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

  • Hui Fang1Email author and

    Affiliated with

    • Hongbo Duan1

      Affiliated with

      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:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equ1_HTML.gif
      (1.1)
      http://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 http://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 http://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 http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq5_HTML.gif with I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq6_HTML.gif. ℤ denotes the set of all integers, and t j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq7_HTML.gif ( j Z http://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 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq10_HTML.gif, k Z http://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 http://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 ) http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq15_HTML.gif at t = t j http://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 ) http://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 } . http://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 ; http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq18_HTML.gif uniformly for t R http://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 http://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 } ; http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq21_HTML.gif and a 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq22_HTML.gif such that
      http://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 http://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 | , http://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 . http://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 http://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 http://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 http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq27_HTML.gif, I ( x ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq29_HTML.gif, e E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq30_HTML.gif, r > 0 http://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 http://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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equg_HTML.gif
      Let
      http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq34_HTML.gif, there exists y E http://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 http://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 http://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 http://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 http://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 ℤ (the set of all integers), that is,
      j Z | a j | 2 < + , a = { a j } j = + l 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equi_HTML.gif
      The space l 2 http://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 . http://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 } http://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 , http://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 . http://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 } http://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 } http://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 ) http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq43_HTML.gif such that { q n } http://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 ) http://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 ) http://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 . http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq44_HTML.gif, k Z http://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 , http://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 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq47_HTML.gif, k Z http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq48_HTML.gif and q = y http://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 http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equn_HTML.gif
      which implies q ( ± ) = 0 http://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 , http://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 http://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 http://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 . http://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 < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equq_HTML.gif

      Consequently, q H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq53_HTML.gif and { q n } http://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 http://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 . http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq54_HTML.gif and
      http://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 http://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 http://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 , http://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 http://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 , http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq57_HTML.gif.

      Since
      http://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 . http://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 http://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 . http://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 ) http://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 . http://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 http://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 http://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 http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equx_HTML.gif
      Since q ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq63_HTML.gif as t http://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 http://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 . http://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 . http://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 http://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 < + , http://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 < + . http://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 ) http://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 http://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 http://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 ) http://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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equab_HTML.gif
      j Z http://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 ) http://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 http://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 , + ) http://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 ] ) http://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 , http://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 ) . http://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 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq72_HTML.gif and r > 0 http://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 http://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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equad_HTML.gif
      Proof If q H http://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 http://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 http://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 , http://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 . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equae_HTML.gif

      Therefore, as μ > 2 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq76_HTML.gif, there exists r > 0 http://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 http://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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq78_HTML.gif and λ > 1 http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq80_HTML.gif of ℝ and λ large enough such that
      λ | v ( t ) | > 1 , for all  t ( a , b ) . http://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 http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equag_HTML.gif

      Since μ > 2 http://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 http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq73_HTML.gif, φ ( e ) 0 http://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 } http://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 , http://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 ) ) http://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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq85_HTML.gif. Furthermore, q n http://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 } http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq87_HTML.gif and δ = 1 / n http://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 http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq89_HTML.gif, { q n } http://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 } http://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 , http://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 < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equak_HTML.gif
      For any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, there exists δ > 0 http://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 http://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 . http://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
      http://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 http://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 ) , http://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 } http://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 http://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 http://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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq95_HTML.gif of { u n } http://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 ) . http://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 , http://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 http://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 http://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 http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif and n n 0 http://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 , http://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 , http://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 http://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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq100_HTML.gif of { n } http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equas_HTML.gif
      where ℕ denotes the set of all positive integers. So, for k N http://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 http://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 . http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq104_HTML.gif, t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq19_HTML.gif. Since { u n } http://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 } http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif. Therefore, { v n k } http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq108_HTML.gif in H 1 ( R ) http://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 http://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 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq110_HTML.gif. Therefore, v n k http://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 ] http://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 , http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equav_HTML.gif

      that is, u 0 http://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 } http://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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equaw_HTML.gif
      and { q n } http://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 ℝ, where d is the mountain pass value. By Lemma 2.8, there exists a sequence { x n k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq115_HTML.gif in ℤ such that
      ω k : = q n k ( + x n k T ) ω 0 in  H 1 ( R ) . http://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 http://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 http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq118_HTML.gif ( j Z http://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 . http://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 ) http://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 http://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 . http://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 , http://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 . http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq120_HTML.gif, ω k ω http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq122_HTML.gif in H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif, { ω k } http://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 ) http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq124_HTML.gif for some c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq125_HTML.gif and all k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq102_HTML.gif. Also, { ω k } http://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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq127_HTML.gif and, V http://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 ] http://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 http://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 http://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 ] http://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 . http://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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq131_HTML.gif and ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq33_HTML.gif, take J 0 http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbb_HTML.gif
      Since ω k ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq122_HTML.gif in H 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq106_HTML.gif, ω k ω http://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 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_IEq133_HTML.gif, therefore ω k http://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 ] http://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 http://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 http://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 . http://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 ) | , http://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
      http://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 } . http://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
      http://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 ) . http://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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-138/MediaObjects/13661_2012_Article_299_Equbh_HTML.gif

      Thus, φ ( ω ) = 0 http://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

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      Copyright

      © Fang and Duan; licensee Springer. 2012

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