Open Access

Homoclinic orbits for second-order discrete Hamiltonian systems with subquadratic potential

Advances in Difference Equations20132013:228

https://doi.org/10.1186/1687-1847-2013-228

Received: 3 April 2013

Accepted: 11 July 2013

Published: 31 July 2013

Abstract

Under the assumptions that W ( n , x ) is indefinite sign and subquadratic as | x | + and L ( n ) satisfies

lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] > 0

for some constant ν < 2 , we establish a theorem on the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system

[ p ( n ) u ( n 1 ) ] L ( n ) u ( n ) + W ( n , u ( n ) ) = 0 ,

where p ( n ) and L ( n ) are N × N real symmetric matrices for all n Z , and p ( n ) is always positive definite.

MSC:39A11, 58E05, 70H05.

Keywords

homoclinic solutiondiscrete Hamiltonian systemsubquadraticcritical point

1 Introduction

Consider the second-order self-adjoint discrete Hamiltonian system
[ p ( n ) u ( n 1 ) ] L ( n ) u ( n ) + W ( n , u ( n ) ) = 0 ,
(1.1)

where n Z , u R N , u ( n ) = u ( n + 1 ) u ( n ) is the forward difference, p , L : Z R N × N and W : Z × R N R , W ( n , x ) is continuously differentiable in x for every n Z .

As usual, we say that a solution u ( n ) of system (1.1) is homoclinic (to 0) if u ( n ) 0 as n ± . In addition, if u ( n ) 0 then u ( n ) is called a nontrivial homoclinic solution.

The existence and multiplicity of nontrivial homoclinic solutions for problem (1.1) have been extensively investigated in the literature with the aid of critical point theory and variational methods; see, for example, [113]. Most of them treat the case where W ( n , x ) is superquadratic as | x | .

Compared to the superquadratic case, as far as the author is aware, there are a few papers [10, 12, 13] concerning the case where W ( n , x ) has subquadratic growth at infinity. Specifically, [12] and [10] dealt with the existence and multiplicity of homoclinic solutions for (1.1) under the following assumptions on L:

( L ) L ( n ) is an N × N real symmetric positive definite matrix for all n Z and there exists a constant β > 0 such that
( L ( n ) x , x ) β | x | 2 , ( n , x ) Z × R N ;
( L ν ) L ( n ) is an N × N real symmetric positive definite matrix for all n Z and there exists a constant ν < 2 such that
lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] > 0 ,

respectively. In the above two cases, since L ( n ) is positive definite, the variational functional associated with system (1.1) is bounded from below, techniques based on the genus properties have been well applied. In particular, Clark’s theorem is an efficacious tool to prove the existence and multiplicity of homoclinic solutions for system (1.1). However, if L ( n ) is not global positive definite on , the problem is far more difficult as 0 is a saddle point rather than a local minimum of the variational functional, which is strongly indefinite and it is not easy to obtain the boundedness of the Palais-Smale sequence. In a recent paper [13], based on a new direct sum decomposition of the ‘work space’, Tang and Lin proved the following theorem by using a linking theorem which was developed in [14].

Theorem 1.1 [13]

Assume that p ( n ) is an N × N real symmetric positive definite matrix for all n Z , L and W satisfy the following assumptions:

( L ν ) L ( n ) is an N × N real symmetric matrix for all n Z and there exists a constant ν < 2 such that
lim inf | n | + [ | n | ν 2 inf | x | = 1 ( L ( n ) x , x ) ] > 0 ;
(W1) there exist constants max { 1 , 2 / ( 3 ν ) } < γ 1 < γ 2 < 2 and a 1 , a 2 0 such that
| W ( n , x ) | a 1 | x | γ 1 + a 2 | x | γ 2 , ( n , x ) Z × R N ;
(W2) there exists a function φ C ( [ 0 , + ) , [ 0 , + ) ) such that
| W ( n , x ) | φ ( | x | ) , ( n , x ) Z × R N ,

where φ ( s ) = O ( s γ 3 1 ) as s 0 + , max { 1 , 2 / ( 3 ν ) } < γ 3 < 2 ;

(W3) there exist constants b 1 > 0 , b 2 , b 3 0 and max { 1 , 2 / ( 3 ν ) } < γ 6 < γ 5 < γ 4 < 2 such that
2 W ( n , x ) W ( n , x ) b 1 | x | γ 4 b 2 | x | γ 5 b 3 | x | γ 6 , ( n , x ) Z × R N ;
(W4) there exist constants b 4 > 0 , b 5 , b 6 0 and max { 1 , 2 / ( 3 ν ) } < γ 7 < γ 8 < γ 9 < 2 such that
W ( n , x ) b 4 | x | γ 7 b 5 | x | γ 8 b 6 | x | γ 9 , ( n , x ) Z × R N ;

(W5) W ( n , x ) = W ( n , x ) , ( n , x ) Z × R N .

Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

We remark that the condition ‘positive definite’ is removed in ( L ν ), i.e., L ( n ) is not required to be global positive definite on . The main goal of this paper is to weaken conditions (W1), (W2), (W3) and (W4) of Theorem 1.1 under assumption ( L ν ).

To state our result, we first introduce the following assumptions:

(W1) there exist constants σ i [ 0 , 2 ν ) , a i 0 and max { 1 , 2 ( 1 + σ i ) / ( 3 ν ) } < γ i < 2 with i = 1 , 2 such that
| W ( n , x ) | i = 1 2 a i ( 1 + | n | σ i ) | x | γ i , ( n , x ) Z × R N ;
(W2) there exist two constants max { 1 , 2 ( 1 + σ i ) / ( 3 ν ) } < γ i + 2 < 2 , i = 1 , 2 and two functions φ 1 , φ 2 C ( [ 0 , + ) , [ 0 , + ) ) such that
| W ( n , x ) | i = 1 2 ( 1 + | n | σ i ) φ i ( | x | ) , ( n , x ) Z × R N ,

where φ i ( s ) = O ( s γ i + 2 1 ) as s 0 + , i = 1 , 2 ;

(W3) there exist constants b 1 > 0 , b 2 0 and 1 < γ 6 < γ 5 < 2 such that
2 W ( n , x ) W ( n , x ) b 1 | x | γ 5 b 2 | x | γ 6 , ( n , x ) Z × R N ;
(W4) there exist constants b 3 > 0 , b 4 0 and 1 < γ 7 < γ 8 < 2 such that
W ( n , x ) b 3 | x | γ 7 b 4 | x | γ 8 , ( n , x ) Z × R N .

We are now in a position to state the main result of this paper.

Theorem 1.2 Assume that p ( n ) is an N × N real symmetric positive definite matrix for all n Z , L and W satisfy ( L ν ), (W1), (W2), (W3), (W4) and (W5). Then system (1.1) possesses infinitely many nontrivial homoclinic solutions.

2 Preliminaries

In what follows, we always assume that p ( n ) is a real symmetric positive definite matrix for all n Z . As done in [13], we define
l ( n ) = inf x R N , | x | = 1 ( L ( n ) x , x )
(2.1)
and
Z 1 = { n Z : l ( n ) 0 } , Z 2 = { n Z : l ( n ) > 0 } .
(2.2)
Then by ( L ν ), l ( n ) is bounded from below and so Z 1 is a finite set and
l : = min { l ( n ) : n Z 2 } > 0 .
(2.3)
Define
L ˜ ( n ) = { l I N , n Z 1 , L ( n ) , n Z 2 ; l ˜ ( n ) = { l , n Z 1 , l ( n ) , n Z 2 .
(2.4)
Then, it follows from (2.1), (2.2), (2.3) and (2.4) that
( L ˜ ( n ) x , x ) l ˜ ( n ) | x | 2 l | x | 2 , ( n , x ) Z × R N .
(2.5)
Let
S = { { u ( n ) } n Z : u ( n ) R N , n Z } , E = { u S : n Z [ ( p ( n + 1 ) u ( n ) , u ( n ) ) + ( L ˜ ( n ) u ( n ) , u ( n ) ) ] < + } ,
and for u , v E , let
( u , v ) = n Z [ ( p ( n + 1 ) u ( n ) , v ( n ) ) + ( L ˜ ( n ) u ( n ) , v ( n ) ) ] .
Then E is a Hilbert space with the above inner product, and the corresponding norm is
u = { n Z [ ( p ( n + 1 ) u ( n ) , u ( n ) ) + ( L ˜ ( n ) u ( n ) , u ( n ) ) ] } 1 / 2 , u E .
As usual, for 1 q < + , set
l q ( Z , R N ) = { { u ( n ) } n Z : u ( n ) R N , n Z , n Z | u ( n ) | q < + }
and
l ( Z , R N ) = { { u ( n ) } n Z : u ( n ) R N , n Z , sup n Z | u ( n ) | < + } ,
and their norms are defined by
u q = ( n Z | u ( n ) | q ) 1 / q , u l q ( Z , R N ) ; u = sup n Z | u ( n ) | , u l ( Z , R N ) ,

respectively.

Lemma 2.1 [[9], Lemma 2.2]

For u E , one has
u 1 ( l + 4 α ) l 4 u ,
(2.6)

where α = inf { ( p ( n ) x , x ) : n Z , x R N , | x | = 1 } .

Set
b ( u , v ) = n Z [ ( p ( n + 1 ) u ( n ) , v ( n ) ) + ( L ( n ) u ( n ) , v ( n ) ) ] , u , v E .
(2.7)

Lemma 2.2 [[13], Lemma 2.3]

Suppose that L satisfies ( L ν ). Then
  1. (i)
    b ( u , v ) is a bilinear function on E, and there exists a constant C 0 > 0 such that
    | b ( u , v ) | C 0 u v , u , v E ;
    (2.8)
     
  2. (ii)
    b ( u , u ) = u 2 n Z 1 ( ( L ˜ ( n ) L ( n ) ) u ( n ) , u ( n ) ) , u E .
    (2.9)
     
By ( L ν ), there exist an integer N 0 > max { | n | : n Z 1 } and M 0 > 0 such that
| n | ν 2 inf | x | = 1 ( L ( n ) x , x ) M 0 , | n | N 0 ,
which implies
| n | ν 2 ( L ( n ) x , x ) M 0 | x | 2 , | n | N 0 , x R N .
(2.10)
Lemma 2.3 Suppose that L satisfies ( L ν ). Then, for σ [ 0 , 2 ν ) and 1 q ( 2 ( 1 + σ ) / ( 3 ν ) , 2 ) , E is compactly embedded in l q ( Z , R N ) ; moreover,
| n | > N ( 1 + | n | σ ) | u ( n ) | q K ( σ , q ) N κ u q , u E , N N 0
(2.11)
and
n Z ( 1 + | n | σ ) | u ( n ) | q [ ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 + K ( σ , q ) N κ ] u q , u E , N N 0 ,
(2.12)
where
κ = ( 3 ν ) q 2 ( 1 + σ ) 2 > 0 , K ( σ , q ) = 2 [ 2 ( 2 q ) ( 3 ν ) q 2 ( 1 + σ ) ] 1 q 2 M 0 q / 2 .
(2.13)
Proof Let r = [ ( 3 ν ) q 2 ( 1 + σ ) ] / ( 2 q ) . Then r > 0 . For u E and N N 0 , it follows from (2.10), (2.13) and the Hölder inequality that
| n | > N ( 1 + | n | σ ) | u ( n ) | q 2 ( | n | > N | n | [ ( 2 ν ) q 2 σ ] / ( 2 q ) ) 1 q 2 ( | n | > N | n | 2 ν | u ( n ) | 2 ) q 2 = 2 ( | n | > N | n | ( r + 1 ) ) 1 q 2 ( | n | > N | n | 2 ν | u ( n ) | 2 ) q 2 2 ( 2 r N r ) 1 q 2 [ 1 M 0 | n | > N ( L ( n ) u ( n ) , u ( n ) ) ] q 2 2 1 + ( 2 q ) / 2 M 0 q / 2 r ( 2 q ) / 2 N κ u q = K ( σ , q ) N κ u q .
This shows that (2.11) holds. Hence, from (2.5), (2.11) and the Hölder inequality, one has
n Z ( 1 + | n | σ ) | u ( n ) | q = | n | N ( 1 + | n | σ ) | u ( n ) | q + | n | > N ( 1 + | n | σ ) | u ( n ) | q ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 ( | n | N l ˜ ( n ) | u ( n ) | 2 ) q 2 + K ( σ , q ) N κ u q ( | n | N ( 1 + | n | σ ) 2 / ( 2 q ) [ l ˜ ( n ) ] q / ( 2 q ) ) 1 q 2 u q + K ( σ , q ) N κ u q .

This shows that (2.12) holds.

Finally, we prove that E is compactly embedded in l q ( Z , R N ) . Let { u k } E be a bounded sequence. Then by (2.6), there exists a constant Λ > 0 such that
u k 1 ( l + 4 α ) l 4 u k Λ , k N .
(2.14)
Since E is reflexive, { u k } possesses a weakly convergent subsequence in E. Passing to a subsequence if necessary, it can be assumed that u k u 0 in E. It is easy to verify that
lim k u k ( n ) = u 0 ( n ) , n Z .
(2.15)
For any given number ε > 0 , we can choose N ε > 0 such that
2 q 1 K ( σ , q ) N ε κ { [ ( l + 4 α ) l 4 Λ ] q + u 0 q } < ε .
(2.16)
It follows from (2.15) that there exists k 0 N such that
| n | N ε | u k ( n ) u 0 ( n ) | q < ε for  k k 0 .
(2.17)
On the other hand, it follows from (2.11), (2.14) and (2.16) that
| n | > N ε | u k ( n ) u 0 ( n ) | q 2 q 1 | n | > N ε ( | u k ( n ) | q + | u 0 ( n ) | q ) 2 q 1 K ( σ , q ) N ε κ ( u k q + u 0 q ) 2 q 1 K ( σ , q ) N ε κ { [ ( l + 4 α ) l 4 Λ ] q + u 0 q } ε , k N .
(2.18)
Since ε is arbitrary, combining (2.17) with (2.18), we get
u k u 0 q q = n Z | u k ( n ) u 0 ( n ) | q 0 as  k .

This shows that { u k } possesses a convergent subsequence in l q ( Z , R N ) . Therefore, E is compactly embedded in l q ( Z , R N ) for 1 q ( 2 ( 1 + σ ) / ( 3 ν ) , 2 ) . □

Lemma 2.4 Suppose that L and W satisfy ( L ν ) and (W1). Then, for u E ,
n Z | W ( n , u ( n ) ) | ϕ 1 ( N ) u γ 1 + ϕ 2 ( N ) u γ 2 , N N 0 ,
(2.19)
where
κ 1 = ( 3 ν ) γ 1 2 ( 1 + σ 1 ) 2 , κ 2 = ( 3 ν ) γ 2 2 ( 1 + σ 2 ) 2 ;
(2.20)
ϕ 1 ( N ) = a 1 [ ( | n | N ( 1 + | n | σ 1 ) 2 / ( 2 γ 1 ) [ l ˜ ( n ) ] γ 1 / ( 2 γ 1 ) ) 1 γ 1 2 + K ( σ 1 , γ 1 ) N κ 1 ] ,
(2.21)
ϕ 2 ( N ) = a 2 [ ( | n | N ( 1 + | n | σ 2 ) 2 / ( 2 γ 2 ) [ l ˜ ( n ) ] γ 2 / ( 2 γ 2 ) ) 1 γ 2 2 + K ( σ 2 , γ 2 ) N κ 2 ] .
(2.22)
Proof For N N 0 , it follows from (W1), (2.12), (2.20), (2.21) and (2.22) that
n Z | W ( n , u ( n ) ) | i = 1 2 a i n Z ( 1 + | n | σ i ) | u ( n ) | γ i i = 1 2 a i [ ( | n | N ( 1 + | n | σ i ) 2 / ( 2 γ i ) [ l ˜ ( n ) ] γ i / ( 2 γ i ) ) 1 γ i 2 + K ( σ i , γ i ) N κ i ] u γ i = ϕ 1 ( N ) u γ 1 + ϕ 2 ( N ) u γ 2 .

This shows that (2.19) holds. □

Lemma 2.5 Assume that L and W satisfy ( L ν ), (W1) and (W2). Then the functional f : E R defined by
f ( u ) = 1 2 b ( u , u ) n Z W ( n , u ( n ) ) , u E
(2.23)
is well defined and of class C 1 ( E , R ) and
f ( u ) , v = b ( u , v ) n Z ( W ( n , u ( n ) ) , v ( n ) ) , u , v E .
(2.24)

Furthermore, the critical points of f in E are the solutions of system (1.1) with u ( ± ) = 0 .

Proof Lemmas 2.2 and 2.4 imply that f defined by (2.23) is well defined on E. Next, we prove that (2.24) holds. By (W2), there exist M 1 , M 2 > 0 such that
φ i ( | x | ) M i | x | γ 2 + i 1 , x R N , | x | 1 , i = 1 , 2 .
(2.25)
For any u , v E , there exists an integer N 1 > N 0 such that | u ( n ) | + | v ( n ) | < 1 for | n | > N 1 . Then, for any sequence { θ n } n Z R with | θ n | < 1 for n Z and any number h ( 0 , 1 ) , by (W2), (2.11) and (2.25), we have
n Z max h [ 0 , 1 ] | ( W ( n , u ( n ) + θ n h v ( n ) ) , v ( n ) ) | | n | N 1 max h [ 0 , 1 ] | W ( n , u ( n ) + θ n h v ( n ) ) | | v ( n ) | + | n | > N 1 max h [ 0 , 1 ] | W ( n , u ( n ) + θ n h v ( n ) ) | | v ( n ) | | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i | n | > N 1 ( 1 + | n | σ i ) ( | u ( n ) | + | v ( n ) | ) γ 2 + i 1 | v ( n ) | | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i | n | > N 1 ( 1 + | n | σ i ) | v ( n ) | γ 2 + i + i = 1 2 M i ( | n | > N 1 ( 1 + | n | σ i ) | u ( n ) | γ 2 + i ) 1 1 γ 2 + i × ( | n | > N 1 ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i | n | N 1 max | x | u + v | W ( n , x ) | | v ( n ) | + i = 1 2 M i K ( σ i , γ 2 + i ) N 1 κ 2 + i ( u γ 2 + i 1 + v γ 2 + i 1 ) v < + ,
(2.26)
where κ 2 + i = [ γ 2 + i ( 3 ν ) 2 ( 1 + σ i ) ] / 2 > 0 , i = 1 , 2 . Then by (2.23), (2.26) and Lebesgue’s dominated convergence theorem, we have
f ( u ) , v = lim h 0 + f ( u + h v ) f ( u ) h = lim h 0 + [ b ( u , v ) + h b ( v , v ) 2 n Z ( W ( n , u ( n ) + θ n h v ( n ) ) , v ( n ) ) ] = b ( u , v ) n Z ( W ( n , u ( n ) ) , v ( n ) ) .

This shows that (2.24) holds. In view of the proof of [[13], Lemma 2.6], the critical points of f in E are the solutions of system (1.1) with u ( ± ) = 0 . □

Let us prove now that f is continuous. Let u k u in E. Then there exists a constant δ > 0 such that
u ( l + 4 α ) l 4 δ , u k ( l + 4 α ) l 4 δ , k = 1 , 2 , .
(2.27)
It follows from (2.6) that
u δ , u k δ , k = 1 , 2 , .
(2.28)
By (W2), there exist M 3 , M 4 > 0 such that
φ i ( | x | ) M 2 + i | x | γ 2 + i 1 , x R N , | x | δ , i = 1 , 2 .
(2.29)
From (2.11), (2.24), (2.27), (2.28), (2.29), (W2) and the Hölder inequality, we have
| f ( u k ) f ( u ) , v | | b ( u k u , v ) | + n Z | ( W ( n , u k ( n ) ) W ( n , u ( n ) ) , v ( n ) ) | C 0 u k u v + | n | N | W ( n , u k ( n ) ) W ( n , u ( n ) ) | | v ( n ) | + | n | > N ( | W ( n , u k ( n ) ) | + | W ( n , u ( n ) ) | ) | v ( n ) | o ( 1 ) + i = 1 2 M 2 + i | n | > N ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i 1 + | u ( n ) | γ 2 + i 1 ) | v ( n ) | o ( 1 ) + i = 1 2 M 2 + i ( | n | > N ( 1 + | n | σ i ) | u k ( n ) | γ 2 + i ) 1 1 γ 2 + i ( | n | > N ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i + i = 1 2 M 2 + i ( | n | > N ( 1 + | n | σ i ) | u ( n ) | γ 2 + i ) 1 1 γ 2 + i ( | n | > N ( 1 + | n | σ i ) | v ( n ) | γ 2 + i ) 1 γ 2 + i o ( 1 ) + i = 1 2 M 2 + i K ( σ i , γ 2 + i ) N κ 2 + i ( u k γ 2 + i 1 + u γ 2 + i 1 ) v = o ( 1 ) , k + , N + , v E ,

which implies the continuity of f . The proof is complete.  □

Lemma 2.6 [14]

Let X be an infinite dimensional Banach space and let f C 1 ( X , R ) be even, satisfy the (PS)-condition, and f ( 0 ) = 0 . If X = X 1 X 2 (direct sum), where X 1 is finite dimensional, and f satisfies
  1. (i)

    f is bounded from below on X 2 ;

     
  2. (ii)

    for each finite dimensional subspace X ˜ X , there are positive constants ρ = ρ ( X ˜ ) and σ = σ ( X ˜ ) such that f | B ρ X ˜ 0 and f | B ρ X ˜ σ , where B ρ = { x X : x = ρ } .

     

Then f possesses infinitely many nontrivial critical points.

3 Proof of the theorem

Proof of Theorem 1.2 For u E , we define two functions as follows:
u ( n ) = { u ( n ) , n Z 1 , 0 , n Z 2 ; u + ( n ) = { 0 , n Z 1 , u ( n ) , n Z 2 .
(3.1)
Set
X 1 = { u : u E } , X 2 = { u + : u E } .
(3.2)
Then X : = E = X 1 X 2 (direct sum) and dim ( X 1 ) < + . Obviously, (W1) and (W5) imply f ( 0 ) = 0 and f is even. In view of Lemma 2.5, f C 1 ( E , R ) . In what follows, we first prove that f satisfies the (PS)-condition. Assume that { u k } k N E is a (PS)-sequence: { f ( u k ) } k N is bounded and f ( u k ) 0 as k + . From (2.23), (2.24) and (W3), we have
f ( u k ) , u k 2 f ( u k ) = n Z [ 2 W ( n , u k ( n ) ) ( W ( n , u k ( n ) ) , u k ( n ) ) ] b 1 n Z | u k ( n ) | γ 5 b 2 n Z | u k ( n ) | γ 6 = b 1 u k γ 5 γ 5 b 2 u k γ 6 γ 6 .
It follows that there exists a constant C 1 > 0 such that
b 1 u k γ 5 γ 5 b 2 u k γ 6 γ 6 C 1 ( 1 + u k ) .
(3.3)
Since dim ( X 1 ) < + , it follows that there exists a constant C 2 > 0 such that
u k 2 2 = ( u k , u k ) l 2 u k γ 5 u k γ 5 C 2 u k 2 u k γ 5 ,
(3.4)
where γ 5 = γ 5 / ( γ 5 1 ) . Combining (3.3) with (3.4), one has
n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) = n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) C 3 u k 2 2 C 4 ( 1 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) .
(3.5)
From (2.19), (2.23) and (3.5), we obtain
u k 2 = n Z [ ( p ( n + 1 ) u k ( n ) , u k ( n ) ) + ( L ˜ ( n ) u k ( n ) , u k ( n ) ) ] = b ( u k , u k ) + n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) = n Z 1 ( ( L ˜ ( n ) L ( n ) ) u k ( n ) , u k ( n ) ) + 2 f ( u k ) + 2 n Z W ( n , u k ( n ) ) C 5 ( 1 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) + 2 ϕ 1 ( N 0 ) u k γ 1 + 2 ϕ 2 ( N 0 ) u k γ 2 C 6 ( 1 + u k γ 1 + u k γ 2 + u k 2 / γ 5 + u k 2 γ 6 / γ 5 ) .
(3.6)
Since 1 < γ 1 < γ 2 < 2 , 1 < γ 6 < γ 5 < 2 , it follows from (3.6) that { u k } is bounded. Let A > 0 such that
u k 1 ( l + 4 α ) l 4 u k A , k N .
(3.7)
So, passing to a subsequence if necessary, it can be assumed that u k u 0 in E. It is easy to verify that
lim k u k ( n ) = u 0 ( n ) , n Z .
(3.8)
By (W2), there exist M 5 , M 6 > 0 such that
φ i ( | x | ) M 4 + i | x | γ 2 + i 1 , x R N , | x | A , i = 1 , 2 .
(3.9)
For any given number ε > 0 , we can choose an integer N 3 > N 0 such that
M 4 + i K ( σ i , γ 2 + i ) N 3 κ 2 + i { [ ( l + 4 α ) l 4 A ] γ 2 + i + u 0 γ 2 + i } < ε , i = 1 , 2 .
(3.10)
It follows from (3.8) and the continuity of W ( n , x ) on x that there exists k 0 N such that
n = N 2 N 2 | W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) | | u k ( n ) u 0 ( n ) | < ε for  k k 0 .
(3.11)
On the other hand, it follows from (2.11), (3.7), (3.9), (3.10) and (W2) that
| n | > N 2 | W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) | | u k ( n ) u 0 ( n ) | | n | > N 2 [ | W ( n , u k ( n ) ) | + | W ( n , u 0 ( n ) ) | ] ( | u k ( n ) | + | u 0 ( n ) | ) i = 1 2 | n | > N 2 ( 1 + | n | σ i ) [ φ i ( | u k ( n ) | ) + φ i ( | u 0 ( n ) | ) ] ( | u k ( n ) | + | u 0 ( n ) | ) i = 1 2 M 4 + i | n | > N 2 ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i 1 + | u 0 ( n ) | γ 2 + i 1 ) ( | u k ( n ) | + | u 0 ( n ) | ) 2 i = 1 2 M 4 + i | n | > N 2 ( 1 + | n | σ i ) ( | u k ( n ) | γ 2 + i + | u 0 ( n ) | γ 2 + i ) i = 1 2 2 M 4 + i K ( σ i , γ 2 + i ) N 2 κ 2 + i ( u k γ 2 + i + u 0 γ 2 + i ) i = 1 2 2 M 4 + i K ( σ i , γ 2 + i ) N 2 κ 2 + i { [ ( l + 4 α ) l 4 A ] γ 2 + i + u 0 γ 2 + i } 4 ε , k N .
(3.12)
Since ε is arbitrary, combining (3.11) with (3.12), we get
n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) 0 as  k .
(3.13)
It follows from (2.24) that
f ( u k ) f ( u 0 ) , u k u 0 = b ( u k u 0 , u k u 0 ) n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) = u k u 0 2 n Z 1 ( ( L ˜ ( n ) L ( n ) ) ( u k u 0 ) , u k u 0 ) n Z ( W ( n , u k ( n ) ) W ( n , u 0 ( n ) ) , u k ( n ) u 0 ( n ) ) .
(3.14)

Since f ( u k ) f ( u 0 ) , u k u 0 0 , it follows from (3.8), (3.13) and (3.14) that u k u 0 in E. Hence, f satisfies the (PS)-condition.

Next, for u X 2 , it follows from (2.9), (2.19) and (2.23) that
f ( u ) = 1 2 b ( u , u ) n Z W ( n , u ( n ) ) = 1 2 u 2 n Z W ( n , u ( n ) ) 1 2 u 2 ϕ 1 ( N 0 ) u γ 1 ϕ 2 ( N 0 ) u γ 2 +
(3.15)

as u + and u X 2 , since 1 < γ 1 < γ 2 < 2 .

Finally, we prove that assumption (ii) in Lemma 2.6 holds. Let X ˜ X be any finite dimensional subspace. Then there exist constants c 0 = c ( X ˜ ) > 0 and c = c ( X ˜ ) > 0 such that
c 0 u u γ i c u , i = 7 , 8 , u X ˜ .
(3.16)
From (2.9), (2.23), (3.16) and (W4), one has
f ( u ) = 1 2 b ( u , u ) n Z W ( n , u ( n ) ) 1 2 u 2 b 3 n Z | u ( n ) | γ 7 + b 4 n Z | u ( n ) | γ 8 = 1 2 u 2 b 3 u γ 7 γ 7 + b 4 u γ 8 γ 8 1 2 u 2 b 3 c 0 γ 7 u γ 7 + b 4 c γ 8 u γ 8 , u X ˜ .
Since 1 < γ 7 < γ 8 < 2 , the above estimation implies that there exist ρ = ρ ( b 3 , b 4 , c 0 , c ) = ρ ( X ˜ ) > 0 and σ = σ ( b 3 , b 4 , c 0 , c ) = σ ( X ˜ ) > 0 such that
f ( u ) 0 , u B ρ X ˜ ; f ( u ) σ , u B ρ X ˜ .

This shows that assumption (ii) in Lemma 2.6 holds. By Lemma 2.6, f has infinitely many critical points which are homoclinic solutions for system (1.1). □

4 Example

In this section, we give an example to illustrate our result.

Example 4.1 In system (1.1), let p ( n ) be an N × N real symmetric positive definite matrix for all n Z , L ( n ) = ( 1 + sin 2 n ) ( | n | 4 / 5 6 ) I N , and let
W ( n , x ) = ( 1 + sin 2 n ) [ ( 1 + | n | 1 / 9 ) | x | 5 / 4 3 | x | 3 / 2 + ( 1 + | n | 1 / 2 ) | x | 7 / 4 ] .
(4.1)
Then L satisfies ( L ν ) with ν = 6 / 5 , and
W ( n , x ) = ( 1 + sin 2 n ) [ 5 4 ( 1 + | n | 1 / 9 ) | x | 3 / 4 x 9 2 | x | 1 / 2 x + 7 4 ( 1 + | n | 1 / 2 ) | x | 1 / 4 x ] , | W ( n , x ) | 5 ( 1 + | n | 1 / 9 ) | x | 5 / 4 + 5 ( 1 + | n | 1 / 2 ) | x | 7 / 4 , ( n , x ) Z × R N , | W ( n , x ) | 7 ( 1 + | n | 1 / 9 ) | x | 1 / 4 + 8 ( 1 + | n | 1 / 2 ) | x | 3 / 4 , ( n , x ) Z × R N , 2 W ( n , x ) W ( n , x ) 1 4 | x | 7 / 4 3 | x | 3 / 2 , ( n , x ) Z × R N
and
W ( n , x ) | x | 5 / 4 6 | x | 3 / 2 , ( n , x ) Z × R N .
Thus all the conditions of Theorem 1.2 are satisfied with
5 4 = γ 1 = γ 3 = γ 7 < γ 6 = γ 8 = 3 2 < γ 5 = γ 4 = γ 2 = 7 4 ; a 1 = a 2 = 5 ; b 1 = 1 4 , b 2 = 3 , b 3 = 1 , b 4 = 6 ; σ 1 = 1 9 , σ 2 = 1 2 ; φ 1 ( s ) = 7 s 1 / 4 , φ 2 ( s ) = 8 s 3 / 4 .

Hence, by Theorem 1.2, system (1.1) has infinitely many nontrivial homoclinic solutions. However, one can see that W ( n , x ) defined by (4.1) does not satisfy (W1) and (W2).

Declarations

Acknowledgements

The author would like to express their thanks to the referees for their helpful suggestions. This work is partially supported by the NNSF (No. 11171351) of China and supported by the Scientific Research Fund of Hunan Provincial Education Department (08A053) and supported by the Hunan Provincial Natural Science Foundation of China (No. 11JJ2005).

Authors’ Affiliations

(1)
Department of Mathematics, Huaihua College, Huaihua, P.R. China

References

  1. Agarwal RP: Difference Equations and Inequalities: Theory, Methods and Applications. 2nd edition. Dekker, New York; 2000.MATHGoogle Scholar
  2. Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations. Kluwer Academic, Dordrecht; 1996.View ArticleMATHGoogle Scholar
  3. Chen P, Tang XH, Agarwal RP: Homoclinic solutions for second order differential equations generated by impulses. Adv. Math. Sci. Appl. 2011, 21: 447-465.MathSciNetMATHGoogle Scholar
  4. Deng XQ, Cheng G: Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign. Acta Appl. Math. 2008, 103: 301-314. 10.1007/s10440-008-9237-zMathSciNetView ArticleMATHGoogle Scholar
  5. Guo C, O’Regan D, Xu Y, Agarwal RP: Existence and multiplicity of homoclinic orbits of a second-order differential difference equation via variational methods. Sci. Publ. State Univ. Novi Pazar Ser. A, Appl. Math. Inf. Mech. 2012, 4: 1-15.Google Scholar
  6. Guo C, O’Regan D, Xu Y, Agarwal RP: Homoclinic orbits for a singular second-order neutral differential equation. J. Math. Anal. Appl. 2010, 366: 550-560. 10.1016/j.jmaa.2009.12.038MathSciNetView ArticleMATHGoogle Scholar
  7. Ma M, Guo ZM: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 2006, 323(1):513-521. 10.1016/j.jmaa.2005.10.049MathSciNetView ArticleMATHGoogle Scholar
  8. Ma M, Guo ZM: Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal. 2007, 67: 1737-1745. 10.1016/j.na.2006.08.014MathSciNetView ArticleMATHGoogle Scholar
  9. Lin XY, Tang XH: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 2011, 373: 59-72. 10.1016/j.jmaa.2010.06.008MathSciNetView ArticleMATHGoogle Scholar
  10. Lin, XY, Tang, XH: Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential (to appear)Google Scholar
  11. Tang XH, Lin XY, Xiao L: Homoclinic solutions for a class of second order discrete Hamiltonian systems. J. Differ. Equ. Appl. 2010, 16: 1257-1273. 10.1080/10236190902791635MathSciNetView ArticleMATHGoogle Scholar
  12. Tang XH, Lin XY: Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 2011, 17: 1617-1634. 10.1080/10236191003730514MathSciNetView ArticleMATHGoogle Scholar
  13. Tang XH, Lin XY: Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 2013, 19: 796-813. 10.1080/10236198.2012.691168MathSciNetView ArticleMATHGoogle Scholar
  14. Ding YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 1995, 25(11):1095-1113. 10.1016/0362-546X(94)00229-BMathSciNetView ArticleMATHGoogle Scholar

Copyright

© Lin; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.