Open Access

Existence of nontrivial weak homoclinic orbits for second-order impulsive differential equations

Boundary Value Problems20122012:138

DOI: 10.1186/1687-2770-2012-138

Received: 25 July 2012

Accepted: 7 November 2012

Published: 26 November 2012

Abstract

A sufficient condition is obtained for the existence of nontrivial weak homoclinic orbits of second-order impulsive differential equations by employing the mountain pass theorem, a weak convergence argument and a weak version of Lieb’s lemma.

1 Introduction

Fečkan [1], Battelli and Fečkan [2] studied the existence of homoclinic solutions for impulsive differential equations by using perturbation methods. Tang et al. [36] studied the existence of homoclinic solutions for Hamiltonian systems via variational methods. In recent years, many researchers have paid much attention to multiplicity and existence of solutions of impulsive differential equations via variational methods (for example, see [712]). However, few papers have been published on the existence of homoclinic solutions for second-order impulsive differential equations via variational methods.

In this paper, we consider the following impulsive differential equations:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ2_HTML.gif
(1.2)

where V : R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq1_HTML.gif is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq2_HTML.gif, V ( t , 0 ) = V ( t , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq3_HTML.gif with V ( t , x ) = ( V / x ) ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq4_HTML.gif, and I C ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq5_HTML.gif with I ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq6_HTML.gif. denotes the set of all integers, and t j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq7_HTML.gif ( j Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq8_HTML.gif) are impulsive points. Moreover, there exist a positive integer p and a positive constant T such that 0 < t 0 < t 1 < < t p 1 < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq9_HTML.gif, t l + k p = t l + k T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq10_HTML.gif, k Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq11_HTML.gif, l = 0 , 1 , , p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq12_HTML.gif. q ( t j + ) = lim h 0 + q ( t j + h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq13_HTML.gif and q ( t j ) = lim h 0 + q ( t j h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq14_HTML.gif represent the right and left limits of q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq15_HTML.gif at t = t j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq16_HTML.gif respectively.

We say that a function q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq17_HTML.gif is a weak homoclinic orbit of Eqs. (1.1) and (1.2) if q satisfies (1.1) and
q { q C ( R , R ) : j = + | q ( t j ) | 2 < + , q L 2 ( R ) , q ( ± ) = 0 , q ( k T ) = 0 , k Z } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equa_HTML.gif

Motivated by the works of Nieto and Regan [7], Smets and Willem [13], in this paper we study the existence of nontrivial weak homoclinic orbits of (1.1)-(1.2) by using the mountain pass theorem, a weak version of Lieb’s lemma and a weak convergence argument. Our method is different from those of [8, 9].

The main result is the following.

Theorem 1.1 Assume that Eqs. (1.1) and (1.2) satisfy the following conditions:

(H1) There exists a positive number T such that
V ( t + T , x ) = V ( t , x ) , V ( t + T , x ) = V ( t , x ) , ( t , x ) R 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equb_HTML.gif

(H2) lim x 0 V ( t , x ) x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq18_HTML.gif uniformly for t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif;

(H3) There exists a constant μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq20_HTML.gif such that
x V ( t , x ) μ V ( t , x ) > 0 , ( t , x ) R × R { 0 } ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equc_HTML.gif
(H4) There exist constants a 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq21_HTML.gif and a 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq22_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equd_HTML.gif
(H5) There exists a constant b, with 0 < b < μ 2 ( μ + 2 ) T p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq23_HTML.gif, such that
| I ( x ) | b | x | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Eque_HTML.gif
and
2 0 x I ( t ) d t I ( x ) x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equf_HTML.gif

Then there exists a nontrivial weak homoclinic orbit of Eqs. (1.1) and (1.2).

Remark 1.1 (H2) implies that q ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq24_HTML.gif is an equilibrium of (1.1)-(1.2).

Remark 1.2 Set V ( t , x ) = ( 2 + sin t ) x 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq25_HTML.gif, I ( x ) = x 10 π p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq26_HTML.gif. It is easy to see that V ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq27_HTML.gif, I ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq28_HTML.gif satisfy (H1)-(H5).

2 Proof of main results

Lemma 2.1 (Mountain pass lemma [14])

Let E be a Banach space and φ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq29_HTML.gif, e E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq30_HTML.gif, r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq31_HTML.gif be such that e > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq32_HTML.gif and
b : = inf y = r φ ( y ) > φ ( 0 ) φ ( e ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equg_HTML.gif
Let
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equh_HTML.gif

Then, for each ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq34_HTML.gif, there exists y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq35_HTML.gif such that

(V1) d 2 ε φ ( y ) d + 2 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq36_HTML.gif;

(V2) dist ( y , E ) 2 δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq37_HTML.gif;

(V3) φ ( y ) 8 ε δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq38_HTML.gif.

In what follows, l 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq39_HTML.gif denotes the space of sequences whose second powers are summable on (the set of all integers), that is,
j Z | a j | 2 < + , a = { a j } j = + l 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equi_HTML.gif
The space l 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq39_HTML.gif is equipped with the following norm:
a l 2 = ( j Z | a j | 2 ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equj_HTML.gif

We now prove some technical lemmas.

Lemma 2.2 The space
H : = { q C ( R , R ) : { q ( t j ) } j = + l 2 , q L 2 ( R ) , q ( ± ) = 0 , q ( k T ) = 0 , k Z } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ3_HTML.gif
(2.1)
is a Hilbert space with the inner product
( q 1 , q 2 ) H = R q 1 ( t ) q 2 ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ4_HTML.gif
(2.2)
and the corresponding norm
q H = ( R | q ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ5_HTML.gif
(2.3)
Proof Let { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif be a Cauchy sequence in H, then { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq41_HTML.gif is a Cauchy sequence in L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq42_HTML.gif and there exists y L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq43_HTML.gif such that { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq41_HTML.gif converges to y in L 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq42_HTML.gif. Define the function q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq17_HTML.gif as follows:
q ( t ) = k T t y ( s ) d s , k T t < ( k + 1 ) T , k Z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equk_HTML.gif
It is easy to see that
lim h 0 + q ( k T h ) = ( k 1 ) T k T y ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equl_HTML.gif
Since q n ( k T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq44_HTML.gif, k Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq45_HTML.gif, we have
| ( k 1 ) T k T y ( s ) d s | = | ( k 1 ) T k T y ( s ) d s [ q n ( k T ) q n ( ( k 1 ) T ) ] | = | ( k 1 ) T k T [ y ( s ) q n ( s ) ] d s | ( k 1 ) T k T | y ( s ) q n ( s ) | d s T 1 2 [ ( k 1 ) T k T | y ( s ) q n ( s ) | 2 d s ] 1 2 T 1 2 [ R | y ( s ) q n ( s ) | 2 d s ] 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equm_HTML.gif

which implies that ( k 1 ) T k T y ( s ) d s = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq46_HTML.gif, that is, q ( k T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq47_HTML.gif, k Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq45_HTML.gif. Therefore, q is continuous. Thus, q C ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq48_HTML.gif and q = y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq49_HTML.gif.

Noticing that, for k T t < ( k + 1 ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq50_HTML.gif, we have
| q ( t ) | 2 = | k T t y ( s ) d s | 2 [ k T ( k + 1 ) T | y ( s ) | d s ] 2 T k T ( k + 1 ) T | y ( s ) | 2 d s = T k T + | y ( s ) | 2 d s T ( k + 1 ) T + | y ( s ) | 2 d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equn_HTML.gif
which implies q ( ± ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq51_HTML.gif. On the other hand, since
j = + | q ( t j ) | 2 = l = 0 p 1 k = + | q ( t l + k p ) | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equo_HTML.gif
and k T < t l + k p = t l + k T < ( k + 1 ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq52_HTML.gif ( l = 0 , 1 , , p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq12_HTML.gif), we have
| q ( t l + k p ) | 2 = | k T t l + k p y ( s ) d s | 2 T k T ( k + 1 ) T | y ( s ) | 2 d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equp_HTML.gif
Therefore,
j = + | q ( t j ) | 2 l = 0 p 1 k = + T k T ( k + 1 ) T | y ( s ) | 2 d s = p T R | y ( s ) | 2 d s < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equq_HTML.gif

Consequently, q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif and { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif converges to q in H. The proof is complete. □

Lemma 2.3 For any q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif, the following inequalities hold:
| q | : = sup t R | q ( t ) | T 1 2 q H , | q | 2 : = [ R | q ( t ) | 2 d t ] 1 2 T q H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equr_HTML.gif
Furthermore, q H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq54_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equs_HTML.gif
Proof For any t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif, there exists an integer k such that ( k 1 ) T t < k T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq55_HTML.gif. Then it follows from Cauchy-Schwarz inequality that
| q ( t ) | = | q ( k T ) q ( t ) | t k T | q ( s ) | d s ( k 1 ) T k T | q ( s ) | d s T 1 2 ( ( k 1 ) T k T | q ( s ) | 2 d s ) 1 2 T 1 2 q H , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equt_HTML.gif

which implies | q | T 1 2 q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq56_HTML.gif.

Furthermore, from the above argument, we have
k = + ( k 1 ) T k T | q ( t ) | 2 d t T 2 k = + ( k 1 ) T k T | q ( t ) | 2 d t = T 2 q H 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equu_HTML.gif

that is, | q | 2 T q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq57_HTML.gif.

Since
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equv_HTML.gif
Finally, we obtain that
j = + | q ( t j ) | 2 = l = 0 p 1 k = + | q ( t l + k T ) | 2 = l = 0 p 1 k = + | t l + k T ( k + 1 ) T q ( s ) d s | 2 l = 0 p 1 k = + [ k T ( k + 1 ) T | q ( s ) | d s ] 2 p T k = + k T ( k + 1 ) T | q ( s ) | 2 d s = p T q H 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equw_HTML.gif

The proof is complete. □

Define the functional φ : H R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq58_HTML.gif as follows:
φ ( q ) = 1 2 R | q ( t ) | 2 d t R V ( t , q ( t ) ) d t + j = + 0 q ( t j ) I ( s ) d s , q H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ6_HTML.gif
(2.4)
Lemma 2.4 If (H1)-(H5) hold, then φ C 1 ( H , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq59_HTML.gif and
φ ( q ) , h = R q ( t ) h ( t ) d t R V ( t , q ( t ) ) h ( t ) d t + j = + I ( q ( t j ) ) h ( t j ) , h H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ7_HTML.gif
(2.5)
Proof From the continuity of V, V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq60_HTML.gif and (H2)-(H3), we see that, for each γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq61_HTML.gif, there exists C γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq62_HTML.gif, such that
| V ( t , x ) | C γ | x | , | V ( t , x ) | 1 2 C γ | x | 2 , t R , | x | γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equx_HTML.gif
Since q ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq63_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq64_HTML.gif, there exists ρ γ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq65_HTML.gif such that
| q ( t ) | γ , whenever  | t | ρ γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equy_HTML.gif
Therefore, we have
| V ( t , q ( t ) ) | C γ | q ( t ) | , | V ( t , q ( t ) ) | 1 2 C γ | q ( t ) | 2 , for all  | t | ρ γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equz_HTML.gif
It follows from (H5) that, q , h H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq66_HTML.gif,
| j = + I ( q ( t j ) ) h ( t j ) | j = + | I ( q ( t j ) ) | | h ( t j ) | j = + b | q ( t j ) | | h ( t j ) | b ( j = + | q ( t j ) | 2 ) 1 2 ( j = + | h ( t j ) | 2 ) 1 2 < + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaa_HTML.gif
and
j = + | 0 q ( t j ) I ( s ) d s | j = + min { 0 , q ( t j ) } max { 0 , q ( t j ) } | I ( s ) | d s b 2 j = + | q ( t j ) | 2 < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ8_HTML.gif
(2.6)

Thus, φ and the right hand of (2.5) is well defined on H. By the definition of Fréchet derivative, it is easy to see that φ C 1 ( H , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq59_HTML.gif and (2.5) holds. □

Lemma 2.5 If q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif is a critical point of the functional φ, then q satisfies (1.1).

Proof If q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif is a critical point of the functional φ, then for any h C 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq67_HTML.gif, we have
0 = φ ( q ) , h = R q ( t ) h ( t ) d t R V ( t , q ( t ) ) h ( t ) d t + j = + I ( q ( t j ) ) h ( t j ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equab_HTML.gif
j Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq68_HTML.gif, take h C 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq67_HTML.gif such that h ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq69_HTML.gif for any t ( , t j ] [ t j + 1 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq70_HTML.gif, and h C 0 ( [ t j , t j + 1 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq71_HTML.gif. Therefore, we have
0 = t j t j + 1 q ( t ) h ( t ) d t t j t j + 1 V ( t , q ( t ) ) h ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equac_HTML.gif
by the definition of the weak derivative, which implies
q ( t ) + V ( t , q ( t ) ) = 0 a.e. on  ( t j , t j + 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ9_HTML.gif
(2.7)

Hence, the critical point q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif of the functional φ satisfies (1.1). The proof is complete. □

Lemma 2.6 Under the assumptions (H1)-(H5), there exists e H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq72_HTML.gif and r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq31_HTML.gif such that e H > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq73_HTML.gif and
b : = inf y H = r φ ( y ) > φ ( 0 ) φ ( e ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equad_HTML.gif
Proof If q H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif and q H 1 T 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq74_HTML.gif, then, by Lemma 2.3, | q | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq75_HTML.gif. Hence, by (H5) and Lemma 2.3, we have
j = + 0 q ( t j ) I ( s ) d s j = + min { 0 , q ( t j ) } max { 0 , q ( t j ) } | I ( s ) | d s 1 2 j = + b | q ( t j ) | 2 1 2 b T p q H 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ10_HTML.gif
(2.8)
and
j = + I ( q ( t j ) ) q ( t j ) j = + | I ( q ( t j ) ) | | q ( t j ) | j = + b | q ( t j ) | 2 b T p q H 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ11_HTML.gif
(2.9)
It follows from (2.8), (H4) and Lemma 2.3 that
φ ( q ) = 1 2 q H 2 R V ( t , q ( t ) ) d t + j = + 0 q ( t j ) I ( s ) d s 1 2 q H 2 a 1 R | q ( t ) | μ d t 1 2 b T p q H 2 1 2 q H 2 a 1 | q | μ 2 R | q ( t ) | 2 d t 1 2 b T p q H 2 1 2 ( 1 b T p ) q H 2 a 1 T μ + 2 2 q H μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equae_HTML.gif

Therefore, as μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq20_HTML.gif and b < μ 2 ( μ + 2 ) T p < 1 T p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq76_HTML.gif, there exists r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq31_HTML.gif such that inf q H = r φ ( q ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq77_HTML.gif.

Now, let v H { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq78_HTML.gif and λ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq79_HTML.gif. Then there exists a subset ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq80_HTML.gif of and λ large enough such that
λ | v ( t ) | > 1 , for all  t ( a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaf_HTML.gif
Since V ( t , λ v ( t ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq81_HTML.gif, by (2.4), (H4) and Lemma 2.3, we have
φ ( λ v ) λ 2 2 R | v ( t ) | 2 d t a b V ( t , λ v ( t ) ) d t + j = + 0 λ v ( t j ) I ( s ) d s λ 2 2 v H 2 a 0 λ μ a b | v ( t ) | μ d t + λ 2 2 b T p v H 2 = λ 2 2 ( 1 + b T p ) v H 2 a 0 λ μ a b | v ( t ) | μ d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equag_HTML.gif

Since μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq20_HTML.gif, the right-hand member is negative of λ sufficiently large, and there exists e : = λ v H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq82_HTML.gif such that e H > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq73_HTML.gif, φ ( e ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq83_HTML.gif. The proof is complete. □

Lemma 2.7 Under the assumptions (H1)-(H5), there exists a bounded sequence { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif in H such that
φ ( q n ) d , φ ( q n ) 0 , dist ( q n , H ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equah_HTML.gif

where d : = inf γ Γ sup t [ 0 , 1 ] φ ( γ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq84_HTML.gif, Γ = { γ C ( [ 0 , 1 ] , H ) : γ ( 0 ) = 0 , γ ( 1 ) = e } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq85_HTML.gif. Furthermore, q n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq86_HTML.gif does not converge to 0 in measure.

Proof All we have to prove is that any sequence { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif obtained by taking ε = 1 / n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq87_HTML.gif and δ = 1 / n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq88_HTML.gif in Lemma 2.1 is bounded and q n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq86_HTML.gif does not converge to 0 in measure. For n sufficiently large, it follows from (H3), (H5), (2.4), (2.5), (2.8) and (2.9) that
d + 1 + q n H φ ( q n ) 1 μ φ ( q n ) , q n = ( 1 2 1 μ ) R | q n ( t ) | 2 d t R [ V ( t , q n ( t ) ) 1 μ V ( t , q n ( t ) ) q n ( t ) ] d t + j = + 0 q n ( t j ) I ( s ) d s 1 μ j = + I ( q n ( t j ) ) q n ( t j ) = ( 1 2 1 μ ) q n H 2 1 μ R [ μ V ( t , q n ( t ) ) V ( t , q n ( t ) ) q n ( t ) ] d t + j = + 0 q n ( t j ) I ( s ) d s 1 μ j = + I ( q n ( t j ) ) q n ( t j ) ( 1 2 1 μ ) q n H 2 b T p 2 q n H 2 b T p μ q n H 2 = ( 1 2 1 μ b T p 2 b T p μ ) q n H 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equai_HTML.gif

Since b < μ 2 ( μ + 2 ) T p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq89_HTML.gif, { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif is bounded in H.

Let a 2 : = sup n N { q n H } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq90_HTML.gif. By (H2) and (H3), we have
1 2 V ( t , u ) u V ( t , u ) = o ( u 2 ) , as  u 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaj_HTML.gif
which implies
a 3 : = sup | u | T 1 2 a 2 1 2 V ( t , u ) u V ( t , u ) u 2 < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equak_HTML.gif
For any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq34_HTML.gif such that, for | u | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq91_HTML.gif, we have
| 1 2 V ( t , u ) u V ( t , u ) | ε u 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equal_HTML.gif
Therefore, by Lemma 2.3, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ12_HTML.gif
(2.10)
If q n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq86_HTML.gif converges to 0 in measure on R, then it follows from (H5) and (2.10) that
0 < d = φ ( q n ) 1 2 φ ( q n ) , q n + o ( 1 ) = R [ 1 2 V ( t , q n ) q n V ( t , q n ) ] d t + j = + 0 q n ( t j ) I ( s ) d s 1 2 j = + I ( q n ( t j ) ) q n ( t j ) + o ( 1 ) meas { | q n ( t ) | > δ } T a 2 2 a 3 + ε T 2 a 2 2 + 1 2 j = + [ 2 0 q n ( t j ) I ( s ) d s I ( q n ( t j ) ) q n ( t j ) ] + o ( 1 ) meas { | q n ( t ) | > δ } T a 2 2 a 3 + ε T 2 a 2 2 + o ( 1 ) = o ( 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equam_HTML.gif

a contradiction. The proof is complete. □

The following lemma is similar to a weak version of Lieb’s lemma [15], which will play an important role in the proof of Theorem 1.1.

Lemma 2.8 If { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq92_HTML.gif is bounded in H and u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq93_HTML.gif does not converge to 0 in measure, then there exist a sequence { x n k } Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq94_HTML.gif and a subsequence { u n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq95_HTML.gif of { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq96_HTML.gif such that
u n k ( + x n k T ) u 0 in H 1 ( R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equan_HTML.gif
Proof If
lim n sup q Z sup t [ q T T , q T + T ] | u n ( t ) | = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equao_HTML.gif
then, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, there exists n 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq97_HTML.gif such that, for n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq98_HTML.gif, we have
sup q Z sup t [ q T T , q T + T ] | u n ( t ) | ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equap_HTML.gif
Therefore, for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif and n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq98_HTML.gif, we have
| u n ( t ) | ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaq_HTML.gif
which implies
lim n meas { t R : | u n ( t ) | > ε } = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equar_HTML.gif
a contradiction. Therefore, there exist a constant ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq99_HTML.gif and a subsequence { n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq100_HTML.gif of { n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq101_HTML.gif such that
sup x Z sup t [ x T T , x T + T ] | u n k ( t ) | > ρ , k N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equas_HTML.gif
where denotes the set of all positive integers. So, for k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq102_HTML.gif, there exists x n k Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq103_HTML.gif such that
sup t [ x n k T T , x n k T + T ] | u n k ( t ) | > ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equat_HTML.gif
Let v n k ( t ) = u n k ( t + x n k T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq104_HTML.gif, t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif. Since { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq92_HTML.gif is bounded in H, by Lemma 2.3, it is easy to see that { v n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq105_HTML.gif is bounded in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif. Therefore, { v n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq107_HTML.gif has a subsequence which weakly converges to u in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif. Without loss of generality, we assume that v n k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq108_HTML.gif in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif. Thus, v n k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq109_HTML.gif in H 1 ( [ T , T ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq110_HTML.gif. Therefore, v n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq111_HTML.gif uniformly converges to u in [ T , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq112_HTML.gif. Noticing that
sup t [ T , T ] | v n k ( t ) | = sup t [ T , T ] | u n k ( t + x n k T ) | = sup t [ x n k T T , x n k T + T ] | u n k ( t ) | > ρ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equau_HTML.gif
we have
sup t [ T , T ] | u ( t ) | ρ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equav_HTML.gif

that is, u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq113_HTML.gif. □

Proof of Theorem 1.1 By Lemma 2.7, there exists a bounded { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq114_HTML.gif in H such that
φ ( q n ) d , φ ( q n ) 0 , dist ( q n , H ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaw_HTML.gif
and { q n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq40_HTML.gif does not converge to 0 in measure on , where d is the mountain pass value. By Lemma 2.8, there exists a sequence { x n k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq115_HTML.gif in such that
ω k : = q n k ( + x n k T ) ω 0 in  H 1 ( R ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equax_HTML.gif
For any fixed k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq102_HTML.gif, set s = t + x n k T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq116_HTML.gif and h k ( s ) : = h ( s x n k T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq117_HTML.gif. Then s j : = t j + x n k T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq118_HTML.gif ( j Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq8_HTML.gif) are impulsive points and
h k H = ( R | h k ( s ) | 2 d s ) 1 2 = ( R | h ( s ) | 2 d s ) 1 2 = h H . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equay_HTML.gif
For any h C 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq67_HTML.gif with h ( k T ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq119_HTML.gif, we have
φ ( ω k ) , h = R ω k ( t ) h ( t ) d t R V ( t , ω k ( t ) ) h ( t ) d t + j = + I ( ω k ( t j ) ) h ( t j ) = R [ q n k ( t + x n k T ) h ( t ) V ( t , q n k ( t + x n k T ) ) h ( t ) ] d t + j = + I ( q n k ( t j + x n k T ) ) h ( t j ) = R [ q n k ( s ) h ( s x n k T ) V ( s x n k T , q n k ( s ) ) h ( s x n k T ) ] d s + j = + I ( q n k ( s j ) ) h ( s j x n k T ) = R [ q n k ( s ) h ( s x n k T ) V ( s , q n k ( s ) ) h ( s x n k T ) ] d s + j = + I ( q n k ( s j ) ) h ( s j x n k T ) = R [ q n k ( s ) h k ( s ) V ( s , q n k ( s ) ) h k ( s ) ] d s + j = + I ( q n k ( s j ) ) h k ( s j ) = φ ( q n k ) , h k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaz_HTML.gif
Hence, we have
| φ ( ω k ) , h | = | φ ( q n k ) , h k | φ ( q n k ) h k H = φ ( q n k ) h H , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equba_HTML.gif
which implies
φ ( ω k ) , h 0 as  k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ13_HTML.gif
(2.11)
Since H H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq120_HTML.gif, ω k ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq121_HTML.gif in H, therefore
R ω k h R ω h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ14_HTML.gif
(2.12)
As ω k ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq122_HTML.gif in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif, { ω k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq123_HTML.gif is bounded in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif and hence | ω k | c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq124_HTML.gif for some c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq125_HTML.gif and all k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq102_HTML.gif. Also, { ω k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq126_HTML.gif uniformly converges to ω on supp ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq127_HTML.gif and, V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq60_HTML.gif being uniformly continuous on supp ( h ) × [ c , c ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq128_HTML.gif, V ( t , ω k ) h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq129_HTML.gif uniformly converges to V ( t , ω ) h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq130_HTML.gif on supp ( h ) × [ c , c ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq128_HTML.gif. By the Lebesgue dominated convergence theorem, this implies that
R V ( t , ω k ) h R V ( t , ω ) h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ15_HTML.gif
(2.13)
For any h H https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq131_HTML.gif and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, take J 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq132_HTML.gif sufficiently large such that
( j = J 0 + 1 + | h ( t j ) | 2 ) 1 2 ε , ( j = J 0 1 | h ( t j ) | 2 ) 1 2 ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbb_HTML.gif
Since ω k ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq122_HTML.gif in H 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif, ω k ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq122_HTML.gif in H 1 ( [ t J 0 , t J 0 ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq133_HTML.gif, therefore ω k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq134_HTML.gif uniformly converges to ω in [ t J 0 , t J 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq135_HTML.gif. By the continuity of I, there exists K > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq136_HTML.gif such that, when k > K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq137_HTML.gif, we have
| j = J 0 J 0 [ I ( ω k ( t j ) ) I ( ω ( t j ) ) ] h ( t j ) | ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbc_HTML.gif
Since
| I ( ω k ( t j ) ) | b | ω k ( t j ) | , | I ( ω ( t j ) ) | b | ω ( t j ) | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbd_HTML.gif
it follows from Lemma 2.3 that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Eqube_HTML.gif
Similarly, we have
( j = J 0 1 [ I ( ω k ( t j ) ) I ( ω ( t j ) ) ] 2 ) 1 2 2 b T p max { sup k ω k H , ω H } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbf_HTML.gif
By the Cauchy-Schwarz inequality, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbg_HTML.gif
Therefore,
lim k j = + I ( ω k ( t j ) ) h ( t j ) = j = + I ( ω ( t j ) ) h ( t j ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ16_HTML.gif
(2.14)
From (2.11)-(2.14), we have
φ ( ω ) , h = lim k φ ( ω k ) , h = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbh_HTML.gif

Thus, φ ( ω ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq138_HTML.gif and ω is a nontrivial weak homoclinic orbit of (1.1)-(1.2). □

Declarations

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 10971085). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, Kunming University of Science and Technology

References

  1. Fečkan M: Chaos in singularly perturbed impulsive O.D.E. Boll. Unione Mat. Ital, B 1996, 10: 175–198.
  2. Battelli F, Fečkan M: Chaos in singular impulsive O.D.E. Nonlinear Anal. 1997, 28: 655–671. 10.1016/0362-546X(95)00182-UMathSciNetView Article
  3. Tang XH, Xiao L: Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 2009, 351: 586–594. 10.1016/j.jmaa.2008.10.038MathSciNetView Article
  4. Tang XH, Xiao L: Homoclinic solutions for a class of second-order Hamiltonian systems. Nonlinear Anal. 2009, 71: 1140–1152. 10.1016/j.na.2008.11.038MathSciNetView Article
  5. Tang XH, Lin XY: Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 2009, 354: 539–549. 10.1016/j.jmaa.2008.12.052MathSciNetView Article
  6. Tang XH, Lin XY: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb. A 2011, 141: 1103–1119. 10.1017/S0308210509001346MathSciNetView Article
  7. Nieto J, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680–690. 10.1016/j.nonrwa.2007.10.022MathSciNetView Article
  8. Zhang H, Li ZX: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal., Real World Appl. 2011, 1: 39–51.View Article
  9. Han X, Zhang H: Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system. J. Comput. Appl. Math. 2011, 235: 1531–1541. 10.1016/j.cam.2010.08.040MathSciNetView Article
  10. Chen H, Sun J: An application of variational method to second-order impulsive differential equation on the half-line. Appl. Math. Comput. 2010, 217: 1863–1869. 10.1016/j.amc.2010.06.040MathSciNetView Article
  11. 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 Anal. 2010, 73: 440–449. 10.1016/j.na.2010.03.035MathSciNetView Article
  12. Luo Z, Xiao J, Xu J: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations. Nonlinear Anal. 2012, 75: 2249–2255. 10.1016/j.na.2011.10.023MathSciNetView Article
  13. Smets D, Willem M: Solitary waves with prescribed speed on infinite lattices. J. Funct. Anal. 1997, 149: 266–275. 10.1006/jfan.1996.3121MathSciNetView Article
  14. Brezis H, Nirenberg L: Remarks on finding critical points. Commun. Pure Appl. Math. 1991, 64: 939–963.MathSciNetView Article
  15. Lieb EH: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 1983, 74: 441–448. 10.1007/BF01394245MathSciNetView Article

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© Fang and Duan; licensee Springer. 2012

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