Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 470375, 19 pages
doi:10.1155/2010/470375
Research Article
Existence of Homoclinic Solutions for a Class of
Nonlinear Difference Equations
Peng Chen and X. H. Tang
School of Mathematical Sciences and Computing Technology, Central South University, Changsha,
Hunan 410083, China
Correspondence should be addressed to X. H. Tang, [email protected]
Received 5 May 2010; Accepted 2 August 2010
Academic Editor: Jianshe Yu
Copyright q 2010 P. Chen and X. H. Tang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
By using the critical point theory, we establish some existence criteria to guarantee that the
nonlinear difference equation ΔpnΔxn − 1
δ
− qnxn
δ
fn, xn has at least one
homoclinic solution, where n ∈
Z, xn ∈ R, and f : Z × R → R is non periodic in n.Our
conditions on the nonlinear term fn, xn are rather relaxed, and we generalize some existing
results in the literature.
1. Introduction
Consider the nonlinear difference equation of the form
Δ
p
/
0. f : Z × R → R. As usual, we say that a solution un of 1.1 is homoclinic to 0
if un → 0asn →±∞. In addition, if un
/
≡ 0, then un is called a nontrivial homoclinic
solution.
Difference equations have attracted the interest of many researchers in the past
twenty years since they provided a natural description of several discrete models. Such
discrete models are often investigated in various fields of science and technology such
as computer science, economics, neural network, ecology, cybernetics, biological systems,
optimal control, and population dynamics. These studies cover many of the branches of
2 Advances in Difference Equations
difference equation, such as stability, attractiveness, periodicity, oscillation, and boundary
value problem. Recently, there are some new results on periodic solutions of nonlinear
difference equations by using the critical point theory in the literature; see 1–3.
In general, 1.1 may be regarded as a discrete analogue of a special case of the
following second-order differential equation:
p
t
ϕ
x
f
n − 1
− q
n
u
n
f
n, u
n
0, 1.3
where n ∈ Z, u ∈ R, p, q : Z →
R,andf : Z × R → R.
Theorem A see 13. Assume that p, q, and f satisfy the following conditions:
p pn > 0 for all n ∈ Z;
q qn > 0 for all n ∈ Z and lim
|n|→∞
qn∞;
f1 there is a constant μ>2 such that
0 <μ
x
AR For every n ∈ Z, W is continuously differentiable in x, and there is a constant μ>
2 such that
0 <μW
n, x
≤
∇W
n, x
,x
, ∀
n, x
∈ Z ×
R
N
\
{
0
}
. 1.5
However, it seems that results on the existence of homoclinic solutions of 1.1 by
critical point method have not been considered in the literature. The main purpose of this
|n|→∞
qn∞;
F1 Fn, xF
1
n, x − F
2
n, x, for every n ∈ Z, F
1
and F
2
are continuously differentiable
in x, and there is a bounded set J ⊂ Z such that
F
2
n, x
≥ 0, ∀
n, x
∈ J × R,
|
x
|
≤ 1,
1
q
n
, ∀
n, x
∈ Z ×
R \
{
0
}
; 1.7
F3 F
2
n, 0 ≡ 0, and there is a constant ∈ δ 1,μ such that
xf
2
n, x
≤ F
2
n, x
, ∀
n, x
∈ Z × R. 1.8
o
|
x
|
δ
as x −→ 0 1.9
uniformly in n ∈ Z.Then1.1 possesses a nontrivial homoclinic solution.
Remark 1.3. Obviously, both conditions F
1
and F
1
are weaker than f
1
. Therefore, both Theorems
1.1 and 1.2 generalize Theorem A by relaxing conditions f
1
and f
2
.
When Fn, x is subquadratic at infinity, as far as the authors are aware, there is
no research about the existence of homoclinic solutions of 1.1. Motivated by the paper
16, the intention of this paper is that, under the assumption that Fn, x is indefinite sign
and subquadratic as |n|→∞, we will establish some existence criteria to guarantee that
1.1 has at least one homoclinic solution by using minimization theorem in critical point
theory.
Now we present the basic hypothesis on p, q, and F in order to announce the results
in this paper.
, ∀
n, x
∈ Z × R,
|
x
|
≤ 1,
|
F
n, x
|
≤ a
2
n
|
x
|
γ
2
, ∀
n, x
∈ Z × R,
n, x
∈ Z × R 1.11
where ϕsOs
γ
1
−1
as |s|≤c, c is a positive constant.
F6 There exist n
0
∈ Z and two constants η>0 and γ
3
∈ 1,δ 1 such that
F
n
0
,x
≥ η
|
x
|
γ
3
, ∀x ∈ R,
|
x
|
M
|
x
|
γ
3
≤ V
x
≤ M
|
x
|
γ
1
, ∀x ∈ R,
|
x
|
≤ 1,
0 <V
x
≤ M
|
x
|
∈ R,n∈ Z
}
,
E
u ∈ S :
n∈Z
p
n
Δu
n − 1
δ1
q
n
u
n
δ1
< ∞
un
|
p
< ∞
,
l
∞
Z, R
u ∈ S :sup
n∈Z
|
u
n
|
< ∞
,
2.3
6 Advances in Difference Equations
and their norms are defined by
u
p
, ∀u ∈ l
∞
Z, R
, 2.4
respectively.
For any n
1
,n
2
∈ Z with n
1
<n
2
,weletZn
1
,n
2
n
1
,n
2
∩ Z, and for function
f : Z → R and a ∈ R,weset
Z
f
n
u
1
δ 1
u
δ1
−
n∈Z
F
n, u
n
. 2.6
If p, q, and F1, F1
, or F4 holds, then I ∈ C
1
E, R, and one can easily check that
I
u
v
n
− f
n, u
n
v
n
∀u, v ∈ E.
2.7
Furthermore, the critical points of I in E are classical solutions of 1.1 with u±∞0.
We will obtain the critical points of I by using the Mountain Pass Theorem. We recall
it and a minimization theorem as follows.
Lemma 2.1 see 15, 17. Let E be a real Banach space and I ∈ C
1
E, R satisfy (PS)-condition.
Suppose that I satisfies the following conditions:
i I00;
ii there exist constants ρ, α > 0 such that I|
∂B
ρ
0
≥ α;
δ1
∞
≤
u
δ1
, 2.9
where q inf
n∈Z
qn.
Proof. Since u ∈ E, it follows that lim
|n|→∞
|un| 0. Hence, there exists n
∗
∈ Z such that
|
u
n
∗
|
max
n∈Z
|
u
n
δ1
≥ q
u
δ1
∞
. 2.11
The proof is completed.
Lemma 2.3. Assume that F2 and F3 hold. Then for every n, x ∈ Z × R,
i s
−μ
F
1
n, sx is nondecreasing on 0, ∞;
ii s
−
F
2
n, sx is nonincreasing on 0, ∞.
The proof of Lemma 2.3 is routine and so we omit it.
Lemma 2.4 see 18. Let E be a real Banach space and I ∈ C
1
E, R satisfy the (PS)-condition. If
I is bounded from below, then c inf
E
I is a critical value of I.
3. Proofs of Theorems
Proof of Theorem 1.1. In our case, it is clear that I00. We show that I satisfies the PS-
condition. Assume that {u
E
∗
≤ c for k ∈ N. 3.1
From 2.6, 2.7, 3.1, F2,andF3,weobtain
δ 1
c
δ 1
c
u
k
≥
δ 1
I
u
k
−
δ 1
n, u
k
n
−
1
u
k
n
f
2
n, u
k
n
−
δ 1
n∈Z
F
u
k
δ1
,k∈ N.
3.2
8 Advances in Difference Equations
It follows that there exists a constant A>0 such that
u
k
≤ A for k ∈ N. 3.3
Then, u
k
is bounded in E. Going if necessary to a subsequence, we can assume that u
k
u
0
in E. For any given number ε>0, by F1, we can choose ζ>0 such that
f
n, x
≤ εq
k
n
|
δ1
1
q
n
q
n
|
u
k
n
|
δ1
≤
ζ
δ1
A
δ1
u
k
, ∀n ∈ Z. 3.7
Hence, we have by 3.6 and 3.7
|
u
0
n
|
≤ ζ, for
|
n
|
≥ Π. 3.8
It follows from 3.7 and the continuity of fn, x on x that there exists k
0
∈ N such that
Π
n−Π
f
n, u
k
n
>Π
f
n, u
k
n
− f
n, u
0
n
|
u
k
n
− u
0
n
|
u
k
n
|
|
u
0
n
|
≤ ε
|n|>Π
q
n
|
u
k
n
|n|>Π
q
n
|
u
k
n
|
δ1
|
u
0
n
|
δ1
≤ 2ε
u
k
k
n
− f
n, u
0
n
|
u
k
n
− u
0
n
|
−→ 0ask −→ ∞ .
3.11
It follows from 2.7 and the H¨older’s inequality that
I
Δu
k
n − 1
− Δu
0
n − 1
n∈Z
q
n
u
k
n
δ
u
k
n
− u
n∈Z
q
n
u
0
n
δ
u
k
n
− u
0
n
−
n∈Z
f
n, u
k
0
δ1
−
n∈Z
p
n
Δu
k
n − 1
δ
Δu
0
n − 1
−
n∈Z
q
n
u
k
q
n
u
0
n
δ
u
k
n
−
n∈Z
f
n, u
k
n
− f
n, u
0
n∈Z
p
n
Δu
0
n − 1
δ1
1/δ1
n∈Z
p
n
Δu
k
n − 1
δ1
δ/δ1
−
n∈Z
p
n
Δu
k
n − 1
δ1
1/δ1
n∈Z
p
n
Δu
0
n − 1
δ1
δ/δ1
δ/δ1
−
n∈Z
f
n, u
k
n
− f
n, u
0
n
,u
k
n
− u
0
n
u
0
n
δ1
1/δ1
×
n∈Z
p
n
Δu
k
n − 1
δ1
q
n
u
n
δ1
1/δ1
×
n∈Z
p
n
Δu
0
n − 1
δ1
q
n
u
0
n
n
u
k
δ1
u
0
δ1
−
u
0
u
k
δ
−
u
k
u
u
k
δ
−
u
0
δ
u
k
−
u
0
−
n∈Z
u
k
−I
u
0
,u
k
−u
0
→0, it follows from 3.11 and 3.12 that u
k
→ u
0
in E. Hence,
I satisfies the PS-condition.
Advances in Difference Equations 11
We now show that there exist constants ρ, α > 0 such that I satisfies assumption ii of
Lemma 2.1.ByF1, there exists η ∈ 0, 1 such that
f
n, x
≤
1
2
|
x
|
δ1
for n ∈ Z \ J, x ∈ R,
|
x
|
≤ η.
3.14
Set
M sup
F
1
n, x
q
n
| n ∈ J, x ∈ R,
|
x
|
1
,
n∈J,u
n
/
0
F
1
n,
u
n
|
u
n
|
|
u
n
|
μ
≤ M
δ 1
n∈J
q
n
|
u
n
|
δ1
.
3.17
Set α 1/2δ 1qυ
δ1
. Hence, from 2.6, 3.14, 3.17, q,andF1, we have
I
u
1
δ 1
u
F
n, u
n
≥
1
δ 1
u
δ1
−
1
2
δ 1
n∈Z\J
q
n
|
u
n
|
u
n
|
δ1
−
1
2
δ 1
n∈J
q
n
|
u
n
|
δ1
≥
1
2
n
{n∈
−2,2
:
|
u
n
|
≤1}
F
2
n, u
n
≤
{n∈
−2,2
2
n−2
max
|x|≤1
|
F
2
n, x
|
≤
u
∞
2
n−2
max
|
x
|
1
|
F
2
|x|1
|
F
2
n, x
|
2
n−2
max
|x|≤1
|
F
2
n, x
|
M
1
u
M
2
,
|
≤1
|
F
2
n, x
|
.
3.20
Take ω ∈ E such that
|
ω
n
|
⎧
⎨
⎩
1, for
|
n
|
≤ 1,
0, for
|
n
where m
1
n−1
F
1
n, ωn > 0. By 2.6, 3.19, 3.21,and3.22, we have for σ>1
I
σω
1
δ 1
σω
δ1
n∈Z
F
2
n, σω
n
− F
1
n, σω
n
≤
σ
δ1
δ 1
ω
δ1
M
1
σ
ω
M
2
− mσ
μ
.
3.23
Advances in Difference Equations 13
Since μ>≥ δ 1andm>0, 3.23 implies that there exists σ
g ∈ C
0, 1
,E
: g
0
0,g
1
e
. 3.25
Hence, there exists u
∗
∈ E such that
I
u
∗
d, I
u
∗
x
|
δ
for
n, x
∈ Z × R,
|
x
|
≤ η.
3.27
Since Fn, 0 ≡ 0, it follows that
|
F
n, x
|
≤
1
2
δ 1
q
n
δ1
−
n∈Z
F
n, u
n
≥
1
δ 1
u
δ1
−
1
2
δ 1
n∈Z
q
n
u
δ1
α.
3.29
Equation 3.29 shows that u ρ implies that Iu ≥ α, that is, assumption ii of Lemma 2.1
holds. The proof of Theorem 1.2 is completed.
14 Advances in Difference Equations
Proof of Theorem 1.4. In view of Lemma 2.4, I ∈ C
1
E, R. We first show that I is bounded from
below. By F4, 2.6,andH¨older inequality, we have
I
u
1
δ 1
u
δ1
−
n∈Z
F
n, u
n
u
n
|
>1
F
n, u
n
≥
1
δ 1
u
δ1
−
Z
|
u
n
|
n
|
u
n
|
γ
2
≥
1
δ 1
u
δ1
− q
−γ
1
/δ1
⎛
⎝
Z
|
u
n
|
≤1
q
n
u
n
δ1
⎞
⎠
γ
1
/δ1
− q
−γ
1
/δ1
⎛
⎝
Z
|
u
n
|
>1
|
u
n
|
δ1γ
2
−γ
1
/γ
1
q
n
u
n
δ1
⎞
⎠
γ
1
/δ1
≥
δ1δ1−γ
1
⎞
⎠
δ1−γ
1
/δ1
u
γ
1
− q
−γ
1
/δ1
u
γ
2
−γ
1
∞
⎛
⎝
Z
u
δ1
− q
−γ
1
/δ1
⎛
⎝
Z
|
u
n
|
≤1
|
a
1
n
|
δ1/δ1−γ
1
|
>1
|
a
2
n
|
δ1/δ1−γ
1
⎞
⎠
δ1−γ
1
/δ1
u
γ
2
≥
1
δ 1
u
δ1
γ
2
.
3.30
Advances in Difference Equations 15
Since 1 <γ
1
<γ
2
<δ 1, 3.30 implies that Iu → ∞ as u→∞. Consequently, I is
bounded from below.
Next, we prove that I satisfies the PS-condition. Assume that {u
k
}
k∈N
⊂ E is a
sequence such that {Iu
k
}
k∈N
is bounded and I
u
k
→ 0ask → ∞. Then by 2.6, 2.9,
and 3.30, there exists a constant A>0 such that
u
k
∞
Hence, we have, by 3.31 and 3.32,
u
0
∞
≤ A. 3.33
By F5, there exists M
2
> 0 such that
ϕ
|
x
|
≤ M
2
|
x
|
γ
1
−1
, ∀x ∈ R,
|
x
|
≤ A.
3.34
For any given number ε>0, by F5, we can choose an integer Π > 0 such that
n
− f
n, u
0
n
|
u
k
n
− u
0
n
|
<ε, for k ≥ k
0
.
3.36
16 Advances in Difference Equations
On the other hand, it follows from 3.31, 3.33, 3.34, 3.35,andF5 that
n
|
≤
|
n
|
>Π
|
b
n
|
ϕ
|
u
k
n
|
ϕ
|
|
>Π
|
b
n
|
|
u
k
n
|
γ
1
−1
|
u
0
n
|
γ
1
−1
|
|
u
k
n
|
γ
1
|
u
0
n
|
γ
1
≤ 2M
2
q
−γ
1
/δ1
⎛
⎝
γ
1
≤ 2M
2
q
−γ
1
/δ1
⎛
⎝
|
n
|
>Π
|
b
n
|
δ1/δ1−γ
1
⎞
⎠
δ1−γ
1
1
u
0
γ
1
ε, k ∈ N.
3.37
Since ε is arbitrary, combining 3.36 with 3.37,weget
n∈Z
f
n, u
k
n
− f
n, u
0
n
,u
≥
u
k
δ
−
u
0
δ
u
k
−
u
0
−
n∈Z
k
−I
u
0
,u
k
−u
0
→0, it follows from 3.38 and 3.39 that u
k
→ u
0
in E. Hence,
I satisfies PS-condition.
By Lemma 2.4, c inf
E
Iu is a critical value of I, that is, there exists a critical point
u
∗
∈ E such that Iu
∗
c.
Advances in Difference Equations 17
Finally, we show that u
∗
/
0. Let u
0
n
s
δ1
δ 1
u
0
δ1
− F
n
0
,su
0
n
0
≤
s
δ1
δ 1
u
0
δ1
− ηs
that F5 holds with a
1
na
2
nbn|an|. In addition, by F7 and F8, we have
F
n
0
,x
a
n
0
V
x
≥ M
a
n
0
|
x
|
|
x
|
μ
2
−
2 −
|
n
|
|
x
|
1
−
2 −
|
n
|
|
x
|
2
1
|
x
|
μ
1
a
2
|
x
|
μ
2
,F
2
n, x
q
n
2 −
|
n
|
n
⎛
⎝
m
1
i1
a
i
|
x
|
μ
i
−
m
2
j1
b
j
|
x
|
j
⎞
⎠
, 4.3
F
1
n, x
q
n
m
1
i1
a
i
|
x
|
μ
i
,F
2
n, x
q
n
m
sin n
1
|
n
|
|
x
|
3/2
.
4.5
Then
f
n, x
4 cos n
3
1
|
n
|
|
x
|
−2/3
, ∀
n, x
∈ Z × R,
|
x
|
≤ 1,
|
F
n, x
|
≤
2
|
x
|
3/2
1
|
n
|
, ∀
n, x
∈ Z × R,
, ∀
n, x
∈ Z × R.
4.6
We can choose n
0
such that
cos n
0
> 0, sin n
0
> 0. 4.7
Let
η
cos n
0
sin n
0
1
|
n
0
|
.
4.8
Then
F
n
a
2
n
b
n
2
1
|
n
|
,ϕ
s
8s
1/3
9s
1/2
12
.
4.10
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