Open Access

Solvability of a second-order Hamiltonian system with impulsive effects

Boundary Value Problems20132013:151

DOI: 10.1186/1687-2770-2013-151

Received: 24 April 2013

Accepted: 30 May 2013

Published: 25 June 2013

Abstract

In this paper, a class of second-order Hamiltonian systems with impulsive effects are considered. By using critical point theory, we obtain some existence theorems of solutions for the nonlinear impulsive problem. We extend and improve some recent results.

MSC:334B18, 34B37, 58E05.

Keywords

Hamiltonian system impulsive critical point theory

1 Introduction and main results

Consider the second-order Hamiltonian systems with impulsive effects
{ u ¨ ( t ) = F ( t , u ( t ) ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , Δ u ˙ i ( t j ) = I i j ( u i ( t j ) ) , i = 1 , 2 , , N ; j = 1 , 2 , , p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ1_HTML.gif
(1.1)
where 0 = t 0 < t 1 < t 2 < < t p < t p + 1 = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq1_HTML.gif, u ( t ) = ( u 1 ( t ) , u 2 ( t ) , , u N ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq2_HTML.gif, I i j : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq3_HTML.gif ( i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif) are continuous and F : [ 0 , T ] × R n R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq6_HTML.gif satisfies the following assumption:
  1. (A)
    F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq7_HTML.gif is measurable in t for every x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq8_HTML.gif and continuously differentiable in x for a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif and there exist a C ( R + , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq10_HTML.gif, b L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq11_HTML.gif such that
    | F ( t , x ) | a ( | x | ) b ( t ) , | F ( t , x ) | a ( | x | ) b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equa_HTML.gif
     

for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq12_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq13_HTML.gif.

When I i j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq14_HTML.gif ( i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif), (1.1) is the Hamiltonian system
{ u ¨ ( t ) = F ( t , u ( t ) ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ2_HTML.gif
(1.2)

In the past years, the existence of solutions for the second-order Hamiltonian systems (1.2) has been studied extensively via modern variational methods by many authors (see [113]).

When the gradient F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq15_HTML.gif is bounded, that is, there exists g L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq16_HTML.gif such that
| F ( t , x ) | g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equb_HTML.gif
for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq17_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif, Mawhin-Willem in [1] proved the existence of solutions for problem (1.2) under the condition
lim | x | + 0 T F ( t , x ) d t = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equc_HTML.gif
or
lim | x | + 0 T F ( t , x ) d t = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equd_HTML.gif
When the gradient F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq15_HTML.gif is bounded sublinearly, that is, there exist f , g L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq18_HTML.gif and α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq19_HTML.gif such that
| F ( t , x ) | f ( t ) | x | α + g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Eque_HTML.gif
for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq17_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif, Tang [2] proved the existence of solutions for problem (1.2) under the condition
lim | x | + | x | 2 α 0 T F ( t , x ) d t = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equf_HTML.gif
or
lim | x | + | x | 2 α 0 T F ( t , x ) d t = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equg_HTML.gif

which generalizes Mawhin-Willem’s results.

For I i j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq20_HTML.gif, problem (1.1) gives less results (see [1416]). In [14], Zhou and Li extended the results of [2] to impulsive problem (1.1); they proved the following theorems.

Theorem A [14]

Assume that (A) and the following conditions are satisfied:

(h1) There exist f , g L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq21_HTML.gif and α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq22_HTML.gif such that
| F ( t , x ) | f ( t ) | x | α + g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equh_HTML.gif

for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq12_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq13_HTML.gif.

(h2) lim | x | + | x | 2 α 0 T F ( t , x ) d t = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq23_HTML.gif.

(h3) For any i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif,
t I i j ( t ) 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equi_HTML.gif

Then problem (1.1) has at least one weak solution.

Theorem B [14]

Suppose that (A) and the condition (h1) of Theorem  A hold. Assume that:

(h4) lim | x | + | x | 2 α 0 T F ( t , x ) d t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq24_HTML.gif.

(h5) For any i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif,
t I i j ( t ) 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equj_HTML.gif
(h6) There exist a i j , b i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq25_HTML.gif and β i j ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq26_HTML.gif such that
| I i j ( t ) | a i j + b i j | t | α β i j , t R , i = 1 , 2 , , N ; j = 1 , 2 , , p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equk_HTML.gif

Then problem (1.1) has at least one weak solution.

Let
F ( t , x ) = f ( t ) | x | 1 + α + C ( x ) λ 2 | x | 2 + ( h ( t ) , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ3_HTML.gif
(1.3)

where C ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq27_HTML.gif is convex in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq28_HTML.gif (e.g., C ( x ) = λ 2 ( | x 1 | 4 + | x 2 | 2 + + | x N | 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq29_HTML.gif), λ < 4 π 2 T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq30_HTML.gif, f L 1 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq31_HTML.gif satisfying 0 T f ( t ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq32_HTML.gif (e.g., f ( t ) = 2 T 3 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq33_HTML.gif), 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq34_HTML.gif, h L 1 ( 0 , T ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq35_HTML.gif and x = ( x 1 , x 2 , , x N ) R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq36_HTML.gif. It is easy to see that F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq7_HTML.gif satisfies the condition (h2) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem A.

Let
F ( t , x ) = f ( t ) | x | 1 + α + C ( x ) λ 2 | x | 2 + ( h ( t ) , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ4_HTML.gif
(1.4)

where C ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq27_HTML.gif satisfies that the gradient C ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq37_HTML.gif is Lipschitz continuous and monotone in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq28_HTML.gif (e.g., C ( x ) = λ 2 ( | x 1 | + | x 2 | 2 + + | x N | 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq38_HTML.gif), 0 < λ < 4 π 2 T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq39_HTML.gif, f L 1 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq31_HTML.gif satisfies 0 T f ( t ) d t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq32_HTML.gif (e.g., f ( t ) = T 3 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq40_HTML.gif), 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq41_HTML.gif, h L 1 ( 0 , T ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq42_HTML.gif and x = ( x 1 , x 2 , , x N ) R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq43_HTML.gif. It is easy to see that F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq7_HTML.gif satisfies the condition (h4) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem B.

In this paper, we further study the existence of solutions for impulsive problem (1.1). Our main results are the following theorems.

Theorem 1.1 Suppose that F ( t , x ) = H ( t , x ) + G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq44_HTML.gif satisfies assumption (A) and the following conditions hold:

(H1) There exist f , g L 1 ( 0 , T ; R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq21_HTML.gif and α [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq22_HTML.gif such that
| H ( t , x ) | f ( t ) | x | α + g ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ5_HTML.gif
(1.5)

for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq17_HTML.gif and a.e. t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif.

(H2) There exists a positive number λ < 4 π 2 T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq30_HTML.gif such that
( G ( x ) G ( y ) , x y ) λ | x y | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ6_HTML.gif
(1.6)

for all x , y R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq45_HTML.gif.

(H3)
lim | x | + | x | 2 α 0 T F ( t , x ) d t = + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ7_HTML.gif
(1.7)
(H4) For any i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif,
t I i j ( t ) 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ8_HTML.gif
(1.8)

Then impulsive problem (1.1) has at least one weak solution.

Remark 1.1 Theorem 1.1 generalizes Theorem A, which is a special case of our Theorem 1.1 corresponding to G ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq46_HTML.gif.

Example 1.1 Let N = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq47_HTML.gif, p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq48_HTML.gif. Consider the following impulsive problem:
{ u ¨ ( t ) = F ( t , u ( t ) ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , Δ u ˙ i ( t 1 ) = I i 1 ( u i ( t 1 ) ) , i = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equl_HTML.gif
where
F ( t , x ) = ( 2 T 3 t ) | x | 3 2 λ cos x 1 + ( h ( t ) , x ) , h L 1 ( 0 , T ; R 3 ) , I i 1 ( t ) = t 1 3 ( i = 1 , 2 , 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equm_HTML.gif
Take
G ( x ) = λ cos x 1 , x = ( x 1 , x 2 , x 3 ) R 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equn_HTML.gif
which is bounded and
H ( t , x ) = ( 2 T 3 t ) | x | 3 2 + ( h ( t ) , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equo_HTML.gif

α = 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq49_HTML.gif, f ( t ) = 3 2 ( 2 T 3 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq50_HTML.gif, g ( t ) = | h ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq51_HTML.gif. Then all the conditions of Theorem 1.1 are satisfied. According to Theorem 1.1, the above problem has at least one weak solution. However, F does not satisfy the condition (h1) in Theorem A. Therefore, our result improves and generalizes the Theorem A.

Theorem 1.2 Suppose that F ( t , x ) = H ( t , x ) + G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq44_HTML.gif satisfies assumption (A) and the condition (H1) of Theorem  1.1. Furthermore, assume that

(H5) There exist 0 A < 4 π 2 T 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq52_HTML.gif, B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq53_HTML.gif such that
| G ( x ) G ( y ) | A | x y | + B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ9_HTML.gif
(1.9)

for all x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq12_HTML.gif.

(H6)
lim | x | + | x | 2 α 0 T F ( t , x ) d t = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ10_HTML.gif
(1.10)
(H7) For any i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif,
t I i j ( t ) 0 , t R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ11_HTML.gif
(1.11)
(H8) There exist a i j , b i j > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq25_HTML.gif and β i j ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq26_HTML.gif such that
| I i j ( t ) | a i j + b i j | t | α β i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ12_HTML.gif
(1.12)

for every t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq54_HTML.gif, i = 1 , 2 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq4_HTML.gif; j = 1 , 2 , , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq5_HTML.gif.

Then impulsive problem (1.1) has at least one weak solution.

Remark 1.2 Theorem 1.2 generalizes Theorem B, which is a special case of our Theorem 1.2 corresponding to G ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq46_HTML.gif.

Example 1.2 Let N = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq47_HTML.gif, p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq48_HTML.gif. Consider the following impulsive problem:
{ u ¨ ( t ) = F ( t , u ( t ) ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , Δ u ˙ i ( t 1 ) = I i 1 ( u i ( t 1 ) ) , i = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equp_HTML.gif
where
F ( t , x ) = ( T 3 t ) | x | 4 3 λ 2 | x 1 | 2 + ( h ( t ) , x ) , h L 1 ( 0 , T ; R 3 ) , I i 1 ( t ) = t 1 9 ( i = 1 , 2 , 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equq_HTML.gif
Take
G ( x ) = λ 2 | x 1 | 2 , x = ( x 1 , x 2 , x 3 ) R 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equr_HTML.gif
which is bounded from above, and
H ( t , x ) = ( T 3 t ) | x | 4 3 + ( h ( t ) , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equs_HTML.gif

α = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq55_HTML.gif, f ( t ) = 4 3 ( T 3 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq56_HTML.gif, g ( t ) = | h ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq51_HTML.gif, a i 1 = b i 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq57_HTML.gif, β i 1 = 1 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq58_HTML.gif ( i = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq59_HTML.gif). Then all the conditions of Theorem 1.2 are satisfied. According to Theorem 1.2, the above problem has at least one weak solution. However, F does not satisfy the condition (h4) in Theorem B. Therefore, our result improves and generalizes Theorem B.

Theorem 1.3 Suppose that F ( t , x ) = H ( t , x ) + G ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq44_HTML.gif and I i j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq60_HTML.gif satisfy the assumptions (A), (H1), (H2), (H7) and (H8). Furthermore, assume that

(H9)
lim | x | + | x | 2 α F ( t , x ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ13_HTML.gif
(1.13)

uniformly for all t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif.

Then impulsive problem (1.1) has at least one weak solution.

Example 1.3 Let N = 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq47_HTML.gif, p = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq48_HTML.gif. Consider the following impulsive problem:
{ u ¨ ( t ) = F ( t , u ( t ) ) , a.e.  t [ 0 , T ] , u ( 0 ) u ( T ) = u ˙ ( 0 ) u ˙ ( T ) = 0 , Δ u ˙ i ( t 1 ) = I i 1 ( u i ( t 1 ) ) , i = 1 , 2 , 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equt_HTML.gif
where
F ( t , x ) = | x | 1 + α λ 2 | x 1 | 2 , I i 1 ( t ) = t α 3 ( i = 1 , 2 , 3 ) , 0 < α < 1 , 0 < λ < 4 π 2 T 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equu_HTML.gif
Take
H ( t , x ) = | x | 1 + α , G ( x ) = λ 2 | x 1 | 2 , x = ( x 1 , x 2 , x 3 ) R 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equv_HTML.gif

Then all the conditions of Theorem 1.3 are satisfied. According to Theorem 1.3, the above problem has at least one weak solution. However, F ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq7_HTML.gif is neither superquadratic in X nor subquadratic in X.

2 Preliminaries

Let H T 1 = { u : [ 0 , T ] R N : u  is absolutely continuous , u ( 0 ) = u ( T )  and  u ˙ L 2 ( 0 , T ; R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq61_HTML.gif with the inner product
( u , v ) = 0 T ( u ( t ) , v ( t ) ) d t + 0 T ( u ˙ ( t ) , v ˙ ( t ) ) d t , u , v H T 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equw_HTML.gif
inducing the norm
u = ( 0 T | u ( t ) | 2 d t + 0 T | u ˙ ( t ) | 2 d t ) 1 2 , u H T 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equx_HTML.gif
The corresponding functional ϕ on H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq62_HTML.gif given by
ϕ ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T F ( t , u ( t ) ) d t + j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ14_HTML.gif
(2.1)
is continuously differentiable and weakly lower semi-continuous on H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq62_HTML.gif. For the sake of convenience, we denote
ϕ ( u ) = ϕ 1 ( u ) + ϕ 2 ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equy_HTML.gif
where
ϕ 1 ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T F ( t , u ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equz_HTML.gif
and
ϕ 2 ( u ) = j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaa_HTML.gif
For any u , v H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq63_HTML.gif, we have
ϕ ( u ) , v = 0 T ( u ˙ ( t ) , v ˙ ( t ) ) d t + j = 1 p i = 1 N I i j ( u i ( t j ) ) v i ( t j ) + 0 T ( F ( t , u ( t ) ) , v ( t ) ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ15_HTML.gif
(2.2)
Definition 2.1 We say that a function u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq64_HTML.gif is a weak solution of (1.1) if the identity
0 T ( u ˙ ( t ) , v ˙ ( t ) ) d t + j = 1 p i = 1 N I i j ( u i ( t j ) ) v i ( t j ) + 0 T ( F ( t , u ( t ) ) , v ( t ) ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equab_HTML.gif

holds for any v H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq65_HTML.gif.

It is well known that the solutions of impulsive problem (1.1) correspond to the critical point of ϕ.

Definition 2.2 [1]

Let X be a Banach space, ϕ C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq66_HTML.gif and c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq67_HTML.gif.
  1. (1)

    ϕ is said to satisfy the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq68_HTML.gif-condition on X if the existence of a sequence { x n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq69_HTML.gif such that ϕ ( x n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq70_HTML.gif and ϕ ( x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq71_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq72_HTML.gif implies that c is a critical value of ϕ.

     
  2. (2)

    ϕ is said to satisfy the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition on X if any sequence { x n } X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq69_HTML.gif for which ϕ ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq74_HTML.gif is bounded and ϕ ( x n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq71_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq72_HTML.gif possesses a convergent subsequence in X.

     

Remark 2.1 It is clear that the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition implies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq68_HTML.gif-condition for each c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq67_HTML.gif.

Lemma 2.1 [1]

If ϕ is weakly lower semi-continuous on a reflexive Banach space X (i.e., if u k u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq75_HTML.gif, then lim inf k ϕ ( u k ) ϕ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq76_HTML.gif) and has a bounded minimizing sequence, then ϕ has a minimum on X.

Remark 2.2 The existence of a bounded minimizing sequence will be in particular ensured when ϕ is coercive, i.e., such that
ϕ ( u ) + if  u + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equac_HTML.gif

Lemma 2.2 [1]

Let X be a Banach space and let ϕ C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq77_HTML.gif. Assume that X splits into a direct sum of closed subspaces X = X X + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq78_HTML.gif with dim X < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq79_HTML.gif and
sup S R ϕ < inf X + ϕ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equad_HTML.gif
where
S R = { u X : u = R } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equae_HTML.gif
Let
B R = { u X : u R } , M = { h C ( B R , X ) : h ( s ) = s if s B R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaf_HTML.gif
and
c = inf h M max s B R ϕ ( h ( s ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equag_HTML.gif

Then if ϕ satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq68_HTML.gif-condition, c is a critical value of ϕ.

Lemma 2.3 [1]

If the sequence { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq80_HTML.gif converges weakly to u in H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq62_HTML.gif, then { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq80_HTML.gif converges uniformly to u on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq81_HTML.gif.

Lemma 2.4 [1]

If u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif and 0 T u ( t ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq83_HTML.gif, then
0 T | u ( t ) | 2 d t T 2 4 π 2 0 T | u ˙ ( t ) | 2 d t ( Wirtinger’s inequality ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equah_HTML.gif
and
u 2 T 12 0 T | u ˙ ( t ) | 2 d t ( Sobolev’s inequality ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equai_HTML.gif
Lemma 2.5 There exists C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq84_HTML.gif such that if u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif, then
u C u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaj_HTML.gif
Moreover, if 0 T u ( t ) d t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq83_HTML.gif, then
u C u ˙ L 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equak_HTML.gif

where u = max t [ 0 , T ] | u ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq85_HTML.gif and u L 2 = ( 0 T | u ( t ) | 2 d t ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq86_HTML.gif.

3 Proof of main results

For u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif, let u ¯ = 1 T 0 T u ( t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq87_HTML.gif and u ˜ ( t ) = u ( t ) u ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq88_HTML.gif.

Proof of Theorem 1.1 It follows from (H1) and Sobolev’s inequality that
| 0 T [ H ( t , u ( t ) ) H ( t , u ¯ ) ] d t | = | 0 T 0 1 ( H ( t , u ¯ + s u ˜ ( t ) ) , u ˜ ( t ) ) d s d t | 0 T 0 1 f ( t ) | u ¯ + s u ˜ ( t ) | α | u ˜ ( t ) | d s d t + 0 T 0 1 g ( t ) | u ˜ ( t ) | d s d t 0 T 2 f ( t ) ( | u ¯ | α + | u ˜ ( t ) | α ) | u ˜ ( t ) | d t + 0 T g ( t ) | u ˜ ( t ) | d t 2 ( | u ¯ | α + u ˜ α ) u ˜ 0 T f ( t ) d t + u ˜ 0 T g ( t ) d t 3 ( 4 π 2 λ T 2 ) 4 π 2 T u ˜ 2 + 4 π 2 T 3 ( 4 π 2 λ T 2 ) | u ¯ | 2 α ( 0 T f ( t ) d t ) 2 + 2 u ˜ α + 1 0 T f ( t ) d t + u ˜ 0 T g ( t ) d t 4 π 2 λ T 2 16 π 2 0 T | u ˙ ( t ) | 2 d t + C 1 | u ¯ | 2 α + C 2 ( 0 T | u ˙ ( t ) | 2 d t ) α + 1 2 + C 3 ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ16_HTML.gif
(3.1)
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif and some positive constants C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq89_HTML.gif, C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq90_HTML.gif and C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq91_HTML.gif. By (H2) and Wirtinger’s inequality, we have
0 T [ G ( u ( t ) ) G ( u ¯ ) ] d t = 0 T 0 1 ( G ( u ¯ + s u ˜ ( t ) ) G ( u ¯ ) , u ˜ ( t ) ) d s d t = 0 T 0 1 1 s ( G ( u ¯ + s u ˜ ( t ) ) G ( u ¯ ) , s u ˜ ( t ) ) d s d t 0 T 0 1 ( λ ) s | u ˜ ( t ) | 2 d s d t = λ 2 0 T | u ˜ ( t ) | 2 d t λ T 2 8 π 2 0 T | u ˙ ( t ) | 2 d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ17_HTML.gif
(3.2)

for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif.

From (H4), we obtain
ϕ 2 ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equal_HTML.gif
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq82_HTML.gif. Therefore we have
ϕ ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T [ F ( t , u ( t ) ) F ( t , u ¯ ) ] d t + 0 T F ( t , u ¯ ) d t + ϕ 2 ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T [ H ( t , u ( t ) ) H ( t , u ¯ ) ] d t + 0 T [ G ( u ( t ) ) G ( u ¯ ) ] d t + 0 T F ( t , u ¯ ) d t + ϕ 2 ( u ) 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T [ H ( t , u ( t ) ) H ( t , u ¯ ) ] d t + 0 T [ G ( u ( t ) ) G ( u ¯ ) ] d t + 0 T F ( t , u ¯ ) d t 1 2 0 T | u ˙ ( t ) | 2 d t 4 π 2 λ T 2 16 π 2 0 T | u ˙ ( t ) | 2 d t C 1 | u ¯ | 2 α C 2 ( 0 T | u ˙ ( t ) | 2 d t ) α + 1 2 C 3 ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 λ T 2 8 π 2 0 T | u ˙ ( t ) | 2 d t + 0 T F ( t , u ¯ ) d t = 4 π 2 λ T 2 16 π 2 0 T | u ˙ ( t ) | 2 d t + | u ¯ | 2 α ( | u ¯ | 2 α 0 T F ( t , u ¯ ) d t C 1 ) C 2 ( 0 T | u ˙ ( t ) | 2 d t ) α + 1 2 C 3 ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ18_HTML.gif
(3.3)
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq92_HTML.gif. As u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq93_HTML.gif if and only if ( | u ¯ | 2 + u ˙ ( t ) L 2 2 ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq94_HTML.gif, (3.3) and (H3) imply that
ϕ ( u ) + as  u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equam_HTML.gif

By Lemma 2.1 and Remark 2.1, ϕ has a minimum point on H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq62_HTML.gif, which is a critical point of ϕ. Therefore, we complete the proof of Theorem 1.1. □

Lemma 3.1 Assume that the conditions of Theorem  1.2 hold. Then ϕ satisfies the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition.

Proof Let { ϕ ( u n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq95_HTML.gif be bounded and ϕ ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq96_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq72_HTML.gif. From (H1) and Lemma 2.4, we have
| 0 T ( H ( t , u n ( t ) ) , u ˜ n ( t ) ) d t | 0 T | H ( t , u n ( t ) ) | | u ˜ n ( t ) | d t 0 T f ( t ) | u ¯ n + u ˜ ( t ) | α | u ˜ n ( t ) | d t + 0 T g ( t ) | u ˜ n ( t ) | d t 2 ( | u ¯ n | α + u ˜ n α ) u ˜ n 0 T f ( t ) d t + u ˜ n 0 T g ( t ) d t 3 ( 4 π 2 A T 2 ) 2 π 2 T u ˜ n 2 + 2 π 2 T 3 ( 4 π 2 A T 2 ) | u ¯ n | 2 α ( 0 T f ( t ) d t ) 2 + 2 u ˜ n α + 1 0 T f ( t ) d t + u ˜ n 0 T g ( t ) d t 4 π 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t + C 4 | u ¯ n | 2 α + C 5 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 + C 6 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ19_HTML.gif
(3.4)
for all large n and some positive constants C 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq97_HTML.gif, C 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq98_HTML.gif and C 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq99_HTML.gif. It follows from (H5) and Lemma 2.4 that
| 0 T ( G ( u n ( t ) ) , u ˜ n ( t ) ) d t | = | 0 T ( G ( u n ( t ) ) G ( u ¯ n ) , u ˜ n ( t ) ) d t | 0 T | G ( u n ( t ) ) G ( u ¯ n ) | | u ˜ n ( t ) | d t 0 T ( A | u n ( t ) u ¯ n | + B ) | u ˜ n ( t ) | d t = A 0 T | u ˜ n ( t ) | 2 d t + B 0 T | u ˜ n ( t ) | d t A 0 T | u ˜ n ( t ) | 2 d t + B T ( 0 T | u ˜ n ( t ) | 2 d t ) 1 2 A T 2 4 π 2 0 T | u ˙ n ( t ) | 2 d t + B T T 2 π ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ20_HTML.gif
(3.5)
By (3.4), (3.5), (H8) and Young’s inequality, we have
u ˜ n ϕ ( u n ) , u ˜ n = 0 T | u ˙ n ( t ) | 2 d t + 0 T ( F ( t , u n ( t ) ) , u ˜ n ( t ) ) d t + j = 1 p i = 1 N I i j ( u n i ( t ) ) u ˜ n i ( t ) = 0 T | u ˙ n ( t ) | 2 d t + 0 T ( H ( t , u n ( t ) ) , u ˜ n ( t ) ) d t + 0 T ( G ( u n ( t ) ) , u ˜ n ( t ) ) d t + j = 1 p i = 1 N I i j ( u n i ( t ) ) u ˜ n i ( t ) 4 π 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t C 4 | u ¯ n | 2 α C 5 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 ( C 6 + B T T 2 π ) ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 j = 1 p i = 1 N ( a i j + b i j | u n i ( t ) | α β i j ) | u ˜ n i ( t ) | 4 π 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t C 4 | u ¯ n | 2 α C 5 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 ( C 6 + B T T 2 π ) ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 a p N u ˜ n b j = 1 p i = 1 N 2 ( | u ¯ n | α β i j + u ˜ n α β i j ) u ˜ n 4 π 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t C 4 | u ¯ n | 2 α C 5 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 ( C 6 + B T T 2 π ) ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 a p N T 12 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 b j = 1 p i = 1 N β i j | u ¯ n | 2 α 2 b j = 1 p i = 1 N 2 β i j 2 u ˜ n 2 2 β i j 2 b j = 1 p i = 1 N u ˜ n α β i j + 1 4 π 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t C 4 | u ¯ n | 2 α C 5 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 ( C 6 + B T T 2 π ) ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 a p N T 12 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 b j = 1 p i = 1 N β i j | u ¯ n | 2 α b j = 1 p i = 1 N ( 2 β i j ) ( T 12 0 T | u ˙ n ( t ) | 2 d t ) 1 2 β i j 2 b j = 1 p i = 1 N ( T 12 0 T | u ˙ n ( t ) | 2 d t ) α β i j + 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ21_HTML.gif
(3.6)
where a = max { a i j , i = 1 , 2 , , N ; j = 1 , 2 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq100_HTML.gif, b = max { b i j , i = 1 , 2 , , N ; j = 1 , 2 , , p } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq101_HTML.gif. From Wirtinger’s inequality, we obtain
0 T | u ˙ n ( t ) | 2 d t u ˜ n 2 ( 1 + T 2 4 π 2 ) 0 T | u ˙ n ( t ) | 2 d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ22_HTML.gif
(3.7)
The inequalities (3.6) and (3.7) imply that
C 7 | u ¯ n | α ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 C 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ23_HTML.gif
(3.8)
for all large n and some positive constants C 7 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq102_HTML.gif and C 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq103_HTML.gif. It follows from (H5), Cauchy-Schwarz’s inequality and Wirtinger’s inequality that
| 0 T [ G ( u n ( t ) ) G ( u ¯ n ) ] d t | = | 0 T 0 1 ( G ( u ¯ n + s u ˜ n ( t ) ) G ( u ¯ n ) , u ˜ n ( t ) ) d s d t | 0 T 0 1 ( A s | u ˜ n ( t ) | + B ) | u ˜ n ( t ) | d s d t = A 2 0 T | u ˜ n ( t ) | 2 d t + B 0 T | u ˜ n ( t ) | d t A 2 0 T | u ˜ n ( t ) | 2 d t + B T ( 0 T | u ˜ n ( t ) | 2 d t ) 1 2 A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t + B T T 2 π ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ24_HTML.gif
(3.9)
Like in the proof of Theorem 1.1, we get
| 0 T [ H ( t , u n ( t ) ) H ( t , u ¯ n ) ] d t | 4 π 2 A T 2 16 π 2 0 T | u ˙ n ( t ) | 2 d t + C 1 | u ¯ n | 2 α + C 2 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 + C 3 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ25_HTML.gif
(3.10)
From (H7), we have
ϕ 2 ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equan_HTML.gif
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq104_HTML.gif. Since { ϕ ( u n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq95_HTML.gif is bounded, from (3.9) and (3.10), there exists a constant C such that
C ϕ ( u n ) = 1 2 0 T | u ˙ n ( t ) | 2 d t + 0 T [ F ( t , u n ( t ) ) F ( t , u ¯ n ) ] d t + 0 T F ( t , u ¯ n ) d t + ϕ 2 ( u n ) 1 2 0 T | u ˙ n ( t ) | 2 d t + 0 T [ H ( t , u n ( t ) ) H ( t , u ¯ n ) ] d t + 0 T [ G ( u n ( t ) ) G ( u ¯ n ) ] d t + 0 T F ( t , u ¯ n ) d t 12 π 2 + A T 2 16 π 2 0 T | u ˙ n ( t ) | 2 d t + C 1 | u ¯ n | 2 α + C 2 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 + C 3 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 + B T T 2 π ( | u ˙ n ( t ) | 2 d t ) 1 2 + 0 T F ( t , u ¯ n ) d t | u ¯ n | 2 α ( | u ¯ n | 2 α 0 T F ( t , u ¯ n ) d t + C 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ26_HTML.gif
(3.11)
for all large n and some constant C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq105_HTML.gif. By the above inequality and (H6), we know that { | u ¯ n | } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq106_HTML.gif is bounded. In fact, if not, without loss of generality, we may assume that | u ¯ n | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq107_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq72_HTML.gif. Then, from (3.8) and the above inequality, we have
lim n inf | u ¯ n | 2 α 0 T F ( t , u ¯ n ) d t > , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equao_HTML.gif
which contradicts (H6). Hence { | u ¯ n | } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq106_HTML.gif is bounded. Furthermore, { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq108_HTML.gif is bounded from (3.7) and (3.8). Hence, there exists a subsequence of u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq109_HTML.gif defined by u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq110_HTML.gif such that
u n u in  H T 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equap_HTML.gif
By Lemma 2.3, we have
u n u in  C [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaq_HTML.gif
On the other hand, we get
ϕ ( u n ) ϕ ( u ) , u n u = 0 T | u ˙ n ( t ) u ˙ ( t ) | 2 d t + 0 T ( F ( t , u n ( t ) ) F ( t , u ( t ) ) , u n ( t ) u ( t ) ) d t + j = 1 p i = 1 N [ I i j ( u n i ( t j ) ) I i j ( u i ( t j ) ) ] ( u n i ( t j ) u i ( t j ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ27_HTML.gif
(3.12)
It follows from the above equality, (A) and the continuity of I i j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq111_HTML.gif that
u n u in  H T 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equar_HTML.gif

Thus, we conclude that ϕ satisfies the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition. □

Now, we give the proof of our Theorem 1.2.

Proof of Theorem 1.2 Let W be the subspace of H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq112_HTML.gif given by
W = { u H T 1 | u ¯ = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equas_HTML.gif
Then H T 1 = W R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq113_HTML.gif. Firstly, we show that
ϕ ( u ) + as  u W , u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ28_HTML.gif
(3.13)
In fact, for u W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq114_HTML.gif, then u ¯ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq115_HTML.gif, by the proof of Theorem 1.1, we have
| 0 T [ H ( t , u ( t ) ) H ( t , 0 ) ] d t | 4 π 2 A T 2 16 π 2 0 T | u ˙ ( t ) | 2 d t + C 2 ( 0 T | u ˙ ( t ) | 2 d t ) α + 1 2 + C 3 ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ29_HTML.gif
(3.14)
Like in the proof of Lemma 3.1, we obtain
| 0 T [ G ( u ( t ) ) G ( 0 ) ] d t | = | 0 T 0 1 ( G ( s u ) G ( 0 ) , u ) d s d t | A 2 0 T | u ( t ) | 2 d t + B 0 T | u ( t ) | d t A T 2 8 π 2 0 T | u ˙ n ( t ) | 2 d t + B T T 2 π ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ30_HTML.gif
(3.15)
By (H8) and Lemma 2.5, we find
| ϕ 2 ( u ) | = | j = 1 p i = 1 N 0 u i ( t j ) I i j ( t ) d t | j = 1 p i = 1 N 0 u i ( t j ) ( a i j + b i j | t | α β i j ) d t a p N u + b j = 1 p i = 1 N u α β i j + 1 a p N C ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 + b j = 1 p i = 1 N ( C 2 0 T | u ˙ ( t ) | 2 d t ) α β i j + 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ31_HTML.gif
(3.16)
for all u W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq116_HTML.gif. By (3.14), (3.15) and (3.16), we have
ϕ ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T [ F ( t , u ( t ) ) F ( t , 0 ) ] d t + 0 T F ( t , 0 ) d t + ϕ 2 ( u ) = 1 2 0 T | u ˙ ( t ) | 2 d t + 0 T [ H ( t , u ( t ) ) H ( t , 0 ) ] d t + 0 T [ G ( u ( t ) ) G ( 0 ) ] d t + 0 T F ( t , 0 ) d t + ϕ 2 ( u ) ( 4 π 2 A T 2 16 π 2 ) 0 T | u ˙ ( t ) | 2 d t C 2 ( 0 T | u ˙ ( t ) | 2 d t ) α + 1 2 C 3 ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 B T T 2 π ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 a p N C ( 0 T | u ˙ ( t ) | 2 d t ) 1 2 b j = 1 p i = 1 N ( C 2 0 T | u ˙ ( t ) | 2 d t ) α β i j + 1 2 d t + 0 T F ( t , 0 ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ32_HTML.gif
(3.17)

for all u W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq116_HTML.gif.

By Lemma 2.4, we have u u ˙ L 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq117_HTML.gif on W. Hence (3.13) follows from (3.17).

On the other hand, by (H7), we get
ϕ 2 ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ33_HTML.gif
(3.18)
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq118_HTML.gif. Therefore, from (3.18) and (H6), we obtain
ϕ ( u ) = 0 T F ( t , u ) d t + ϕ 2 ( u ) | u | 2 α ( | u | 2 α 0 T F ( t , u ) d t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equat_HTML.gif

as | u | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq119_HTML.gif in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq28_HTML.gif. It follows from Lemma 2.2 and Lemma 3.1 that problem (1.1) has at least one weak solution. □

Proof of Theorem 1.3 First we prove that ϕ satisfies the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition. Suppose that U n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq120_HTML.gif is a P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif sequence for ϕ, that is, ϕ ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq96_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq72_HTML.gif and ϕ ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq121_HTML.gif is bounded. In a way similar to the proof of Theorem 1.1, we have
| 0 T ( H ( t , u n ( t ) ) , n ˜ n ( t ) ) d t | 4 π 2 λ T 2 16 π 2 0 T | u ˙ n ( t ) | 2 d t + C 1 | u ¯ n | 2 α + C 2 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 + C 3 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equau_HTML.gif
and
0 T ( G ( u n ( t ) ) , u ˜ n ( t ) ) d t λ T 2 4 π 2 0 T | u ˙ n ( t ) | 2 d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equav_HTML.gif
for all u H T 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq118_HTML.gif. Hence we have
u ˜ n ϕ ( u n ) , u ˜ n = 0 T | u ˙ n ( t ) | 2 d t + 0 T ( F ( t , u n ( t ) ) , u ˜ n ( t ) ) d t + j = 1 p i = 1 N I i j ( u n i ( t ) ) u ˜ n i ( t ) = 0 T | u ˙ n ( t ) | 2 d t + 0 T ( H ( t , u n ( t ) ) , u ˜ n ( t ) ) d t + 0 T ( G ( u n ( t ) ) , u ˜ n ( t ) ) d t + j = 1 p i = 1 N I i j ( u n i ( t ) ) u ˜ n i ( t ) 3 ( 4 π 2 λ T 2 ) 16 π 2 0 T | u ˙ n ( t ) | 2 d t C 1 | u ¯ n | 2 α C 2 ( 0 T | u ˙ n ( t ) | 2 d t ) α + 1 2 C 3 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 a p N T 12 ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 b j = 1 p i = 1 N β i j | u ¯ n | 2 α b j = 1 p i = 1 N ( 2 β i j ) ( T 12 0 T | u ˙ n ( t ) | 2 d t ) 1 2 β i j 2 b j = 1 p i = 1 N ( T 12 0 T | u ˙ n ( t ) | 2 d t ) α β i j + 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaw_HTML.gif
From (3.7) and the above inequalities, we obtain
C 9 | u ¯ n | α ( 0 T | u ˙ n ( t ) | 2 d t ) 1 2 C 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equ34_HTML.gif
(3.19)

for some positive constants C 9 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq122_HTML.gif and C 10 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq123_HTML.gif.

By (H9) there exists M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq124_HTML.gif such that
| x | 2 α F ( t , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equax_HTML.gif
for all | x | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq125_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif, which implies that
F ( t , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equay_HTML.gif
for all | x | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq125_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif. It follows from assumption (A) that
F ( t , x ) a 0 b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equaz_HTML.gif

for all | x | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq126_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif, where a 0 = max 0 s M a ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq127_HTML.gif.

Let γ ( t ) = a 0 b ( t ) L 1 ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq128_HTML.gif, then
F ( t , x ) γ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equba_HTML.gif

for all x R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq17_HTML.gif and t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq9_HTML.gif.

By the boundedness of ϕ ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq121_HTML.gif, (H7) and (3.19), there exists a constant C 11 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq129_HTML.gif such that
C 11 ϕ ( u n ) = 1 2 0 T | u ˙ n ( t ) | 2 d t + 0 T F ( t , u n ( t ) ) d t + ϕ 2 ( u ) 1 2 0 T | u ˙ n ( t ) | 2 d t + 0 T F ( t , u n ( t ) ) d t ( C 9 | u ¯ n | α + C 10 ) 2 + 0 T γ ( t ) d t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_Equbb_HTML.gif

which implies that | u ¯ n | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq130_HTML.gif is bounded. Like in the proof of Lemma 3.1, we know that ϕ satisfies the P . S . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-151/MediaObjects/13661_2013_Article_409_IEq73_HTML.gif condition.

Furthermore, we can prove Theorem 1.3 using the same way as in the proof of Theorem 1.2. Here, we omit it. □

Declarations

Acknowledgements

We express our gratitude to the referees for their valuable criticism of the manuscript and for helpful suggestions. Supported by the National Natural Science Foundation of China (11271371, 10971229).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Central South University

References

  1. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View Article
  2. Tang CL: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 1998, 126: 3263–3270.View Article
  3. Ma J, Tang CL: Periodic solutions for some nonautonomous second-order systems. J. Math. Anal. Appl. 2002, 275: 482–494.MathSciNetView Article
  4. Jiang Q, Tang CL: Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 2007, 328: 380–389.MathSciNetView Article
  5. Luan SX, Mao AM: Periodic solutions of nonautonomous second order Hamiltonian systems. Acta Math. Sin. Engl. Ser. 2005, 21: 685–690.MathSciNetView Article
  6. Schechter M: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ. 2006, 223: 290–302.MathSciNetView Article
  7. Tang CL: Periodic solutions of nonautonomous second order systems with γ -quasisubadditive potential. J. Math. Anal. Appl. 1995, 189: 671–675.MathSciNetView Article
  8. Tang CL, Wu XP: Periodic solutions for second order Hamiltonian systems with a change sign potential. J. Math. Anal. Appl. 2004, 292: 506–516.MathSciNetView Article
  9. Tao ZL, Tang CL: Periodic and subharmonic solutions of second-order Hamiltonian systems. J. Math. Anal. Appl. 2004, 293: 435–445.MathSciNetView Article
  10. Tao ZL, Yan SA, Wu SL: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl. 2007, 331: 152–158.MathSciNetView Article
  11. Ye YW, Tang CL: Periodic solutions for some nonautonomous second order Hamiltonian systems. J. Math. Anal. Appl. 2008, 344: 462–471.MathSciNetView Article
  12. Yang RG: Periodic solutions of some nonautonomous second order Hamiltonian systems. Nonlinear Anal. 2008, 69: 2333–2338.MathSciNetView Article
  13. Tang XH, Meng Q: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11: 3722–3733.MathSciNetView Article
  14. Zhou JW, Li YK: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 2010, 72: 1594–1603.MathSciNetView Article
  15. Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680–690.MathSciNetView Article
  16. Tian Y, Ge W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 2008, 51: 509–527.MathSciNetView Article

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© Dai and Guo; licensee Springer. 2013

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