/0362-546X(94)00175-H

  11. 11.

    Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Appl. Math. Comput. 2003, 143: 233-249. 10.1016/S0096-3003(02)00356-9

  12. 12.

    Guo D: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 2005, 63: 618-641. 10.1016/j.na.2005.05.023

  13. 13.

    Xu X, Wang B, O’Regan D: Multiple solutions for sub-linear impulsive three-point boundary value problems. Appl. Anal. 2008, 87: 1053-1066. 10.1080/00036810802429001

  14. 14.

    Jankowski J: Existence of positive solutions to second order four-point impulsive differential problems with deviating arguments. Comput. Math. Appl. 2009, 58: 805-817. 10.1016/j.camwa.2009.06.001

  15. 15.

    Liu Y, O’Regan D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 1769-1775.

  16. 16.

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

  17. 17.

    Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022

  18. 18.

    Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155-162. 10.1016/j.nonrwa.2008.10.044

  19. 19.

    Chen H, Sun J: An application of variational method to second-order impulsive differential equations on the half line. Appl. Math. Comput. 2010, 217: 1863-1869. 10.1016/j.amc.2010.06.040

  20. 20.

    Bai L, Dai B: Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory. Math. Comput. Model. 2011, 53: 1844-1855. 10.1016/j.mcm.2011.01.006

  21. 21.

    Guo D: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space. Appl. Math. Comput. 2002, 125: 59-77. 10.1016/S0096-3003(00)00115-6

  22. 22.

    Krasnoselskii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, Oxford; 1964.

  23. 23.

    Guo D: The number of nontrivial solutions to Hammerstein nonlinear integral equations. Chin. Ann. Math., Ser. B 1986, 7(2):191-204.

  24. 24.

    Zaanen AC: Linear Analysis. Interscience, New York; 1958.

  25. 25.

    Ambrosetti A, Rabinowitz PH: Dual variational method in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7

  26. 26.

    Rabinowitz PH: Variational methods for nonlinear eigenvalue problems. Course of Lectures - CIME 1974. Varenna, Italy

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Acknowledgements

Research was supported by the National Nature Science Foundation of China (No. 10671167).

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Correspondence to Dajun Guo.

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Keywords

  • impulsive differential equation
  • integral equation
  • variational method
  • critical point theory