Open Access

Homoclinic solutions for second order discrete p-Laplacian systems

Advances in Difference Equations20112011:57

https://doi.org/10.1186/1687-1847-2011-57

Received: 23 June 2011

Accepted: 21 November 2011

Published: 21 November 2011

Abstract

Some new existence theorems for homoclinic solutions are obtained for a class of second-order discrete p-Laplacian systems by critical point theory, a homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second-order difference systems. A completely new and effective way is provided for dealing with the existence of solutions for discrete p-Laplacian systems, which is different from the previous study and generalize the results.

2010 Mathematics Subject Classification: 34C37; 58E05; 70H05.

Keywords

homoclinic solutions discrete variational methods p-Laplacian systems

1. Introduction

In this article, we shall be concerned with the existence of homoclinic orbits for the second-order discrete p-Laplacian systems:
Δ ( φ p ( Δ u ( n - 1 ) ) ) = F ( n , u ( n ) ) + f ( n ) , n , u ,
(1.1)

where p > 1, φ p (s) = |s| p-2 s is the Laplacian operator, Δu(n) = u(n + 1) - u(n) is the forward difference operator, F : × is a continuous function in the second variable and satisfies F(n + T, u) = F(n, u) for a given positive integer T. As usual, , and denote the set of all natural numbers, integers and real numbers, respectively. For a, b , denote (a) = {a, a + 1,...}, (a, b) = {a, a + 1,... b} when ab.

Differential equations occur widely in numerous settings and forms both in mathematics itself and in its application to statistics, computing, electrical circuit analysis, biology and other fields, so it is worthwhile to explore this topic. As is known to us, the development of the study of periodic solution and their connecting orbits of differential equations is relatively rapid. Many excellent results were obtained by variational methods [111]. It is well-known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation, its perturbed system probably produce chaotic phenomenon.

On the other hand, we know that a differential equation model is often derived from a difference equation, and numerical solutions of a differential equation have to be obtained by discretizing the differential equation, therefore, the study of periodic solution and connecting orbits of difference equation is meaningful [1224].

It is clear that system (1.1) is a discretization of the following second differential system
d d t ( | u ˙ ( t ) | p - 2 u ˙ ( t ) ) = F ( t , u ( t ) ) + f ( t ) , t , u .
(1.2)
Recently, the following second order self-adjoint difference equation
Δ [ p ( n ) Δ u ( n - 1 ) ] + q ( n ) u ( n ) = f ( n , u ( n ) ) , n , u
(1.3)

has been studied by using variational method. Yu and Guo established the existence of a periodic solution for Equation (1.3) by applying the critical point theory in [15]. Ma and Guo [20] obtained homoclinic orbits as the limit of the subharmonics for Equation (1.3) by applying the Mountain Pass theorem relying on Ekelands variational principle and the diagonal method, their results are based on scalar equation with q(t) ≠ 0, if q(t) = 0, the traditional ways in [20] are inapplicable to our case.

Some special cases of (1.1) have been studied by many researchers via variational methods [1517, 22, 23]. However, to our best knowledge, results on homoclinic solutions for system (1.1) have not been studied. Motivated by [9, 10, 20], the main purpose of this article is to give some sufficient conditions for the existence of homoclinic solutions to system (1.1).

Our main results are the following theorems.

Theorem 1.1 Assume that F and f satisfy the following conditions:

(H1) F(n, x) is T-periodic with respect to n,T > 0 and continuously differentiable in x;

(H2) There are constants b 1 > 0 and ν > 1 such that for all (n, x) × ,
F ( n , x ) F ( n , 0 ) + b 1 | x | ν ;

(H3) f ≠ 0 is a bounded function such that n | f ( n ) | ν ( ν - 1 ) < .

Then, system (1.1) possesses a homoclinic solution.

Theorem 1.2 Assume that F and f satisfy the following conditions:

(H4) F(n, x) = K(n, x) - W(n, x), where K, W is T-periodic with respect to n,T > 0, K(n, x) and W (n, x) are continuously differentiable in x;

(H5) There is a constant μ > p such that for every n , u \{0},
0 < μ W ( n , x ) ( W ( n , x ) , x ) ;

(H6) W(n,x) = o(|x|), as |x| → 0 uniformly with respect to n;

(H7) There exist constants b 2 > 0 and γ (1, p] such that for all (n, u) × ,
K ( n , 0 ) = 0 , K ( n , x ) b 2 | x | γ ;
(H8) There is a constant ϱ [ p , μ ) such that
( x , K ( n , x ) ) ϱ K ( n , x ) , ( n , x ) [ 0 , T ] × ;
(H9) f0 is a bounded function such that
n | f ( n ) | q < min δ p - 1 p , b 2 δ γ - 1 - M 1 δ μ - 1 q C p ,
where 1 p + 1 q = 1 and
M 1 = sup { W ( n , x ) | n [ 0 , T ] , x , | x | = 1 } ,
C is given in (3.4) and δ (0,1] such that
b 2 δ γ - 1 - M 1 δ μ - 1 = max x [ 0 , 1 ] b 2 x γ - 1 - M 1 x μ - 1 .

Then, system (1.1) possesses a nontrivial homoclinic solution.

Remark Obviously, condition (H9) holds naturally when f = 0. Moreover, if b 2(γ - 1) ≤ M (μ - 1), then
δ = b 2 ( γ - 1 ) M ( μ - 1 ) 1 ( μ - γ ) ,
and so condition (H9) can be rewritten as
n | f ( n ) | q < min 1 p b 2 ( γ - 1 ) M ( μ - 1 ) ( p - 1 ) ( μ - γ ) , b 2 ( μ - γ ) μ - 1 b 2 ( γ - 1 ) M ( μ - 1 ) ( γ - 1 ) ( μ - γ ) q C p ,
if b 2(γ - 1) > M(μ - 1), then δ = 1 and b 2δ(γ - 1)- M δ(μ - 1)= b 2 - M, and so condition (H9) can be rewritten as
n | f ( n ) | q < C - p ( min { p - 1 , b 2 - M } ) q .
(1.4)

2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure.

Let S be the vector space of all real sequences of the form
u = { u ( n ) } n = ( , u ( - n ) , u ( - n + 1 ) , , u ( - 1 ) , u ( 0 ) , u ( 1 ) , , u ( n ) , ) ,
namely
S = { u = { u ( n ) } : u ( n ) N , n } .
For each k , let E k denote the Banach space of 2kT-periodic functions on with values in N under the norm
| | u | | E k : = n = - k T k T - 1 ( | Δ u ( n - 1 ) | p + | u ( n ) | p ) 1 p .
In order to receive a homoclinic solution of (1.1), we consider a sequence of systems:
Δ ( φ p ( Δ u ( n - 1 ) ) ) + F ( n , u ( n ) ) = f k ( n ) , n , u ,
(2.1)

where f k : N is a 2kT-periodic extension of restriction of f to the interval [-kT, kT - 1], k . Similar to [20], we will prove the existence of one homoclinic solution of (1.1) as the limit of the 2kT-periodic solutions of (2.1).

For each k , let l 2 k T p ( , N ) denote the Banach space of 2kT-periodic functions on with values in N under the norm
| | u | | l 2 k T p = n [ - k T , k T - 1 ] | u ( n ) | p 1 p , u l 2 k T p .
Moreover, l 2 k T denote the space of all bounded real functions on the interval [-kT, kT - 1] endowed with the norm
| | u | | l 2 k T = max n [ - k T , k T - 1 ] { | u ( n ) | } , u l 2 k T .
Let
I k ( u ) = n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + F ( n , u ( n ) ) + ( f k ( n ) , u ( n ) ) .
(2.2)
Then I k C 1(E k ,) and it is easy to check that
I k ( u ) v = n = - k T k T - 1 [ ( | Δ u ( n - 1 ) | p - 2 Δ u ( n - 1 ) , Δ v ( n - 1 ) ) + ( F ( n , u ( n ) ) , v ( n ) ) + ( f k ( n ) , v k ( n ) ) ] .

Furthermore, the critical points of I k in E k are classical 2kT-periodic solutions of (2.1).

That is, the functional I k is just the variational framework of (2.1).

In order to prove Theorem 1.2, we need the following preparations.

Let η k : E k → [0, +∞) be such that
η k ( u ) = n = - k T k T - 1 [ | Δ u ( n - 1 ) | p + p K ( n , u ) ] 1 p .
(2.3)
Then it follows from (2.2), (2.3), (H4) and (H8) that
I k ( u ) = 1 p η k p ( u ) + n = - k T k T - 1 [ - W ( n , u ( n ) ) + ( f k ( n ) , u ( n ) ) ] ,
(2.4)
and
I k ( u ) u n = - k T k T - 1 [ | Δ u ( n - 1 ) | p + ϱ K ( n , u ( n ) ) ] - n = - k T k T - 1 ( W ( n , u ( n ) ) , u ( n ) ) + n = - k T k T - 1 ( f k ( n ) , u ( n ) )
(2.5)

We will obtain the critical points of I by using the Mountain Pass Theorem. Since the minimax characterisation provides the critical value, it is important for what follows. Therefore, we state these theorems precisely.

Lemma 2.1 [7] Let E be a real Banach space and I C 1(E, ) satisfy (PS)-condition. Suppose that I satisfies the following conditions:
  1. (i)

    I(0) = 0;

     
  2. (ii)

    There exist constants ρ, α > 0 such that I | B p ( 0 ) α ;

     
  3. (iii)

    There exists e E \ B ̄ ρ ( 0 ) such that I(e) < 0.

     
Then I possesses a critical value cα given by
c = inf g Γ max s [ 0 , 1 ] I ( g ( s ) ) ,
where B ρ (0) is an open ball in E of radius ρ centered at 0, and
Γ = { g C ( [ 0 , 1 ] , E } : g ( 0 ) = 0 , g ( 1 ) = e } .

Lemma 2.2 [4] Let E be a Banach space, I : E a functional bounded from below and differentiable on E. If I satisfies the (PS)-condition then I has a minimum on E.

Lemma 2.3 [3] For every n , the following inequalities hold:
W ( n , u ) W n , u | u | | u | μ , i f 0 < | u | 1 ,
(2.6)
W ( n , u ) W n , u | u | | u | μ , i f q u a d | u | 1 .
(2.7)
Lemma 2.4 Set m := inf{W(n, u) : n [0,T], |u| = 1}. Then for every ζ \{0}, u E k \{0}, we have
n = - k T k T - 1 W ( n , ζ u ( n ) ) m | ζ | μ n = - k T k T - 1 | u ( n ) | μ - 2 k T m .
(2.8)

Proof Fix ζ \{0} and u E k \{0}.

Set
A k = { n [ - k T , k T - 1 ] : | ζ u ( n ) | 1 } , B k = { n [ - k T , k T - 1 ] : | ζ u ( n ) | 1 } .
From (2.7), we have
n = - k T k T W ( n , ζ u ( n ) ) n B k W ( n , ζ u ( n ) ) n B k W n , ζ u ( n ) | ζ u ( n ) | | ζ u ( n ) | μ m n B k | ζ u ( n ) | μ m n = - k T k T - 1 | ζ u ( n ) | μ - m n A k | ζ u ( n ) | μ m | ζ | μ n = - k T k T - 1 | u ( n ) | μ - 2 k T m .

3. Existence of subharmonic solutions

In this section, we prove the existence of subharmonic solutions. In order to establish the condition of existence of subharmonic solutions for (2.1), first, we will prove the following lemmas, based on which we can get results of Theorem 1.1 and Theorem 1.2.

Lemma 3.1 Let a, b , a, b ≥ 0 and u E k . Then for every n,t , the following inequality holds:
| u ( n ) | ( a + b + 1 ) - 1 ν t = n - a n + b | u ( t ) | ν 1 ν + max { a + 1 , b } ( a + b + 1 ) 1 p t = n - a n + b | Δ u ( t - 1 ) | p 1 p .
(3.1)
Proof Fix n , for every τ ,
| u ( n ) | | u ( τ ) | + t = τ + 1 n Δ u ( t - 1 ) ,
(3.2)
then by (3.2) and Hö der inequality, we obtain
( a + b + 1 ) | u ( n ) | τ = n - a n + b | u ( τ ) | + τ = n - a n + b t = τ + 1 n | Δ u ( t - 1 ) | τ = n - a n + b | u ( τ ) | + τ = n - a n t = n = a + 1 n | Δ u ( t - 1 ) | + τ = n + 1 n + b t = n + 1 n + b | Δ u ( t - 1 ) | ( a + b + 1 ) ( ν - 1 ) ν t = n - a n + b | u ( t ) | ν 1 ν + max { a + 1 , b } t = n - a n + b | Δ u ( t - 1 ) | ( a + b + 1 ) ( ν - 1 ) ν t = n - a n + b | u ( t ) | ν 1 ν + max { a + 1 , b } t = n - a n + b ( a + b + 1 ) ( p - 1 ) p t = n - a n + b | Δ u ( t - 1 ) | p 1 p ,

which implies that (3.1) holds. The proof is complete.

Corollary 3.1 Let u E k . Then for every n , the following inequality holds:
| | u ( n ) | | l 2 k T T - 1 ν n = - k T k T - 1 | u ( n ) | ν 1 ν + T ( p - 1 ) p n = - k T k T - 1 | Δ u ( n - 1 ) | p 1 p ,
(3.3)
Proof For n [-kT, kT - 1], we can choose n* [-kT, kT - 1] such that u(n*) = max n[-kT, kT-1]|u(n)|. Let a [0,T) and b = T - a - 1 such that -kTn* - an* ≤ n* + bkT - 1. Then by (3.1), we have
| u ( n * ) | T - 1 ν n = n * - a n * + b | u ( n ) | ν 1 ν + T ( p - 1 ) p n = n * - a n * + b | Δ u ( n - 1 ) | p d s 1 p T - 1 ν n = - k T k T - 1 | u ( n ) | ν 1 ν + T ( p - 1 ) p n = - k T k T - 1 | Δ u ( n - 1 ) | p 1 p ,

which implies that (3.3) holds. The proof is complete.

Corollary 3.2 Let u E k . Then for every n , the following inequality holds:
| | u ( n ) | | l 2 k T 2 max { T ( p - 1 ) p , T - 1 } | | u | | E k = ˙ C | | u | | E k .
(3.4)
Proof Let ν = p in (3.3), we have
| | u ( n ) | | l 2 k T p 2 p T - 1 n = - k T k T - 1 | u ( n ) | p + T p - 1 n = - k T k T - 1 | Δ u ( n - 1 ) | p 2 p max { T p - 1 , T - p } n = - k T k T - 1 | Δ u ( n - 1 ) | p + | u ( n ) | p = 2 p max { T p - 1 , T - p } | | u | | E k p ,

which implies that (3.4) holds. The proof is complete.

For the sake of convenience, set Λ = min δ p - 1 p , b 2 δ γ - 1 - M 1 δ μ - 1 . By (H9), we have
n | f ( n ) | q < Λ q C p ,
(3.5)

where C is given in (3.4).

Here and subsequently,
( k 0 ) = ˙ { k : k , k k 0 } .
Lemma 3.2 Assume that F and f satisfy (H1)-(H3). Then for every k , system (2.1) possesses a 2kT-periodic solution u k E k such that
1 p n = - k T k T - 1 | Δ u k ( n - 1 ) | p + b 1 n = - k T k T - 1 | u k | ν M n = - k T k T - 1 | u k | ν 1 ν ,
(3.6)
where
M = n | f ( n ) | ν ( ν - 1 ) ( ν - 1 ) ν .
(3.7)
Proof Set C 0 = n = 0 T F ( n , 0 ) . By (H2), (H3), (2.2), and the Hö der inequality, we have
I k ( u ) = n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + F ( n , u ( n ) ) + ( f k ( n ) , u ( n ) ) n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + F ( n , 0 ) + b 1 | u ( n ) | ν + ( f k ( n ) , u ( n ) ) = 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 n = - k T k T - 1 | u ( n ) | ν + n = - k T k T - 1 ( f k ( n ) , u ( n ) ) + 2 k C 0 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 n = - k T k T - 1 | u ( n ) | ν - n = - k T k T - 1 | f k ( n ) ν ( ν - 1 ) n = - k T k T - 1 | u ( n ) | ν 1 ν + 2 k C 0 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 n = - k T k T - 1 | u ( n ) | ν - M n = - k T k T - 1 | u ( n ) | ν 1 ν + 2 k C 0 .
(3.8)
For any x [0, +∞), we have
b 1 2 x ν - M x - b 1 2 ( ν - 1 ) 2 M b 1 μ ν ( ν - 1 ) : = - D .
It follows from (3.8) that
I k ( u ) 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 2 n = - k T k T - 1 | u ( n ) | ν - D + 2 k C 0 .

Consequently, I k is a functional bounded from below.

Set
ū = 1 2 k T n = - k T k T - 1 u ( n ) , and ũ ( n ) = u ( n ) = ū .
Then by Sobolev's inequality, we have
| | ũ | | l 2 k T C 1 | | Δ u ( n - 1 ) | | l 2 k T p , and | | ũ | | l 2 k T p C 2 | | Δ u ( n - 1 ) | | l 2 k T p .
(3.9)
In view of (3.9), it is easy to verify, for each k , that the following conditions are equivalent:
  1. (i)

    | | u | | E k ;

     
  2. (ii)

    | ū | p + n = - k T k T - 1 | Δ u ( n - 1 ) | p ;

     
  3. (iii)

    n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 2 n = - k T k T - 1 | u ( n ) | ν .

     
Hence, from (3.8), we obtain
I k ( u ) + as | | u | | E k .
Then, it is easy to verify that I k satisfies (PS)-condition. Now by Lemma 2.2, we conclude that for every k there exists u k E k such that
I k ( u k ) = inf u E k I k ( u ) .
Since
I k ( 0 ) = n = - k T k T - 1 F ( n , 0 ) = 2 k C 0 ,
we have I k (u k ) ≤ 2kC 0. It follows from (3.8) that
1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 1 n = - k T k T - 1 | u k | ν M n = - k T k T - 1 | u k ( n ) | p 1 p .

This shows that (3.6) holds. The proof is complete.

Lemma 3.3 Assume that all conditions of Theorem 1.2 are satisfied. Then for every k (k 0), the system (2.1) possesses a 2kT-periodic solution u k E k .

Proof In our case it is clear that I k (0) = 0. First, we show that I k satisfies the (PS) condition. Assume that {u j } jin E k is a sequence such that {I k (u j )} jis bounded and I k ( u j ) 0 , j + . Then there exists a constant C k > 0 such that
| I k ( u j ) | C k , | | I k ( u j ) | | k * C k
(3.10)
for every j . We first prove that {u j } jis bounded. By (2.3) and (H5), we have
η k p ( u j ) p I k ( u j ) + p μ t = - k T k T - 1 ( W ( n , u j ( n ) ) , u j ( n ) ) - p n = - k T k T - 1 ( f k ( n ) , u ( n ) ) ,
(3.11)
From (2.5), (3.5), (3.10) and (3.11), we have
1 - ϱ μ η k p ( u j ) p I k ( u j ) - p μ I k ( u j ) u j - p - p μ n = - k T k T - 1 ( f k ( n ) , u ( n ) ) p C k + p μ I k ( u j ) k * + p - p μ n = - k T k T - 1 f k ( n ) q 1 q u j E k p C k + p C k μ + p ( μ - 1 ) Λ C p - 1 μ u j E k = p C k + D k u j E k , k ( k 0 ) ,
(3.12)
where
D k = p C k μ + p ( μ - 1 ) Λ C p - 1 μ .
Without loss of generality, we can assume that | | u j | | E k 0 . Then from (2.3), (3.3), and (H7), we obtain for j ,
η k p ( u j ) = n = - k T k T - 1 [ | Δ u j ( n - 1 ) | p + p K ( n , u j ) ] n = - k T k T - 1 [ | Δ u j ( n - 1 ) | p + p b 2 | u j ( n ) | γ ] n = - k T k T - 1 | Δ u j ( n - 1 ) | p + p b 2 ( C | | u j ( n ) | | E k ) γ - p n = - k T k T - 1 | u j ( n ) | p min { 1 , p b 2 ( C | | u j ( n ) | | E k ) γ - p } n = - k T k T - 1 | Δ u j ( n - 1 ) | p + t = - k T k T - 1 | u j ( n ) | p = min { 1 , p b 2 ( C | | u j ( n ) | | E k ) γ - p } | | u j | | E k p = min { | | u j | | E k p , p b 2 C γ - p | | u j ( n ) | | E k γ }
(3.13)
Combining (3.12) with (3.13), we have
min u j E k p , p b 2 C γ - p u j ( n ) E k γ μ μ - ϱ ( p C k + D k u j E k )
(3.14)

It follows from (3.14) that {u j } jis bounded in E k , it is easy to prove that {u j } jhas a convergent subsequence in E k . Hence, I k satisfies the Palais-Smale condition.

We now show that there exist constants ρ, α > 0 independent of k such that I k satisfies assumption (ii) of Lemma 2.1 with these constants. If u E k = δ C : = | ρ , then it follows from (3.4) that |u(n)| ≤ δ ≤ 1 for n [-kT, kT - 1] and k (k 0). By Lemma 2.3 and (H9), we have
n = - k T k T - 1 W ( n , u ( n ) ) = n [ - k T , k T - 1 ] | u ( n ) 0 W ( n , u ( n ) ) n [ - k T , k T - 1 ] | u ( n ) 0 W n , u ( n ) u ( n ) u ( n ) μ M 1 n = - k T k T - 1 u ( n ) μ M 1 δ μ - γ n = - k T k T - 1 u ( n ) γ , k ( k 0 ) ,
(3.15)
and
n = - k T k T - 1 | u ( n ) | p δ p - γ n = - k T k T - 1 | u ( n ) | γ , k ( k 0 ) .
(3.16)
Set
α = δ C 1 C p - 1 min δ p - 1 p , b 2 δ γ - 1 - M 1 δ μ - 1 - n | f ( n ) | q .
(3.17)
Hence, from (2.1), (3.4) and (3.15)-(3.17), we have
I k ( u ) = n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + K ( n , u ( n ) ) - W ( n , u ( n ) ) + ( f k ( n ) , u ( n ) ) 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + b 2 n = - k T k T - 1 | u ( n ) | γ - n = - k T k T - 1 W ( n , u ( n ) ) + n = - k T k T - 1 ( f k ( n ) , u ( n ) ) 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + ( b 2 - M 1 δ μ - γ ) n = - k T k T - 1 | u ( n ) | γ - n = - k T k T - 1 | f k ( n ) | q 1 q n = - k T k T - 1 | u ( n ) | p 1 p 1 p n = - k T k T - 1 | Δ u ( n - 1 ) | p + ( b 2 - M 1 δ μ - γ ) n = - k T k T - 1 | u ( n ) | γ - n | f k ( n ) | q 1 q n = - k T k T - 1 | u ( n ) | p 1 p min 1 p , b 2 δ γ - p - M 1 δ μ - p n = - k T k T - 1 | Δ u ( n - 1 ) | p + n = - k T k T - 1 | u ( n ) | p - n | f k ( n ) | q 1 q n = - k T k T - 1 | Δ u ( n - 1 ) | p + n = - k T k T - 1 | u ( n ) | p 1 p = min 1 p , b 2 δ γ - p - M 1 δ μ - p u E k p - u E k n | f k ( n ) | q 1 q = δ C 1 C p - 1 min δ p - 1 p , b 2 δ γ - 1 - M 1 δ μ - 1 - n | f k ( n ) | q 1 q = α , k ( k 0 ) .
(3.18)

(3.18) shows that u E k = ρ implies that I k (u) ≥ α for k (k 0).

Finally, it remains to show that I k satisfies assumption (iii) of Lemma 2.1. Set
a 1 = max { K ( n , x ) | n [ 0 , T ] , x N , | x | = 1 } ,
and
a 2 = max { K ( n , x ) | n [ 0 , T ] , x N , | x | 1 } ,
Then by (H8) and 0 < a 1a 2 < ∞,
K ( n , x ) a 1 | x | ϱ + a 2 , for ( n , x ) × N .
(3.19)
By (2.2), (3.19) and Lemma 2.4, we have for every ζ \ {0} and u E k \ {0}
I k ( ζ u ) = n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + K ( n , ζ u ( n ) ) - W ( n , ζ u ( n ) ) + ζ ( f k ( n ) , u ( n ) ) | ζ | p p n = - k T k T - 1 | Δ u ( n - 1 ) | p + a 1 | ζ | ϱ n = - k T k T - 1 | u ( n ) | ϱ - m | ζ | μ n = - k T k T - 1 | u ( n ) | μ + | ζ | n | f k ( n ) | q 1 q u E k + 2 k T ( m + a 2 ) , k ( k 0 ) .
(3.20)
Take Q E k 0 such that Qk 0 T) = 0 and Q ≠ 0. Since p ϱ < μ and m > 0, (3.20) implies that there exists ξ > 0 such that ξ Q E k 0 > ρ and I k 0 ( ξ Q ) < 0 . Set e k 0 ( n ) = ξ Q ( n ) and
e k ( n ) = e k 0 ( n ) , for n k 0 T , 0 , for  k 0 T < n k T .
(3.21)
Then e k E k , e k E k = e k 0 E k 0 > ρ and I k ( e k ) = I k 0 ( e k 0 ) < 0 for k (k 0). By Lemma 2.1, I k possesses a critical value c k α given by
c k = inf g Γ k max s [ 0 , 1 ] I k ( g ( s ) ) , k ( k 0 ) .
where
Γ k = { g C ( [ 0 , 1 ] , E k ) : g ( 0 ) = 0 , g ( 1 ) = e k } , k ( k 0 ) .
Hence, for k (k 0), there exists u k E k such that
I k ( u k ) = c k , and  I k ( u k ) = 0 .

Then function u k is a desired classical 2kT-periodic solution of (1.1) for k (k 0). Since c k > 0, u k is a nontrivial solution even if f k (n) = 0. The proof is complete.

4. Existence of homoclinic solutions

Lemma 4.1 Let u k E k be the solution of system (2.1) that satisfies (3.6) for k . Then there exists a positive constant d 1 independent of k such that
u k l 2 k T d 1 , k .
Proof By (3.6), we have
b 1 n = - k T k T - 1 | u k ( n ) | ν M n = - k T k T - 1 | u k ( n ) | ν 1 ν ,
which implies that
n = - k T k T - 1 | u k ( n ) | ν M b 1 ν ( ν - 1 ) .
(4.1)
From (3.6), we obtain
n = - k T k T - 1 | Δ u k ( n - 1 ) | p p M M b 1 1 ( ν - 1 ) .
(4.2)
It follows from (3.3), (4.1) and (4.2) that
u k l 2 k T T - 1 ν n = - k T k T - 1 | u ( n ) | ν 1 ν + T ( p - 1 ) p n = - k T k T - 1 | Δ u ( n - 1 ) | p 1 p T - 1 ν M b 1 1 ( ν - 1 ) + T ( p - 1 ) p ( p M ) 1 p M b 1 1 p ( ν - 1 ) : = d 1 .
Lemma 4.2 Let u k E k be the solution of system (1.1) which satisfies Lemma 3.3 for k (k 0). Then there exists a positive constant d 2 independent of k such that
u k l 2 k T d 2 .
(4.3)
Proof For k (k 0), let g k : [0,1] → E k be a curve given by g k (s) = se k where e k is defined by (3.21). Then g k Γ k and I k ( g k ( s ) ) = I k 0 ( g k 0 ( s ) ) for all k (k 0) and s [0,1], Therefore,
c k max s [ 0 , 1 ] I k 0 ( g k 0 ( s ) ) d 0 , k ( k 0 ) .

where d 0 is independent of k.

As I k ( u k ) = 0 , we get from (2.2), (2.5) and (H5)
c k = I k ( u k ) - 1 ϱ I k ( u k ) u k μ ϱ - 1 n = - k T k T - 1 W ( n , u ( n ) ) + ϱ - 1 ϱ n = - k T k T - 1 ( f k ( n ) , u k ( n ) ) ,
and hence
n = - k T k T - 1 W ( n , u k ( n ) ) 1 μ - ϱ ϱ c k - ( ϱ - 1 ) n = - k T k T - 1 ( f k ( n ) , u k ( n ) ) .
Combining the above with (2.4), we have
η k p ( u k ) = p I k ( u k ) + p n = - k T k T - 1 W ( n , u k ( n ) ) - p n = - k T k T - 1 ( f k ( n ) , u k ( n ) ) p μ d 0 μ - ϱ + p ( μ - 1 ) μ - ϱ n = - k T k T - 1 | f k ( n ) | q 1 q u k E k p μ d 0 μ - ϱ + p ( μ - 1 ) Λ C p - 1 ( μ - ϱ ) u k E k , k ( k 0 ) .
(4.4)
Since u k ≠ 0, similar to the proof of (3.13), we have
η k p ( u k ) min { u k E k p , p b C γ - p u k E k γ } , k ( k 0 ) .
(4.5)
From (4.4) and (4.5), we obtain
min { u k E k p , p b C γ - p u k E k γ } p μ d 0 μ - ϱ + p ( μ - 1 ) Λ C p - 1 ( μ - ϱ ) u k E k , k ( k 0 ) .
(4.6)
Since all coefficients of (4.6) are independent of k, we see that there is d 2 > 0 independent of k such that
u k E k d 2 , k ( k 0 ) ,
(4.7)

which, together with (3.4), implies that (4.3) holds. The proof is complete.

5. Proofs of theorems

Proof of Theorem 1.1 The proof is similar to that of [20], but for the sake of completeness, we give the details.

We will show that {u k } kpossesses a convergent subsequence { u k m } in E loc (,) and a nontrivial homoclinic orbit u emanating from 0 such that u k m u as k m → ∞.

Since u k = {u k (t)} is well defined on [- kT, kT - 1] and ||u k || k d for all k , we have the following consequences.

First, let u k = {u k (t)} be well defined on [-T,T - 1]. It is obvious that {u k } is isomorphic to 2T . Thus, there exists a subsequence u k m 1 and u 1 E 1 of {u k } k\{1}such that
u k m 1 - u 1 1 0 .
Second, let u k m 1 be restricted to [- 2T, 2T - 1]. Clearly, u k m 1 is isomorphic to 4T . Thus there exists a further subsequence u k m 2 of u k m 1 satisfying u 2 u k m 2 and u 2 E 2 such that
u k m 2 - u 2 2 0 k m .
Repeat this procedure for all k . We obtain sequence u k m r u k m r - 1 , u p u k m r and there exists u r E r such that
u k m r - u r r 0 , k m , r = 1 , 2 ,
Moreover, we have
u r + 1 - u r r u k m r + 1 - u r + 1 r + u k m r + 1 - u r r 0 ,
which leads to
u r + 1 ( s ) = u r ( s ) , s [ - r T , r T - 1 ] .
So, for the sequence {u r }, we have u r u , r → ∞, where u (s) = u r (s) for s [-rT, rT - 1] and r . Then take a diagonal sequence { u k m } : u k 1 1 , u k 2 2 , u k m m , , since { u k m m } is a sequence of { u k m r } for any r ≥ 1, it follows that
u k m m - u = u k m m - u m m 0 , m .
It shows that
u k m u  as  k m ,  in  E loc ( , ) ,

where u E ( , ) , E ( , ) = u S | u = Σ m = - + ( | Δ u ( n - 1 ) | p + | u ( n ) | p ) < .

By series convergence theorem, u satisfy
u ( n ) 0 , Δ u ( n - 1 ) 0 ,
and
n = - r T r T - 1 { [ | Δ u k m m ( n - 1 ) | p + | u k m m ( n ) | p ] < } = u k m m ,

as |n| → ∞.

Letting n → ∞, r ≥ 1, we have
n = - r T r T - 1 1 p | Δ u k m m ( n - 1 ) | p + F ( n , u k m m ( n ) ) + ( f k ( n ) , u k m m ( n ) ) d 1 ,
as mr, k m r, where d 1 is independent of k, {k m } {k} are chosen as above, we have
n = - k T k T - 1 1 p | Δ u ( n - 1 ) | p + F ( n , u ( n ) ) + ( f ( n ) , u ( n ) ) d 1 .
Letting p → ∞, by the continuity of F(t,u) and I k , which leads to
I ( u ) = n = - + 1 p | Δ u ( n - 1 ) | p + F ( n , u ( n ) ) + ( f ( n ) , u ( n ) ) d 1 , u E ,
and
I ( u ) = 0 .

Clearly, u is a solution of (1.1).

To complete the proof of Theorem 1.2, it remains to prove that u 0 . . By the above argument, we obtain
Δ ( φ p ( Δ u ( n - 1 ) ) ) = F ( n , u ( n ) ) + f ( n ) ,
(5.1)
By (H5) and (H7), it is easy to see that
F ( n , 0 ) = - K ( n , 0 ) + W ( n , 0 ) = 0

This shows that u = 0 is not a solution of (1.1) with f ≠ 0 and so u ≠ 0.

6. Examples

In this section, we give some examples to illustrate our results.

Example 6.1 Consider the second order discrete p-Laplacian systems:
Δ ( φ p ( Δ u ( n - 1 ) ) ) = F ( n , u ( n ) ) + f ( n ) , n , u ,
(6.1)
where
F ( n , x ) = sin 2 π T n + | x | + b 1 | x | 2 , f ( n ) = 1 | n | , ν = 4 3 .

Then it is easy to verify that all conditions of Theorem 1.1 are satisfied. By Theorem 1.1, the system (6.1) has a nontrivial homoclinic solution.

Example 6.2 Consider the second order discrete systems:
Δ 2 u ( n - 1 ) = F ( n , u ( n ) ) + f ( n ) , n , u ,
(6.2)
where
p = 2 , K ( n , x ) = | x | 2 + 3 | x | 5 2 , W ( n , x ) = 1 6 1 + sin n π 2 | x | 4 , f ( n ) = a 1 + | n | .

It is easy to verify that conditions (H4)-(H8) are satisfied with γ = 2 , ϱ = 5 2 , μ = 4 , T = 4 and b 2 = 1.

Noting that
M = sup 1 6 1 + sin n π 2 | x | 4 | n { 0 , 1 , 2 , 3 } , x N , | x | = 1 = 1 3 .
Therefore, b 2(γ - 1) = M(μ - 1) = 1. Since
n | f ( n ) | 2 = a 2 n 1 ( 1 + | n | ) 2 = a 2 π 2 3 - 1 .

so (1.4) holds, i.e., condition (H9) holds if 0 < a 2 < 3 32 ( π 2 - 3 ) . In view of Theorem 1.2, the system (6.2) possesses a nontrivial homoclinic solution.

Declarations

Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript. This study was partially supported by Major Project of Science Research Fund of Education Department in Hunan (No: 11A095).

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science Jishou University
(2)
School of Mathematical Sciences and Computing Technology Central South University
(3)
College of Mathematics and Computer Science Huanggang Normal University

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© He and Chen; licensee Springer. 2011

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