Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 297618, 20 pages
doi:10.1155/2012/297618
Research Article
Infinitely Many Homoclinic Orbits for
2nth-Order Nonlinear Functional Difference
Equations Involving the p-Laplacian
Xiaofei He1, 2
1
2
Department of Mathematics and Computer Science, Jishou University, Hunan, Jishou 416000, China
Zhangjiajie College of Jishou University, Zhangjiajie 427000, China
Correspondence should be addressed to Xiaofei He, hxfcsu@sina.com
Received 14 October 2011; Accepted 18 November 2011
Academic Editor: Donal O’Regan
Copyright q 2012 Xiaofei He. 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 establishing a new proper variational framework and using the critical point theory, we
establish some new existence criteria to guarantee that the 2nth-order nonlinear difference
equation containing both advance and retardation with p-Laplacian Δn rt − nϕp Δn ut − 1 qtϕp ut ft, ut n, . . . , ut, . . . , ut − n, n ∈ Z3, t ∈ Z, has infinitely many homoclinic
orbits, where ϕp s is p-Laplacian operator; ϕp s |s|p−2 s1 < p < ∞ r, q, f are nonperiodic in
t. Our conditions on the potential are rather relaxed, and some existing results in the literature are
improved.
1. Introduction
In this paper, we shall be concerned with the existence of homoclinic orbits of the nonlinear
difference equation
Δn rt − nϕp Δn ut − 1 qtϕp ut
ft, ut n, . . . , ut, . . . , ut − n,
n ∈ Z3, t ∈ Z,
1.1
where Δ is the forward difference operator defined by Δut ut 1 − ut, Δ2 ut ΔΔut. ϕp s is p-Laplacian operator ϕp s |s|p−2 s1 < p < ∞. As usual, we say that
a solution ut of 1.1 is homoclinic to 0 if ut → 0 as t → ±∞. In addition, if ut /
≡ 0, then
ut is called a nontrivial homoclinic solution. It is well known that homoclinic orbits play an
2
Abstract and Applied Analysis
important role in analyzing the chaos of dynamical systems; we refer the interested reader to
1–15.
We may think of 1.1 as being a discrete analogue of the 2nth-order differential equation
n
qtϕp x − ft, xt n, . . . , xt, . . . , xt − n 0,
rtϕp xn
t ∈ R.
1.2
If p 2, 1.1 reduces to the following equation:
Δn rt − nΔn ut − n qtut ft, ut n, . . . , ut, . . . , ut − n,
n ∈ Z3, t ∈ Z.
1.3
Recently, Chen and Tang 2 applied the critical point theory to prove the existence of homoclinic solutions for 1.3.
In some recent papers 1, 2, 8, 9, 16–18, the authors studied the existence of periodic
and homoclinic solutions of second-order nonlinear difference equation by using the critical
point theory. These papers show that the critical point method is an effective approach
to the study of periodic solutions of second-order difference equations. Compared to oneorder or second-order difference equations, the study of higher-order equations has received
considerably less attention see, e.g., 2, 3, 19–21 and references contained therein. But to
the best knowledge of the authors, results on existence of homoclinic solutions of 1.1 have
not been found in the literature.
Motivated by the recent papers 2, 5, 14, 22, the aim of this paper is to consider problem 1.1 in a more general sense. More exactly our results represent the extensions to equations with p-Laplacian. Throughout the paper, for a function G, we let Gi x1 , x2 , . . . , xi , . . . ,
xn denote the partial derivative of G on the i variable.
2. Main Results
Theorem 2.1. Assume that q and F satisfy the following assumptions:
r For every t ∈ Z, rt > 0;
q For every t ∈ Z, qt > 0, and lim|t| → ∞ qt ∞;
F1 There exists a functional Ft, xn t, . . . , x0 t which is continuously differentiable in the
variable from xn to x0 for every t ∈ Z and satisfy
0
F2ni
t i, xni , . . . , xi ft, xn , xn−1 , . . . , x0 , x−1 , . . . , x−n ,
i−n
n
n
ft, xn , xn−1 , . . . , x0 , x−1 , . . . , x−n o |xi |p−1 , as |xi |p−1 −→ 0,
i−n
i−n
n
n
p
p
|Ft, xn , . . . , x0 | o
|xi | , as
|xi | −→ 0
i−n
uniformly in t ∈ Z \ J, where J is defined in (F2);
i−n
2.1
Abstract and Applied Analysis
3
F2 Ft, xn , . . . , x0 Wt, x0 − Ht, xn , . . . , x0 , for every t ∈ Z, W, H are continuously
differentiable in x0 and xn , . . . , x0 , respectively. Moreover, there is a bounded set J ⊂ Z
such that
Ht, xn , . . . , x0 ≥ 0;
2.2
F3 There is a constant μ > p such that
0 < μWt, x0 ≤ W2 t, x0 x0 ,
∀t, x0 ∈ Z × R \ {0};
2.3
F4 Ht, 0, . . . , 0 ≡ 0, and there is a constant ∈ p, μ such that
0
H2ni
t, xn , . . . , x0 x−i ≤ Ht, xn , . . . , x0 ;
2.4
i−n
F5 There exists a constant b > 0 such that
Ht, xn , . . . , x0 ≤ bγ ,
where γ n
i0 |xi |
p 1/p
for t ∈ Z, γ > 1,
2.5
.
F6 Ft, −xn , . . . , −x0 Ft, xn , . . . , x0 , for all t, xn , . . . , x0 ∈ Z × Rn1 .
Then 1.1 possesses an unbounded sequence of homoclinic solutions.
Theorem 2.2. Assume that r, q, and F satisfy (r), (q), (F1), (F3)–(F5) and the following assumption:
F2 Ft, xn , . . . , x0 Wt, x0 − Ht, xn , . . . , x0 , for every t ∈ Z, W, H are continuously
differentiable in x0 and xn , . . . , x0 , respectively, and
|Ft, xn , . . . , x0 | o γ p
where γ n
i0 |xi |
p 1/p
as γ −→ 0,
2.6
uniformly in t ∈ Z.
Then 1.1 possesses an unbounded sequence of homoclinic solutions.
Theorem 2.3. Assume that r, q, and F satisfy (r), (q), (F1) and satisfy the following assumptions:
F7 For any t ∈ Z,
Ft, xn , . . . , x0 ≥ Ft, x0 ≥ 0;
2.7
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Abstract and Applied Analysis
F8 For any r > 0, there exist a ar, b br > 0, and ν < p such that
1
p
ν/p Ft, xn , . . . , x0 a b ni0 |xi |p
≤
0
F2ni
t, xn , . . . , x0 x−i ,
∀t ∈ Z,
n
i−n
1/p
|xi |p
2.8
≥ r;
i0
F9 For any t ∈ Z
lim s−p minFt, sx ∞.
s → ∞
2.9
|x|1
Then there exists an unbounded sequence of homoclinic solutions for 1.1.
3. Preliminaries
To apply critical point theory to study the existence of homoclinic solutions of 1.1, we shall
state some basic notations and lemmas, which will be used in the proofs of our main results.
Let
S {{ut}t∈Z : ut ∈ R, t ∈ Z},
E u∈S:
|rt − 1Δn ut − 1|p qt|ut|p < ∞
3.1
t∈Z
and for u ∈ E, let
1/p
p
p
n
.
|rt − 1Δ ut − 1| qt|ut|
u 3.2
t∈Z
Then E is a uniform convex Banach space with this norm.
Let I : E → R be defined by
Iu 1
up − Ft, ut n, . . . , ut.
p
t∈Z
3.3
If q and F1 hold, then I ∈ C1 E, R and one can easily check that
rt − 1|Δn ut − 1|p−2 Δn vt − 1 qt|ut|p−2 vt
I u, v t∈Z
−ft, ut n, . . . ut, . . . ut − nvt ,
3.4
∀u, v ∈ E.
Abstract and Applied Analysis
5
By using
Δ ut − 1 n
n
−1
k
n
k
k0
ut n − k − 1,
3.5
we can compute the partial derivative as
∂Iu
Δn rt − nϕp Δn ut − 1 qtϕp ut − ft, ut n, . . . , ut, . . . , ut − n.
∂ut
3.6
So, the critical points of I in E are the solutions of 1.1 with u±∞ 0.
Lemma 3.1 see 12. Let E be a real Banach space and I ∈ C1 E, R satisfy (PS)-condition with I
even. Suppose that I satisfies the following conditions:
i I0 0;
ii There exist constants ρ, α > 0 such that I|∂Bρ 0 ≥ α;
iii For each finite dimensional subspace E ⊂ E, there is r rE > 0 such that Iu ≤ 0 for
u ∈ E \ Br 0, where Br 0 is an open ball in E of radius r centered at 0.
Then I possesses an unbounded sequence of critical values.
Lemma 3.2. For u ∈ E,
p
p
βu∞ ≤ βulp ≤ up ,
3.7
where β inft∈Z qt.
Lemma 3.3. Assume that (F3) holds. Then for every t, x ∈ Z × R, s−μ Wt, sx is nondecreasing on
0, ∞.
The proof of Lemmas 3.2 and 3.3 is routine and so we omit it.
4. Proofs of Theorems
Proof of Theorem 2.1. It is clear that I0 0. Our proof is devided into three steps.
Step 1 PS Condition.
Assume that {uk }k∈N ⊂ E is a sequence such that {Iuk }k∈N is bounded and I uk → 0
as k → ∞. Then there exists a constant c > 0 such that
|Iuk | ≤ c,
I uk E∗
≤ c
for k ∈ N.
4.1
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Abstract and Applied Analysis
From 2.2, 2.3, 2.4, 4.1, F3, and F4, we obtain
p I uk , uk
−p
1
Wt, uk t − W2 t, uk tuk t
uk p − p
t∈Z
p Ht, uk t n, . . . , uk t
pc pcuk ≥ pIuk −
t∈Z
0
p H
t i, uk t n i, . . . , uk t iuk t
t∈Z i−n 2ni
−p
1 p
Wt, uk t − W2 t, uk tuk t
uk − p
n∈Z
p Ht, uk t n, . . . , uk t
−
4.2
t∈Z
−
≥
0
p
H
t, uk t n, . . . , uk tuk t − i
t∈Z i−n 2ni
−p
uk p ,
k ∈ N.
By 4.2, there exists a constant A > 0 such that
uk ≤ A
for k ∈ N.
4.3
It can be assumed that uk u0 in E. For any given number ε > 0, by F1, we can choose
ζ > 0 such that
ft, ut n, . . . , ut, . . . , ut − n
≤ εξp−1
for t ∈ Z\J, utn, . . . , ut, . . . , ut−n ∈ Rn1 ,
4.4
1/p
where ξ ni−n |ut i|p ≤ ζ.
Since qt → ∞, we can also choose an integer Π > max{|k| : k ∈ J} such that
qt ≥
2n 1Ap
,
ζp
|t| ≥ Π.
4.5
By 4.2 and 4.4, we have
|uk t|p 1
ζp
ζp
qtuk tp ≤
uk p ≤
p
qt
2n 1
2n 1A
for |t| ≥ Π, k ∈ N.
4.6
Abstract and Applied Analysis
7
Since uk u0 in E, it is easy to verify that uk t converges to u0 t pointwise for all t ∈ Z, that
is
lim uk t u0 t,
k→∞
∀t ∈ Z.
4.7
Hence, we have by 4.6 and 4.7
ζp
2n 1
|u0 t|p ≤
4.8
for |t| ≥ Π.
It follows from 4.7 and the continuity of ft, ut 1, . . . , ut, . . . , ut − n on ut 1,
. . . , ut, . . . , ut − n that there exists k0 ∈ N such that
Π ft, uk t n, . . . , uk t, . . . , uk t − n−ft, u0 t n, . . . , u0 t, . . . , u0 t − n < ε
t−Π
4.9
for k ≥ k0 .
On the other hand, it follows from F1, 2.4, 4.2, 4.4, 4.6, and 4.8 that
ft, uk t n, . . . , uk t, . . . , uk t − n − ft, u0 t n, . . . , u0 t, . . . , u0 t − n
|t|>Π
× |uk t − u0 t|
ft, uk t n, . . . , uk t, . . . , uk t − n
≤
|t|>Π
ft, u0 t n, . . . , u0 t, . . . , u0 t − n × |uk t| |u0 t|
≤ε
|t|>Π
n
|uk t i|
p−1
i−n
≤ 2n 1ε
p−1
|u0 t i|
|uk t| |u0 t|
4.10
i−n
|uk t|p−1 |u0 t|p−1 |uk t| |u0 t|
t∈Z
≤ 4n 2ε
n
|uk t|p |u0 t|p
t∈Z
≤
4n 2ε p
A u0 p .
β
Since ε is arbitrary, combining 4.9 with 4.10, we get
ft, uk t n, . . . , uk t, . . . , uk t − n − ft, u0 t n, . . . , u0 t, . . . , u0 t − n
4.11
t∈Z
|uk t − u0 t| −→ 0
as k −→ ∞.
8
Abstract and Applied Analysis
Using the Hölder’s inequality
4.12
ac bd ≤ ap bp 1/p cq dq 1/q ,
where a, b, c, d are nonnegative numbers and 1/p 1/q 1, p > 1, it follows from 3.3 and
3.4 that
I uk − I u0 , uk − u0
|rt − 1Δn uk t − 1|p−2 Δn uk t − 1, Δn uk t − 1 − Δn u0 t − 1
t∈Z
qt|uk t|p−2 uk t, uk t − u0 t
t∈Z
−
|rt − 1Δn u0 t − 1|p−2 Δn u0 t − 1, Δn uk t − 1 − Δn u0 t − 1
t∈Z
−
qt|u0 t|p−2 u0 n, uk t − u0 t
t∈Z
−
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk n − u0 n
uk p u0 p −
|rt − 1Δn uk t − 1|p−2 Δn uk t − 1, Δn u0 t − 1
−
n∈Z
qt|uk t|p−2 uk t, u0 t
t∈Z
−
|rt − 1Δn u0 t − 1|p−2 Δn u0 t − 1, Δn uk t − 1
t∈Z
−
qt|u0 t|p−2 u0 t, uk t
t∈Z
−
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk n − u0 n
p
|rt − 1Δn u0 t − 1|p
p
≥ uk u0 −
1/p t∈Z
−
qt|u0 t|p
t∈Z
1/q
qt|uk t|p
t∈Z
1/p −
t∈Z
|rt − 1Δn uk t − 1|p
t∈Z
1/p p
|rt − 1Δ uk t − 1|
n
|rt − 1Δn u0 t − 1|p
t∈Z
1/q
1/q
Abstract and Applied Analysis
1/p −
qt|uk t|
p
t∈Z
−
9
1/q
qt|u0 t|
p
t∈Z
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk t − u0 t
1/p
p
p
≥ uk u0 −
p
|rt − 1Δ u0 t − 1| qt|u0 t|
n
p
t∈Z
·
1/q
p
p
n
|rt − 1Δ uk t − 1| qt|uk t|
t∈Z
1/p
p
p
n
|rt − 1Δ uk n − 1| qt|uk t|
−
t∈Z
·
1/q
p
|rt − 1Δ u0 n − 1| qt|u0 t|
n
p
t∈Z
−
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk t − u0 t
uk p u0 p − u0 uk p−1 − uk u0 p−1
−
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk t − u0 t
uk p−1 − u0 p−1 uk − u0 −
ft, uk t n, . . . , uk t, . . . , uk t − n
t∈Z
−ft, u0 t n, . . . , u0 t, . . . , u0 t − n, uk t − u0 t .
4.13
Since I uk → 0 as k → ∞ and uk u0 in E, it follows from 4.11 and 4.13 that
I uk − I u0 , uk − u0 −→ 0 as k → ∞,
4.14
which yields that uk → u as k → ∞. By the uniform convexity of E and the fact that
uk u0 in E, it follows from the Kadec-Klee property that uk → u0 in E. Hence, I satisfies
PS-condition.
10
Abstract and Applied Analysis
Step 2 Condition ii of Lemma 3.1.
By F1, there exists η ∈ 0, 1 such that
n
2−p β |ut i|p
|Ft, ut n, . . . , ut| ≤
4pn 1 i0
for t ∈ Z \ J,
n
|ut i|p
1/p
i0
≤ η.
4.15
Set
M sup{Wt, u | t ∈ J, u ∈ R, |u| 1},
4.16
and δ min{β/4pM 1μ−p , η}. If u β1/p δ : ρ, then by Lemma 3.2, |ut| ≤ δ ≤ η < 1
for t ∈ Z. By 4.16 and Lemma 3.2, we have
ut
Wt, ut ≤
W t,
|ut|μ
|ut|
t∈J
0
t∈J,ut /
≤M
|ut|μ
t∈J
≤ Mδμ−p
|ut|p
t∈J
≤
Mδμ−p qt|ut|p
β t∈J
≤
1
qt|ut|p .
4p t∈J
Set α βδp /2p. Hence, from 3.3, 4.15, 4.17, q, F1, and F2, we have
Iu 1
up − Ft, ut n, . . . , ut
p
t∈Z
1
Ft, ut n, . . . , ut − Ft, ut n, . . . , ut
up −
p
t∈J
t∈Z\J
n
2−p β 1
p
p
≥ u −
− Wt, ut
|ut i|
p
4pn 1 t∈Z\J i0
t∈J
Ht, ut n, . . . , ut
t∈J
≥
β
1
1
qt|ut|p
|ut|p −
up −
p
4p t∈Z
4p t∈J
≥
1
1
1
qt|ut|p −
qt|ut|p
up −
p
4p t∈Z
4p t∈J
4.17
Abstract and Applied Analysis
11
≥
1
1
1
up −
up −
up
p
4p
4p
1
up
2p
α.
4.18
equation 4.18shows that u ρ implies that Iu ≥ α, that is, I satisfies assumption ii of
Lemma 3.1.
Step 3 Condition iii of Lemma 3.1.
Let E be a finite dimensional subspace of E. Assume that dim E m and u1 , u2 , . . . , um
is a base of E .
For any u ∈ E , there exist λi ∈ R, i 1, 2, . . . , m such that
ut m
for t ∈ Z.
λi ui t
4.19
i1
Let
u∗ m
|λi |ui .
4.20
i1
It is easy to verify that · ∗ defined by 4.20 is a norm of E . Since all the norms of a finite
dimensional normed space are equivalent, so there exists a constant d > 0 such that
for u ∈ E .
4.21
i 1, 2, . . . , m.
4.22
u∗ ≤ du∞
Assume that
ui d,
Since ui ∈ E, by Lemma 3.2, we can choose an integer Π1 > 0 such that
|ui t| <
η
,
m
i 1, 2, . . . , m, |t| > Π1 ,
4.23
where η is given in 4.15. Set
Θ
m
m
λi ui t : λi ∈ R, i 1, 2, . . . , m;
|λi | 1 u ∈ E : u∗ d .
i1
i1
4.24
12
Abstract and Applied Analysis
Hence, for u ∈ Θ, let t0 t0 u ∈ Z such that
|ut0 | u∞ .
4.25
Then, by 4.19–4.25, we have
dd
m
m
|λi | |λi |ui u∗
i1
i1
≤ du∞ d|ut0 |
≤d
m
|λi ||ui t0 |,
4.26
u ∈ Θ.
i1
This shows that |ut0 | ≥ 1 and there exists i0 ∈ {1, 2, . . . , m} such that |ui0 t0 | ≥ 1/m, which,
together with 4.23, implies that |t0 | ≤ Π1 .
Set
τ min{Wt, x : |t| ≤ Π1 , |x| 1}.
4.27
Since Wt, x > 0 for all t ∈ Z and x ∈ R \ {0}, and Wt, x is continuous in x, so τ > 0. It
follows from 4.27 and Lemma 3.3 that
Π1
Wt, ut ≥ Wt0 , ut0 n−Π1
ut0 ≥ W t0 ,
|ut0 |μ
|ut0 |
≥ min Wt0 , x |ut0 |μ
|x|1
≥τ
for u ∈ Θ.
For any u ∈ E, it follows from F5 that
Π1
Ht, ut n, . . . , ut
t−Π1
t∈Z−Π1 ,Π1 ,
n
p
i0 |uti| 1/p
>1
t∈Z−Π1 ,Π1 ,
n
i0 |uti|
p 1/p
Ht, ut n, . . . , ut
Ht, ut n, . . . , ut
≤1
4.28
Abstract and Applied Analysis
≤b
13
1/p
t∈Z−Π1 ,Π1 , ni0 |uti|p >1
Π1
max
/p
n
p
|ut i|
i0
Ht, ut n, . . . , ut
n
p 1/p
t−Π1 i0 |uti| ≤1
Π1
≤ n 1/p1 b |ut| t−Π1 t∈Z
≤ n 1/p1 β−/p bu max
n
i0 |uti|
Π1
t−Π1 n
p 1/p
max
i0 |uti|
Ht, ut n, . . . , ut
≤1
p 1/p
Ht, ut n, . . . , ut
≤1
M1 u M2 ,
4.29
where
M1 n 1/p1 β−/p b,
M2 Π1
t−Π1 n
max
i0 |uti|
p 1/p
Ht, ut n, . . . , ut.
≤1
4.30
From 3.3, 3.7, 4.28, and 4.29, we have for u ∈ Θ and σ > 1
Iσu σp
up − Ft, σut n, . . . , σut
p
t∈Z
σp
Ft, σut n, . . . , σut −
Ft, σut n, . . . , σut
up −
p
|t|>Π
|t|≤Π
2
2
≤
n
βσ p
σp
Ft, σut n, . . . , σut
|ut i|p −
up p
4pn 1
|t|>Π i0
|t|≤Π
≤
σ
σ
Ft, σut n, . . . , σut
up up −
p
4p
|t|≤Π
σ
σ
Wt, σut Ht, σut n, . . . , σut
up up −
p
4p
|t|≤Π
|t|≤Π
≤
σ
σ
up up σ M1 u M2 − τσ μ
p
4p
cσp cp σ p
M1 cσ M2 σ − τσ μ .
p
4p
2
p
2
p
2
p
p
2
p
2
p
4.31
14
Abstract and Applied Analysis
Since μ > > p, we deduce that there is σ0 σ0 d, M1 , M2 , τ σ0 E > 1 such that
Iσu < 0 for u ∈ Θ, σ ≥ σ0 .
4.32
That is
Iu < 0
for u ∈ E , u ≥ dσ0 .
4.33
This shows that iii of Lemma 3.1 holds. By Lemma 3.1, I possesses an unbounded sequence
{dk }k∈N of critical values with dk Iuk , where uk is such that I uk 0 for k 1, 2, . . .. If
{uk }k∈N is bounded, then there exists B > 0 such that
uk ≤ B
for k ∈ N.
4.34
By a similar fashion for the proof of 4.15, for the given η in 4.15, there exists Π3 > 0 such
that
|uk t| ≤ η
for |t| ≥ Π3 , k ∈ N.
4.35
Thus, from F2, 3.7, 4.34, and 4.35, we have
1
uk p dk Ft, uk t n, . . . , uk t
p
t∈Z
dk Ft, uk t n, . . . , uk t |t|>Π3
Π3
Ft, uk t n, . . . , uk t
t−Π3
Π3
n
β
≥ dk −
Ht, uk t n, . . . , uk t
|uk t i|p −
4pn 1 |t|>Π i0
n−Π3
4.36
3
≥ dk −
Π3
1
max√ |Ht, uk t n, . . . , uk t|.
uk p −
4p
n−Π3 |uk |≤B/ β
It follows that
dk ≤
Π3
5
max√ |Ht, uk t n, . . . , uk t| < ∞.
uk p 4p
t−Π3 |uk |≤B/ β
4.37
This contradicts the fact that {dk }k∈N is unbounded, and so {uk }k∈N is unbounded.
Proof of Theorem 2.2. In the proof of Theorem 2.1, the condition that Ht, ut n, . . . , ut ≥ 0
1/p
for t, ut n, . . . , ut ∈ J × Rn1 , γ ni0 |ut i|p ≤ 1 in F1 is only used in the proofs
Abstract and Applied Analysis
15
of assumption ii of Lemma 3.1. Therefore, we only prove that assumption ii of Lemma 3.1
still holds using F2’ instead of F2. By F2’, it follows that
2−p β
|Ft, ut n, . . . , ut| ≤
2pn 1
1/p
n
n
p
p
for t ∈ Z,
≤ η.
|ut i|
|ut i|
i0
i0
4.38
If u β1/p η : ρ, then by Lemma 3.2 , |ut| ≤ η for t ∈ Z. Set α βηp /2p. Hence, from 3.3,
4.38, and Lemma 3.2, we have
1
up − Ft, ut n, . . . , ut
p
t∈Z
n
2−p β
1
p
p
≥ u −
|ut i|
p
2pn 1 i0
Iu ≥
1
1
up −
up
p
2p
1
up
2p
4.39
α.
equation 4.39 shows that u ρ implies that Iu ≥ α, that is, assumption ii of Lemma 3.1
holds. The proof of Theorem 2.2 is completed.
Proof of Theorem 2.3. We first show that I satisfies condition C. Assume that {uk }k∈N ⊂ E is
a C sequence of I, that is, {Iuk }k∈N is bounded and 1 uk I uk → 0 as k → ∞.
Then it follows from 3.3 and 3.4 that
C1 ≥ pIuk − I uk , uk
0
F2ni t, uk t n, . . . , uk tuk t − i − pFt, uk t n, . . . , uk t .
4.40
t∈Z i−n
It follows from F8 that there exists η ∈ 0, 1 such that 4.15 holds. By F7 and F8, we
have
0
F2ni
t, ut n, . . . , utut − i > pFt, uk t n, . . . , ut ≥ 0,
i−n
4.41
16
Abstract and Applied Analysis
and for t ∈ Z,
n
i0 |ut
i|p ≥ ηp , we have
⎡
n
Ft, uk t n, . . . , uk t ≤ ⎣a b
|ut i|p
ν/p ⎤
⎦
i0
×
0
4.42
F2ni
t, uk t n, . . . , uk tuk t − i − pFt, uk t n, . . . , uk t .
i−n
It follows from 3.2, 3.3, 4.38, 4.40, 4.41, and 4.42 that
1
uk p Iuk Ft, uk t n, . . . , ut
p
t∈Z
Iuk Ft, uk t n, . . . , ut
n
p 1/p
t∈Z i0 |uti| ≤η
Ft, uk t n, . . . , ut
n
p 1/p
t∈Z i0 |uti| >η
2−p β
≤ Iuk |uk t i|p
2pn 1
n
1/p
t∈Z i0 |uti|p ≤η
⎡
ν/p ⎤
n
p
⎣a b
⎦
|ut i|
i0
n
p 1/p
t∈Z i0 |uti| >η
0
×
F2ni t, uk t n, . . . , uk tuk t − i − pFt, uk t n, . . . , uk t
i−n
⎡
ν/p ⎤
n
1
2
p
⎦
≤ C2 |ut i|
uk ⎣a b
2p
i0
t∈Z
0
×
F2ni t, uk t n, . . . , uk tuk t − i − pFt, uk t n, . . . , uk t
i−n
≤ C2 1
∇Wn, uk n, uk n − pWn, uk n
uk p a buk ν∞
2p
t∈Z
1
uk p C1 a bn 1uk ν∞
2p
1
≤ C2 uk p C1 a β−ν/p bn 1uk ν ,
2p
≤ C2 k ∈ N.
4.43
Since ν < p, it follows from 4.43 that {uk }k∈N is bounded. Similar to the proof of
Theorem 2.1, we can prove that {uk } has a convergent subsequence in E. Hence, I satisfies
condition C.
It is obvious that I is even and I0 0, and so assumption i of Lemma 3.1 holds.
The proof of assumption ii of Lemma 3.1 is the same as in the proof of Theorem 2.2.
Abstract and Applied Analysis
17
Finally, it remains to show that I satisfies assumption iii of Lemma 3.1. Let E be a
finite dimensional subspace of E. Since all norms of a finite dimensional normed space are
equivalent, so there is a constant d > 0 such that 4.21 holds. Assume that dim E m and
u1 , u2 , . . . , um are the bases of E such that 4.22 holds. Let η, Π1 , and Θ be the same as in the
proof of Theorem 2.1. Then 4.23, 4.25, and 4.26 hold. For the Π1 given in the proof of
Theorem 2.1. By F9, there exists σ0 σ0 d, Π1 > 1 such that
s−p minWn, sx ≥ 2dp
|x|1
for s ≥
σ0
, n ∈ Z−Π1 , Π1 .
2
4.44
It follows from F7, F9, 3.3, and 4.44 that
Iσu σp
up − Ft, σut n, . . . , σut
p
t∈Z
≤
σp
up − Ft0 , σut0 p
≤
σp
up − min Ft0 , σ|ut0 |x
|x|1
p
dσp
− dσp |ut0 |p
≤
p
≤
4.45
dσp
− dσp
p
−
dσp
,
q
u ∈ Θ, σ ≥ σ0 .
We deduce that there is σ0 σ0 d, Π1 σ0 E > 1 such that
Iσu < 0 for u ∈ Θ, σ ≥ σ0 .
4.46
That is
Iu < 0
for u ∈ E , u ≥ dσ0 .
4.47
This shows that condition iii of Lemma 3.1 holds. By Lemma 3.1, I possesses an unbounded
sequence {dk }k∈N of critical values with dk Iuk , where uk is such that I uk 0 for
k 1, 2, . . .. Since
1
Ft, uk t n, . . . , uk t ≥ dk
uk p dk p
n∈Z
4.48
and {dk }k∈N is unbounded, it follows that {uk }k∈N is unbounded. The proof is complete.
18
Abstract and Applied Analysis
5. Examples
In this section, we give some examples to illustrate our results.
Example 5.1. In 1.1, let qt → ∞ as |t| → ∞ and
⎡
53|t|/24|t| ⎤
n
⎦.
Ft, ut n, . . . , ut ⎣|ut|4|t| −
u2 t i
5.1
i0
Let p 2, μ 4, 3, J {−3, −2, −1, 0, 1, 2, 3} and
4|t|
Wt, ut |ut|
,
Ht, ut n, . . . , ut n
53|t|/24|t|
u t i
2
.
5.2
i0
Then it is easy to verify that all conditions of Theorem 2.1 are satisfied. By Theorem 2.1, 1.1
has a nontrivial homoclinic solution.
Example 5.2. In 1.1, let rt > 0, qt → ∞ as |t| → ∞ and
⎛
j /p ⎞
m1
m2
n
⎠,
Ft, ut n, . . . , ut ⎝ ak |ut|μk − bj
|ut i|p
k1
j1
5.3
i0
where μ1 > μ2 > · · · > μm1 > 1 > 2 > · · · > m2 > p, ak , bj > 0, k 1, 2, . . . , m1 ; j 1, 2, . . . , m2 .
Let μ μm1 , 1 , and
m1
Wt, ut ak |ut|μk ,
Ht, ut n, . . . , ut m2
j /p
n
p
bj
.
|ut i|
5.4
i0
j1
k1
Then it is easy to verify that all conditions of Theorem 2.2 are satisfied. By Theorem 2.2, 1.1
has a nontrivial homoclinic solution.
Example 5.3. In 1.1, let
⎡
1/p ⎤
n
n
⎦,
Ft, ut n, . . . , ut qt |ut i|p ln⎣1 |ut i|p
i0
i0
where q : Z → 0, ∞ such that qt → ∞ as |t| → ∞. Since
0
F2ni
t, ut n, . . . , utut − i
i−n
⎡
⎡
⎤
1/p ⎤
n
p p1/p
n
n
i|
|ut
i0
⎦
⎦
qt⎣p |ut i|p ln⎣1 |ut i|p
n
p 1/p
1
i0
i0
i0 |ut i|
5.5
Abstract and Applied Analysis
1
Ft, ut n, . . . , ut
≥ p
n
p 1/p
1
i0 |ut i|
≥ 0,
19
∀t, ut n, . . . , ut ∈ Z × Rn1 .
5.6
This shows that F8 holds with a b ν 1. It is easy to verify that assumptions q, F1,
and F7 of Theorem 2.3 are satisfied. By Theorem 2.3, 1.1 has an unbounded sequence of
homoclinic solutions.
Acknowledgment
This work is partially supported by Major Project of Science Research Fund of Education Department in Hunan no: 11A095.
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