Open Access

RETRACTED ARTICLE: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects

Advances in Difference Equations20142014:235

https://doi.org/10.1186/1687-1847-2014-235

Received: 10 June 2014

Accepted: 5 August 2014

Published: 9 September 2014

The Retraction Note to this article has been published in Advances in Difference Equations 2015 2015:204

Abstract

In this paper, we study the existence of periodic solutions of second-order impulsive differential equations at resonance. We prove the existence of periodic solutions under a generalized Landesman-Lazer type condition by using the variational method. The impulses can generate a periodic solution.

Keywords

impulsive differential equationsLandesman-Lazer type conditionvariational method

1 Introduction

We are concerned with periodic boundary value problem of second-order impulsive differential equations at resonance
{ x ( t ) + m 2 x ( t ) + f ( t , x ( t ) ) = e ( t ) , a.e.  t [ 0 , 2 π ] , x ( 0 ) x ( 2 π ) = x ( 0 ) x ( 2 π ) = 0 , x ( t j + ) = x ( t j ) , Δ x ( t j ) : = x ( t j + ) x ( t j ) = I j ( t j , x ( t j ) ) , j = 1 , 2 , , p ,
(1.1)

where m N , f : [ 0 , 2 π ] × R R is a Carathéodory function, e L 1 ( 0 , 2 π ) , 0 < t 1 < t 2 < < t p < 2 π , and I j : [ 0 , 2 π ] × R R is continuous for every j.

When Δ x ( t j ) 0 , problem (1.1) becomes to the well-known periodic boundary value problem at resonance
{ x ( t ) + m 2 x ( t ) + f ( t , x ( t ) ) = e ( t ) , a.e.  t [ 0 , 2 π ] , x ( 0 ) x ( 2 π ) = x ( 0 ) x ( 2 π ) = 0 .
(1.2)
There are many existence results for problem (1.2) in the literature. Let us mention some pioneering works by Lazer [1], Lazer and Leach [2], and Landesman and Lazer [3]. In [3], a key sufficient condition for the existence of solutions of problem (1.2) is the so-called Landesman-Lazer condition,
0 2 π e ( t ) sin ( m t + θ ) d t < 0 2 π [ ( lim inf x + f ( t , x ) ) sin + ( m t + θ ) ( lim sup x f ( t , x ) ) sin ( m t + θ ) ] d t , θ R ,
(1.3)

where sin ± ( m t + θ ) = max { ± sin ( m t + θ ) , 0 } .

It is well known that the theory of impulsive differential equations has been recognized to not only be richer than that of differential equations without impulses, but also to provide a more adequate mathematical model for numerous processes and phenomena studied in physics, biology, engineering, etc. We refer the reader to the book [4]. Recently, the Dirichlet and periodic boundary conditions problems for second-order differential equations with impulses in the derivative and without impulses are studied by some authors via variational method [511]. In this paper, we will investigate problem (1.1) under a more general Landesman-Lazer type condition. Define
F ( t , x ) = 0 x f ( t , s ) d s , F + ( t ) = lim inf x + F ( t , x ) x , F ( t ) = lim sup x F ( t , x ) x
and for j = 1 , 2 , , p ,
J j ( t , x ) = 0 x I j ( t , s ) d s , J j + ( t ) = lim sup x + J j ( t , x ) x , J j ( t ) = lim inf x J j ( t , x ) x .

Throughout this paper, we give the following fundamental assumptions.

( H 1 ) There exists p L 1 ( [ 0 , 2 π ] , [ 0 , + ) ) such that | f ( t , x ) | p ( t ) , for a.e. t [ 0 , 2 π ] and for all x R .

( H 2 ) There exist positive constants c 1 , c 2 , , c p such that for all t , x R ,
| I j ( t , x ) | c j , j = 1 , 2 , , p .
( H 3 ) For all θ R ,
j = 1 p J j + ( t j ) sin + ( m t j + θ ) j = 1 p J j ( t j ) sin ( m t j + θ ) + 0 2 π e ( t ) sin ( m t + θ ) d t < 0 2 π ( F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ) d t .

We now can state the main theorem of this paper.

Theorem 1.1 Assume that the conditions ( H 1 ), ( H 2 ), and ( H 3 ) hold. Then problem (1.1) has at least one 2π-periodic solution.

To demonstrate the impulsive effects clearly, we can take
I j ( t , x ) d j , j = 1 , 2 , , p ,
(1.4)

where d 1 , d 2 , , d p are constants. Hence, J j ± ( t ) = d j .

From Theorem 1.1, we obtain the following result.

Corollary 1.2 Assume that we have the conditions ( H 1 ), (1.4), and the following.

( H 3 ) For all θ R ,
j = 1 p d j sin ( m t j + θ ) + 0 2 π e ( t ) sin ( m t + θ ) d t < 0 2 π ( F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ) d t

hold. Then problem (1.1) has at least one 2π-periodic solution.

Moreover, we have the following corollary.

Corollary 1.3 Assume that we have the conditions ( H 1 ) and the following.

( H 3 ) For all θ R ,
0 2 π e ( t ) sin ( m t + θ ) d t < 0 2 π ( F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ) d t
(1.5)

holds. Then problem (1.2) has at least one 2π-periodic solution.

Remark 1.4 By a simple calculation, one can easily derive
F + ( t ) = lim inf x + F ( t , x ) x lim inf x + f ( t , x ) , F ( t ) = lim sup x F ( t , x ) x lim sup x f ( t , x ) .

A simple example f ( t , x ) = sin t + cos x illustrates it. Thus condition ( H 3 ) generalizes condition (1.3). Hence, our results improve the related results in the literature mentioned above. Moreover, since we consider the problem with impulses, Theorem 1.1 is also a complement of the pioneering works.

Remark 1.5 It is remarkable that Landesman-Lazer condition ( H 3 ) is an ‘almost’ necessary and sufficient condition when F + and F are replaced by f + and f , where f + = lim x + f ( t , x ) , f = lim x f ( t , x ) , and f ( t ) f ( t , x ) f + ( t ) (see [[12], p.70]). If the condition (1.5) is not satisfied, i.e., θ R ,
0 2 π e ( t ) sin ( m t + θ ) d t 0 2 π ( F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ) d t ,
problem (1.2) cannot be guaranteed to have periodic solution. For example, we consider resonant differential equation
x + m 2 x + ( 1 + sin m t ) arctan x = 8 sin m t .
(1.6)
Obviously, f ( t , x ) = ( 1 + sin m t ) arctan x , e ( t ) = 8 sin m t , and F + ( t ) = π 2 ( 1 + sin m t ) , F ( t ) = π 2 ( 1 + sin m t ) . Taking θ = 0 , we have
0 2 π e ( t ) sin m t d t 0 2 π ( F + ( t ) sin + m t F ( t ) sin m t ) d t = 8 π π 2 0 2 π ( 1 + sin m t ) | sin m t | d t 8 π 2 π 2 > 0 .
Then ( H 3 ) is not satisfied. From now on, we prove that (1.6) has not 2π-periodic solution by contradiction. Assume that (1.6) has 2π-periodic solution. Multiplying both sides of (1.6) by sin m t and integrating over [ 0 , 2 π ] , we get
8 π = 0 2 π ( 1 + sin m t ) arctan x sin m t d t 0 2 π | ( 1 + sin m t ) arctan x cos m t | d t π 0 2 π d t = 2 π 2 ,

which is impossible. Hence, problem (1.2) may have no solution if the condition ( H 3 ) is not satisfied. However, as long as ( H 3 ) holds, problem (1.1) will have at least one periodic solution. Therefore, the impulses can generate a periodic solution.

The rest of the paper is organized as follows. In Section 2, we shall state some notations, some necessary definitions, and a saddle theorem due to Rabinowitz. In Section 3, we shall prove Theorem 1.1.

2 Preliminaries

In the following, we introduce some notations and some necessary definitions.

Define
H = { x H 1 ( 0 , 2 π ) : x ( 0 ) = x ( 2 π ) } ,
with the norm
x = ( 0 2 π ( x 2 ( t ) + x 2 ( t ) ) d t ) 1 2 .
Consider the functional φ ( x ) defined on H by
φ ( x ) = 1 2 0 2 π x 2 ( t ) d t m 2 2 0 2 π x 2 ( t ) d t 0 2 π F ( t , x ( t ) ) d t + 0 2 π e ( t ) x ( t ) d t + j = 1 p J j ( t j , x ( t j ) ) .
(2.1)
Similarly as in [7], φ ( x ) is continuously differentiable on H, and
φ ( x ) v ( t ) = 0 2 π x ( t ) v ( t ) d t m 2 0 2 π x ( t ) v ( t ) d t 0 2 π f ( t , x ( t ) ) v ( t ) d t + 0 2 π e ( t ) v ( t ) d t + j = 1 p I j ( t j , x ( t j ) ) v ( t j ) , for  v ( t ) H .
(2.2)

Now, we have the following lemma.

Lemma 2.1 If x H is a critical point of φ, then x is a 2π-periodic solution of (1.1).

The proof of Lemma 2.1 is similar to Lemma 2.1 in [6], so we omit it.

We say that φ satisfies (PS) if every sequence ( x n ) for which φ ( x n ) is bounded in and φ ( x n ) 0 (as n ) possesses a convergent subsequence.

To prove the main result, we will use the following saddle point theorem due to Rabinowitz [13] (or see [12]).

Theorem 2.2 Let φ C 1 ( H , R ) and H = H H + , dim ( H ) < , dim ( H + ) = . We suppose that:
  1. (a)

    There exists a bounded neighborhood D of 0 in H and a constant α such that φ | D α ;

     
  2. (b)

    there exists a constant β > α such that φ | H + β ;

     
  3. (c)

    φ satisfies (PS).

     

Then the functional φ has a critical point in H.

3 The proof of Theorem 1.1

In this section, we first show that the functional φ satisfies the Palais-Smale condition.

Lemma 3.1 Assume that the conditions ( H 1 ), ( H 2 ), and ( H 3 ) hold. Then φ defined by (2.1) satisfies (PS).

Proof Let M > 0 be a constant and { x n } H be a sequence satisfying
| φ ( x n ) | = | 1 2 0 2 π x n 2 d t m 2 2 0 2 π x n 2 d t 0 2 π F ( t , x n ) d t + 0 2 π e ( t ) x n ( t ) d t + j = 1 p J j ( t j , x n ( t j ) ) | M
(3.1)
and
lim n φ ( x n ) = 0 .
(3.2)
We first prove that { x n } is bounded in H by contradiction. Assume that { x n } is unbounded. Let { z k } be an arbitrary sequence bounded in H. It follows from (3.2) that, for any k N ,
lim n | φ ( x n ) z k | lim n φ ( x n ) z k = 0 .
Thus
lim n φ ( x n ) z k = 0 uniformly for  k N .
Hence,
lim n ( 0 2 π ( x n z k m 2 x n z k ) d t 0 2 π ( f ( t , x n ) z k e ( t ) z k ) d t + j = 1 p I j ( t j , x n ( t j ) ) z k ( t j ) ) = 0 .
(3.3)
By ( H 1 ) and ( H 2 ), we have
lim n ( 0 2 π f ( t , x n ) z k e ( t ) z k x n d t j = 1 p I j ( t j , x n ( t j ) ) z k ( t j ) x n ) = 0 .
(3.4)
From (3.3) and (3.4), we obtain
lim n 0 2 π ( x n x n z k m 2 x n x n z k ) d t = 0 .
(3.5)
Set
y n = x n x n .
Then we have
lim n 0 2 π ( y n z k m 2 y n z k ) d t = 0 ,
and furthermore,
lim n i 0 2 π [ ( y n y i ) z k m 2 ( y n y i ) z k ] d t = 0 .
(3.6)
Replacing z k in (3.6) by ( y n y i ) , we get
lim n i ( y n y i 2 ( m 2 + 1 ) y n y i 2 2 ) = 0 .
Due to the compact embedding H L 2 ( 0 , 2 π ) , going to a subsequence,
y n y 0 weakly in  H , y n y 0 in  L 2 ( 0 , 2 π ) .
Therefore,
lim n i y n y i 2 2 = 0 .
Furthermore, we have
lim n i y n y i 2 = 0 ,
which implies ( y n ) is Cauchy sequence in H. Thus, y n y 0 in H. It follows from (3.5) and the usual regularity argument for ordinary differential equations (see [14]) that
y 0 = k 1 sin m t + k 2 cos m t ,
(3.7)

where k 1 2 + k 2 2 = 1 ( m 2 + 1 ) π ( y 0 = 1 ). (Different subsequences of { y n } correspond to different k 1 and k 2 .)

Write (3.7) as
y 0 = 1 ( m 2 + 1 ) π sin ( m t + θ ) ,

where θ satisfies sin θ = k 2 k 1 2 + k 2 2 and cos θ = k 1 k 1 2 + k 2 2 .

Taking z k = 1 ( m 2 + 1 ) π sin ( m t + θ ) , we get, for any n N ,
0 2 π ( x n z k m 2 x n z k ) d t = 0 .
(3.8)
Thus, it follows from (3.3) and (3.8) that
lim n [ 0 2 π ( f ( t , x n ) e ( t ) ) 1 ( m 2 + 1 ) π sin ( m t + θ ) d t j = 1 p I j ( t j , x n ( t j ) ) 1 ( m 2 + 1 ) π sin ( m t j + θ ) ] = 0 .
(3.9)
By ( H 1 ) and ( H 2 ), we obtain
lim n [ 0 2 π ( f ( t , x n ) e ( t ) ) ( 1 ( m 2 + 1 ) π sin ( m t + θ ) y n ) d t j = 1 p I j ( t j , x n ( t j ) ) ( 1 ( m 2 + 1 ) π sin ( m t j + θ ) y n ( t j ) ) ] = 0 .
(3.10)
It follows from (3.9) and (3.10) that
lim n [ 0 2 π ( f ( t , x n ) e ( t ) ) y n d t j = 1 p I j ( t j , x n ( t j ) ) y n ( t j ) ] = 0 .
Hence, replacing z k in (3.3) by y n , we have
lim n 0 2 π ( x n x n x n m 2 x n x n x n ) d t = 0 .
(3.11)
Now, dividing (3.1) by x n , we get
M x n 1 2 0 2 π ( x n 2 x n m 2 x n 2 x n ) d t 0 2 π F ( t , x n ) e ( t ) x n x n + j = 1 p J j ( t j , x n ( t j ) ) x n M x n ,
which yields
0 2 π F ( t , x n ) e ( t ) x n x n M x n + 1 2 0 2 π ( x n 2 x n m 2 x n 2 x n ) d t + j = 1 p J j ( t j , x n ( t j ) ) x n .
(3.12)
Note that x n x n 1 ( m 2 + 1 ) π sin ( m t + θ ) in H. Due to the compact embedding H C ( 0 , 2 π ) and | x n ( t ) | + , we have x n x n 1 ( m 2 + 1 ) π sin ( m t + θ ) in C ( 0 , 2 π ) . Furthermore,
lim n x n ( t ) = { + , t I + : = { t [ 0 , 2 π ] | sin ( m t + θ ) > 0 } , , t I : = { t [ 0 , 2 π ] | sin ( m t + θ ) < 0 } .
Hence, from (3.11) and (3.12), we have
lim inf n 0 2 π F ( t , x n ) e ( t ) x n x n d t lim inf n j = 1 p J j ( t j , x n ( t j ) ) x n ( t j ) x n + ( t j ) x n ( t j ) x n lim sup n j = 1 p J j ( t j , x n ( t j ) ) x n ( t j ) x n + ( t j ) x n lim inf n j = 1 p J j ( t j , x n ( t j ) ) x n ( t j ) x n ( t j ) x n = 1 ( m 2 + 1 ) π j = 1 p J j + ( t j ) sin + ( m t j + θ ) 1 ( m 2 + 1 ) π j = 1 p J j ( t j ) sin ( m t j + θ ) .
(3.13)
Using Fatou’s lemma, we get
lim inf n 0 2 π F ( t , x n ) x n d t = lim inf n [ I + F ( t , x n ) x n x n x n d t I F ( t , x n ) x n x n x n d t ] I + lim inf n F ( t , x n ) x n x n x n d t I lim sup n F ( t , x n ) x n x n x n d t .
Thus, by a simple computation, we have
lim inf n 0 2 π F ( t , x n ) x n d t 1 ( m 2 + 1 ) π 0 2 π [ F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ] d t .
(3.14)
Hence, it follows from (3.13) and (3.14) that
j = 1 p J j + ( t j ) sin + ( m t j + θ ) j = 1 p J j ( t j ) sin ( m t j + θ ) + 0 2 π e ( t ) sin ( m t + θ ) d t 0 2 π [ F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ] d t .
This contradicts ( H 3 ). It implies that the sequence ( x n ) is bounded. Thus, there exists x 0 H such that x n x 0 weakly in H. Due to the compact embedding H L 2 ( 0 , 2 π ) and H C ( 0 , 2 π ) , going to a subsequence,
x n x 0 in  L 2 ( 0 , 2 π ) , x n x 0 in  C ( 0 , 2 π ) .
From (3.3), we obtain
lim n i ( 0 2 π ( ( x n x i ) z k m 2 ( x n x i ) z k ) d t 0 2 π ( f ( t , x n ) f ( t , x i ) ) z k d t + j = 1 p ( I j ( t j , x n ( t j ) ) I j ( t j , x i ( t j ) ) ) z k ( t j ) ) = 0 .
Replacing z k by x n x i in the above equality, we get
lim n i ( 0 2 π ( ( x n x i ) 2 m 2 ( x n x i ) 2 ) d t 0 2 π ( f ( t , x n ) f ( t , x i ) ) ( x n x i ) d t + j = 1 p ( I j ( t j , x n ( t j ) ) I j ( t j , x i ( t j ) ) ) ( x n ( t j ) x i ( t j ) ) ) = 0 .
(3.15)
By ( H 1 ) and ( H 2 ), we have
lim n i 0 2 π ( f ( t , x n ) f ( t , x i ) ) ( x n x i ) d t = 0
(3.16)
and
lim n i j = 1 p ( I j ( t j , x n ( t j ) ) I j ( t j , x i ( t j ) ) ) ( x n ( t j ) x i ( t j ) ) = 0 .
(3.17)
Thus, it follows from (3.15), (3.16), and (3.17) that
lim n i 0 2 π [ ( x n x i ) 2 m 2 ( x n x i ) 2 ] d t = 0 .
Therefore,
lim n i x n x i 2 = 0 ,

which implies x n x 0 in H. It shows that φ satisfies (PS). □

Now, we can give the proof of Theorem 1.1.

Proof of Theorem 1.1 Denote
H = R span { sin t , cos t , sin 2 t , cos 2 t , , sin m t , cos m t }
and
H + = span { sin ( m + 1 ) t , cos ( m + 1 ) t , } .
We first prove that
lim inf x φ ( x ) = , for  x H ,
(3.18)
by contradiction. Assume that there exists a sequence ( x n ) H such that x n (as n ) and there exists a constant c satisfying
lim inf n φ ( x n ) c .
(3.19)
By ( H 1 ), we have
lim n 0 2 π F ( t , x n ) e ( t ) x n x n 2 d t = 0 .
(3.20)
By ( H 2 ), we get
lim n j = 1 p J j ( t j , x n ( t j ) ) x n 2 = 0 .
(3.21)
From (3.19) and the definition of φ, we obtain
lim inf n [ 1 2 0 2 π x n 2 m 2 x n 2 x n 2 d t 0 2 π F ( t , x n ) e ( t ) x n x n 2 d t + j = 1 p J j ( t j , x n ( t j ) ) x n 2 ] 0 .
(3.22)
For x H , we have
0 2 π ( x 2 m 2 x 2 ) d t = x 2 ( m 2 + 1 ) x 2 2 0 .
(3.23)
The equality in (3.23) holds only for
x = 1 ( m 2 + 1 ) π sin ( m t + θ ) , θ R .
Set y n = x n x n . Since dim H < , going to a subsequence, there exists y 0 H such that y n y 0 in H and y n y 0 in L 2 ( 0 , 2 π ) . Then (3.20), (3.21), (3.22), and (3.23) imply that
y 0 = 1 ( m 2 + 1 ) π sin ( m t + θ ) , θ R .
By (3.19), we have, for n large enough,
1 2 0 2 π x n 2 m 2 x n 2 x n d t 0 2 π F ( t , x n ) e ( t ) x n x n d t + j = 1 p J j ( t j , x n ( t j ) ) x n c x n .
(3.24)
It follows from x n H that
0 2 π x n 2 m 2 x n 2 x n 0 .
(3.25)
From (3.24) and (3.25), we get, for n large enough,
c x n 0 2 π F ( t , x n ) e ( t ) x n x n d t + j = 1 p J j ( t j , x n ( t j ) ) x n .
Thus,
lim inf n 0 2 π ( F ( t , x n ) x n e ( t ) ) x n x n d t lim inf n j = 1 p J j ( t j , x n ( t j ) ) x n .
Using an argument similar to the proof of Lemma 3.1, we get
j = 1 p J j + ( t j ) sin + ( m t j + θ ) j = 1 p J j ( t j ) sin ( m t j + θ ) + 0 2 π e ( t ) sin ( m t + θ ) d t 0 2 π ( F + ( t ) sin + ( m t + θ ) F ( t ) sin ( m t + θ ) ) d t ,

which is a contradiction to ( H 3 ).

Then (3.18) holds.

Next, we prove that
lim x φ ( x ) = , for all  x H + ,

and φ is bounded on bounded sets.

Because of the compact embedding of H C ( 0 , 2 π ) and H L 2 ( 0 , 2 π ) , there exists constants m 1 , m 2 such that
x m 1 x , x 2 m 2 x .
Then by ( H 1 ) and ( H 2 ), one has
| φ ( x ) | = | 1 2 0 2 π x 2 d t m 2 2 0 2 π x 2 d t 0 2 π [ F ( t , x ) e ( t ) x ] d t + j = 1 p J j ( t j , x ( t j ) ) | 1 2 x 2 + m 2 2 m 2 2 x 2 + 0 2 π ( | p ( t ) | | x | + | e ( t ) | | x | ) d t + j = 1 p c j | x ( t j ) | 1 + m 2 m 2 2 2 x 2 + m 1 ( p 1 + e 1 ) x + j = 1 p c j m 1 x .
(3.26)

Hence, φ is bounded on bounded sets of H.

Since x H + , we have
x 2 ( ( m + 1 ) 2 + 1 ) x 2 2 .
(3.27)
Thus, from (3.26) and (3.27), we obtain
φ ( x ) = 1 2 0 2 π x 2 d t m 2 2 0 2 π x 2 d t 0 2 π [ F ( t , x ) e ( t ) x ] d t + j = 1 p J j ( t j , x ( t j ) ) 2 m + 1 2 ( ( m + 1 ) 2 + 1 ) x 2 m 1 ( p 1 + e 1 + j = 1 p c j ) x ,
which implies
lim x φ ( x ) = , for all  x H + .

Up to now, the conditions (a) and (b) of Theorem 2.2 are satisfied. According to Lemma 3.1, (c) is also satisfied. Hence, by Theorem 2.2, (1.1) has at least one solution. This completes the proof. □

Notes

Declarations

Acknowledgements

The authors would like to express their thanks to the editor of the journal and the referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper.

Authors’ Affiliations

(1)
School of Science, Jiujiang University
(2)
Department of Scientific Research Management, Jiujiang University

References

  1. Lazer AC: On Schauder’s fixed point theorem and forced second order nonlinear oscillations. J. Math. Anal. Appl. 1968, 21: 421-425. 10.1016/0022-247X(68)90225-4MathSciNetView ArticleMATHGoogle Scholar
  2. Lazer AC, Leach DE: Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura Appl. 1969, 82(4):49-68.MathSciNetView ArticleMATHGoogle Scholar
  3. Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 1970, 19: 60-623.MathSciNetMATHGoogle Scholar
  4. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleMATHGoogle Scholar
  5. Bogun I: Existence of weak solutions for impulsive p -Laplacian problem with superlinear impulses. Nonlinear Anal., Real World Appl. 2012, 13(6):2701-2707. 10.1016/j.nonrwa.2012.03.014MathSciNetView ArticleMATHGoogle Scholar
  6. Ding W, Qian D: Periodic solutions for sublinear systems via variational approach. Nonlinear Anal., Real World Appl. 2010, 11(4):2603-2609. 10.1016/j.nonrwa.2009.09.007MathSciNetView ArticleMATHGoogle Scholar
  7. Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11(1):155-162. 10.1016/j.nonrwa.2008.10.044MathSciNetView ArticleMATHGoogle Scholar
  8. Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10(2):680-690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar
  9. Zhang H, Li Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11(1):67-78. 10.1016/j.nonrwa.2008.10.016MathSciNetView ArticleMATHGoogle Scholar
  10. Zhang X, Meng Q: Nontrivial periodic solutions for delay differential systems via Morse theory. Nonlinear Anal. 2011, 74(5):1960-1968. 10.1016/j.na.2010.11.003MathSciNetView ArticleMATHGoogle Scholar
  11. Tomiczek P: The Duffing equation with the potential Landesman-Lazer condition. Nonlinear Anal. 2009, 70(2):735-740. 10.1016/j.na.2008.01.006MathSciNetView ArticleMATHGoogle Scholar
  12. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.View ArticleMATHGoogle Scholar
  13. Rabinowitz PH CBMS Reg. Conf. Ser. in Math. 65. In Minmax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.View ArticleGoogle Scholar
  14. Fučík S: Solvability of Nonlinear Equations and Boundary Value Problems. D. Reidel Publ. Company, Holland; 1980.MATHGoogle Scholar

Copyright

© Li and Zheng; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.