Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 463480, 14 pages
doi:10.1155/2011/463480
Research Article
Existence of Periodic Solutions for a Class of
Discrete Hamiltonian Systems
Qiongfen Zhang, X. H. Tang, and Qi-Ming Zhang
School of Mathematical Sciences and Computing Technology, Central South University,
Changsha, Hunan 410083, China
Correspondence should be addressed to X. H. Tang, tangxh@csu.edu.cn
Received 9 April 2011; Accepted 12 June 2011
Academic Editor: Mingshu Peng
Copyright q 2011 Qiongfen Zhang et al. 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 applying minimax methods in critical point theory, we prove the existence of periodic solutions
for the following discrete Hamiltonian systems Δ2 ut − 1 ∇Ft, ut 0, where t ∈ Z, u ∈ RN ,
F : Z × RN → R, Ft, x is continuously differentiable in x for every t ∈ Z and is T -periodic in t; T
is a positive integer.
1. Introduction
Consider the following discrete Hamiltonian system:
Δ2 ut − 1 ∇Ft, ut 0,
t ∈ Z,
1.1
where Δ is the forward difference operator defined by Δut ut 1 − ut, Δ2 ut ΔΔut, t ∈ Z, u ∈ RN , F : Z × RN → R, and Ft, x is continuously differentiable in x for
every t ∈ Z and is T -periodic in t; T is a positive integer.
Difference equations usually describe evolution of certain phenomena over the course
of time. For example, if a certain population has discrete generations, the size of the t 1 th
generation xt1 is a function of the tth generation xt. In fact, difference equations provide
a natural description of many discrete models in real world. Since discrete models exist in
various fields of science and technology such as statistics, computer science, electrical circuit
analysis, biology, neural network, optimal control, and so on, it is of practical importance to
investigate the solutions of difference equations. For more details about difference equations,
we refer the readers to the books 1–3.
2
Discrete Dynamics in Nature and Society
In some recent papers 4–15, the authors studied the existence of periodic solutions
and subharmonic solutions of difference equations by applying critical point theory. These
papers show that the critical point theory is an effective method to the study of periodic
solutions for difference equations. In 2007, Xue and Tang 11 investigated the existence of
periodic solutions for 1.1 and obtained the main result.
Theorem A see 11. Suppose that F satisfies the following conditions:
F1 there exists a positive constant T such that Ft T, x Ft, x for all t, x ∈ Z × RN ;
F2 there are constants L1 > 0, L2 > 0, and 0 ≤ α < 1 such that
|∇Ft, x| ≤ L1 |x|α L2 ,
∀t, x ∈ Z1, T × RN ,
1.2
where Za, b : ∩a, b for every a, b ∈ Z with a ≤ b;
F3 |x|−2α Tt1 Ft, x → ∞ as |x| → ∞ for all t ∈ Z1, T .
Then problem 1.1 has at least one periodic solution with period T .
Let
2πt
1 |x|3/2 ht, x,
sin
T
7/4
Ft, x ft|x|
1.3
where f : Z1, T → R, ft T ft, h : Z1, T → RN , and ht T ht. It is easy to see
that
3/4 3 2πt
7 1|x|1/2 |ht|
|∇Ft, x| ≤ ft |x| sin
4
2
T
1.4
3/4
7 N
≤
ft ε |x| aε |ht|, ∀t, x ∈ Z1, T × R ,
4
where ε > 0 and aε is a positive constant and is dependent on ε. The above inequality shows
that there are functions not satisfying condition F2. If we let Tt1 ft 0, α 3/4, T 2,
then we have
|x|−2α
T
Ft, x 2 h1 h2, |x|−3/2 x .
1.5
t1
But the above equality does not satisfy F3. This example shows that it is valuable to further
improve conditions F2 and F3.
Before stating our main results, we first introduce some preliminaries.
2. Preliminaries
Let
HT u : Z −→ RN | ut T ut, t ∈ Z .
2.1
Discrete Dynamics in Nature and Society
3
HT can be equipped with the inner product
u, v
T
ut, vt,
∀u, v ∈ HT ,
2.2
∀u ∈ HT ,
2.3
t1
by which the norm on HT can be reduced by
u T
1/2
|ut|
2
,
t1
where ·, · and | · | denote the usual inner product and the usual norm in RN . It is easy to see
that HT , ·, ·
is a finite-dimensional Hilbert space and linear homeomorphic to RNT . For
any r > 1, define
ur T
1/r
|ut|
r
,
∀u ∈ HT .
2.4
t1
Obviously, u u2 and u is equivalent to ur . Hence, there exist two positive constants
C1 , C2 , which are independent on r, such that
C1 ur ≤ u ≤ C2 ur ,
∀u ∈ HT .
2.5
If we define u∞ supt∈Z1,T |ut|, it is easy to see that for any r > 1,
u∞ ≤ ur ,
∀u ∈ HT .
2.6
For any u ∈ HT , let
ϕu −
T
T
1
|Δut|2 Ft, ut − Ft, 0.
2 t1
t1
2.7
We can compute the Fréchet derivative of 2.7 as
∂ϕu
Δ2 ut − 1 ∇Ft, ut,
∂ut
t ∈ Z1, T .
2.8
Hence, u is a critical point of ϕ on HT if and only if
Δ2 ut − 1 ∇Ft, ut 0,
t ∈ Z1, T , u ∈ RN .
2.9
So, the critical points of ϕ are classical solutions of 1.1. The following lemmas are useful in
our proof.
4
Discrete Dynamics in Nature and Society
Lemma 2.1 see 11. As a subspace of HT , Nk is defined by
Nk : u ∈ HT | −Δ2 ut − 1 λk ut ,
2.10
where λk 2 − 2 cos kω, ω 2π/T, k ∈ Z0, T/2, · and denote the Gauss Function. Then
there hold
j, j ∈ Z0, T/2;
i Nk ⊥ Nj , k /
T/2
ii HT ⊕k0 Nk .
T/2
Lemma 2.2 see 11. Define Hk : ⊕kj0 Nj , Hk⊥ : ⊕jk1 Nj , k ∈ Z0, T/2 − 1; then one has
T
|Δut|2 ≤ λk u2 ,
t1
T
t1
|Δut|2 ≥ λk1 u2 ,
∀u ∈ Hk ;
∀u ∈ Hk⊥ .
2.11
2.12
3. Main Results and Proofs
Theorem 3.1. Suppose that F satisfies (F1) and the following conditions
F2 there are p, q : Z1, T → R , α ∈ 0, 1 such that
|∇Ft, x| ≤ pt|x|α qt,
∀t, x ∈ Z1, T × RN ,
3.1
where Za, b : Z ∩ a, b for every a, b ∈ Z with a ≤ b;
T
2 T
2
F3 lim inf|x| → ∞ |x|−2α
t1 Ft, x > λT/2 2λ1 /2λ1 t1 p t for all t ∈ Z1, T .
Then problem 1.1 has at least one periodic solution with period T .
Theorem 3.2. Suppose that F satisfies (F1) and (F2) with α 1. Moreover, assume the following
conditions hold:
T
pt < λ1 ,
3.2
t1
and
T
T
1/2
F4 lim inf|x| → ∞ |x|−2 Tt1 Ft, x > λT/2 λ1/2
1 t1 ptλ1 λ1 − t1 pt/2λ1 λ1 −
T 2
T
t1 pt
t1 p t for all t ∈ Z1, T .
Then problem 1.1 has at least one periodic solution with period T .
Remark 3.3. It is easy to see that F2 is more general than F2 and F3 is weaker than F3.
Theorem 3.2 is a new result, which completes Theorem A when α 1.
Discrete Dynamics in Nature and Society
5
For the sake of convenience, we denote
1/2
T
2
p t
,
M1 M2 t1
T
pt,
M3 t1
T
qt.
3.3
t1
Proof of Theorem 3.1. First we prove that ϕ satisfies the PS condition. Suppose that a
consequence {un } ⊂ HT is such that −C3 ≤ ϕun ≤ C3 , where C3 > 0 and ϕ un → 0 as
n → ∞. Then for sufficiently large n,
−u ≤ ϕ un , u ≤ u.
3.4
∈ H0 H0⊥ , where H0 N0 , and H0⊥ From Lemma 2.1, we can write u as u u u
T/2
Nk . From F3 , we can choose a1 > 1/λ21 > 0 such that
k1
lim inf |x|
−2α
|x| → ∞
T
t1
Ft, x >
λT/2 a1 √
a1 M12 .
2
3.5
From F2 , 2.6, Hölder inequality, and Young inequality, we have
1
T
T
ut, u
t ds
∇Ft, u s
Ft, ut − Ft, u t1 0
t1
≤
T 1
t1
≤
T 1
ut|α qt |
ut| ds
pt|u s
t1
≤
ut, u
t| ds
|∇Ft, u s
0
T
0
T
pt |u|α |
ut|α |
ut|
ut| qt|
t1
≤ |u|
3.6
t1
T
α
t1
1/2 1/2
T
2
2
p t
M2 ut|
uα1
u∞
|
∞ M3 t1
M1 |u|α u M2 uα1
u∞
∞ M3 √
a1 2 2α
1
M1 |u| M2 ≤ √ u2 uα1 M3 u.
2 a1
2
In a similar way, we have
T
n t ≤
∇Ft, un t, u
t1
a1 λ1 2
1
M1 |un |2α M2 un 2 un α1 M3 un .
2a1 λ1
2
3.7
6
Discrete Dynamics in Nature and Society
n ∈ H0
Let un un u
H0⊥ . From 2.12 and 3.7, we have
T
T
un t un t, Δ
un t ≥ λ1 un 2 ,
Δun t, Δ
Δ
t1
3.8
t1
T
T
un t − ϕ un , u
n ∇Ft, un t, u
n t
Δun t, Δ
t1
t1
3.9
a1 λ1 2
1
M1 |un |2α M2 ≤ un un 2 un α1 M3 un .
2a1 λ1
2
It follows from 3.8 and 3.9 that
a1 λ1 2
λ1
M1 |un |2α ,
un 2 C4 ≤
2
2
3.10
where C4 mins∈0,∞ {a1 λ21 − 1/2a1 λ1 s2 − M2 sα1 − 1 M3 s}. The fact that a1 > 1/λ21 > 0
implies that −∞ < C4 < 0. So it follows from 3.10 that
un 2 ≤ a1 M12 |un |2α − 2C4 λ1 ,
3.11
and so
un ≤
√
a1 M1 |un |α C5 ,
3.12
where C5 > 0. It follows from the boundedness of ϕun , 2.11, 3.6, 3.11, and 3.12 that
C3 ≥ ϕu −
T
T
1
|Δun t|2 Ft, un t − Ft, 0
2 t1
t1
T
T
T
1
un t|2 Ft, un t − Ft, un Ft, un − Ft, 0
|Δ
2 t1
t1
t1
√
a1 2
1
1
≥ − λT/2 M1 |un |2α − M2 un − √ un 2 −
un α1 − M3 un 2
2 a1
2
−
T
t1
Ft, un − Ft, 0
Discrete Dynamics in Nature and Society
7
T
1
1
≥ − λT/2 √
a1 M12 |un |2α − 2C4 λ1 Ft, un − Ft, 0
2
2 a1
t1
√
√
α1
√
a1 2
−
M1 |un |2α − M2 a1 M1 |un |α C5
− M3 a1 M1 |un |α C5
2
√
1
C4 λ1
≥ − λT/2 a1 M12 − a1 M12 |un |2α λT/2 C4 λ1 √
2
a1
√
α1
√
− M2 2α a1 M1
|un |αα1 2α C5α1 − M1 M3 a1 |un |α − M3 C5
T
Ft, un − Ft, 0
t1
√
√
1
λT/2 a1 a1 M12 − M1 M3 a1 |un |−α
2
t1
α1
C4 λ1
αα−1
α √
λT/2 C4 λ1 √ − M3 C5
−M2 2
a1 M1
|un |
a1
|un |−2α
2α
|un |
T
− M2 2α C5α1 −
Ft, un −
T
Ft, 0.
t1
3.13
Inequalities 3.5 and 3.13 imply that {un } is bounded. Hence, {
un } is bounded by 3.12,
and then {un } is bounded. Since HT is finite dimensional, there exists a subsequence of {un }
convergent in HT . Thus, we conclude that PS condition is satisfied.
In order to use the saddle point theorem 16, Theorem 4.6, we only need to verify the
following conditions:
I1 ϕx → ∞ as |x| → ∞ in H0 ;
I2 ϕu → −∞ as u → ∞ in H0⊥ .
In fact, from F3 , we have
T
Ft, x −→ ∞ as |x| −→ ∞ in H0 .
3.14
t1
For any x ∈ H0 , since
T
t1
2
|Δx| 0, we have
ϕx T
Ft, x − Ft, 0.
3.15
t1
It follows from 3.14 and the above inequality that
ϕx −→ ∞ as |x| −→ ∞ in H0 .
Thus I1 is easy to verify.
3.16
8
Discrete Dynamics in Nature and Society
Next, for all u ∈ H0⊥ , from F2 and 2.6, we have
1
T
T
∇Ft, sut, utds
Ft, ut − Ft, 0 t1
t1 0
≤
T
pt|ut|α1 t1
T
qt|ut|
t1
3.17
≤ M2 uα1
∞ M2 u∞
≤ M2 uα1 M2 u.
By 2.7, 2.12, and 3.17, we obtain
ϕu −
T
T
1
|Δut|2 Ft, ut − Ft, 0
2 t1
t1
3.18
1
≤ − λ1 u2 M2 uα1 M2 u.
2
Since λ1 > 0 and α ∈ 0, 1, we have ϕu → −∞ as u → ∞ in H0⊥ . The proof of Theorem 3.1
is complete.
Proof of Theorem 3.2. By 3.2 and F4, we can choose an a2 ∈ R such that
a2 >
lim inf|x|−2
|x| → ∞
T
Ft, x >
t1
1
> 0,
λ1
3.19
√ a2
1
1
1
a2
λT/2 √ M2
M12 .
2
2 a2 2
λ1 − M2
2
It follows from F2 with α 1, 2.6, Hölder inequality, and Young inequality that
1
T
T
ut, u
tds
∇Ft, u s
Ft, ut − Ft, u t1 0
t1
≤
T 1
t1
≤
T 1
t1
ut, u
t|ds
|∇Ft, u s
0
0
pt|u| s|
ut||
ut|ds T 1
t1
qt|
ut|ds
0
T
T
1
pt |u| |
ut|
ut| |
ut| qt|
2
t1
t1
3.20
Discrete Dynamics in Nature and Society
≤ |u|
9
1/2 1/2
T
T
T
1
2
p t
u∞ qt
ut|
u2∞ pt |
2
t1
t1
t1
t1
T
2
M2
u∞
u2∞ M3 2
√
a2 2 2
1
M2
2
u
M1 |u| M3 ≤
u.
√
2 a2
2
2
M1 |u|
u 3.21
In a similar way, we have
T
M2
a2
1
un 2 M12 |un |2 M3 n t ≤
un .
∇Ft, un t, u
2a2
2
2
t1
3.22
From 3.8 and 3.22, we have
λ1 un 2 ≤
T
un t
Δun t, Δ
t1
T
− ϕ un , u
n ∇Ft, un t, u
n t
≤ un 3.23
t1
M2
a2
1
un 2 M12 |un |2 M3 un .
2a2
2
2
It follow from 3.23 that
a2
1
un 2 C6 ≤ M12 |un |2 ,
λ1 − M2 2
2
3.24
where C6 mins∈0,∞ {λ1 a2 − 1/2a2 s2 − 1 M3 s}. The fact that a2 > 1/λ1 > 0 implies
that −∞ < C6 < 0. So it follows from 3.24 that
un 2 ≤
a2
2C6
M2 |un |2 −
,
λ1 − M2 1
λ1 − M2
3.25
and so
un ≤
a2
M1 |un | C7 ,
λ1 − M2
3.26
10
Discrete Dynamics in Nature and Society
where C7 > 0. It follows from the boundedness of ϕun , 2.11, 3.21, 3.25, and 3.26 that
T
T
1
|Δun t|2 Ft, un t − Ft, 0
2 t1
t1
C3 ≥ ϕu −
T
T
T
1
un t|2 Ft, un t − Ft, un Ft, un − Ft, 0
|Δ
2 t1
t1
t1
√
a2 2
1
1
M2
M1 |un |2 − M3 ≥ − λT/2 un 2 −
un un 2 −
√ 2
2 a2
2
2
−
T
Ft, un − Ft, 0
t1
1
1
M2
≥ − λT/2 √ 2
2 a2
2
2C6
a2
M2 |un |2 −
λ1 − M2 1
λ1 − M2
√
T
a2 2
a2
M1 |un |2 − M3
M1 |un | C7 Ft, un − Ft, 0
2
λ1 − M2
t1
T
1
1
a2
1
λT/2 √ M2
|un |2 |un |−2α Ft, un −
M12
2
2
a
2
λ
−
M
2
1
2
t1
√
a2 2
a2
1
C6
−1
λT/2 √ M2
M1 −
−
M1 M3 |un |
2
λ1 − M2
a2
λ1 − M2
−
− M3 C7 −
T
Ft, 0.
t1
3.27
Inequalities 3.20 and 3.27 imply that {un } is bounded. Hence, {
un } is bounded by 3.26,
and then {un } is bounded. Since HT is finite dimensional, there exists a subsequence of {un }
convergent in HT . Thus, we conclude that PS condition is satisfied.
In the following, we prove that ϕ satisfies I1 and I2. In fact, from F4, we have
T
Ft, x −→ ∞ as
|x| −→ ∞ in H0 .
3.28
t1
T
It follows from 3.27 and
ϕx t1
T
t1
Thus I1 is easy to verify.
|Δx|2 0 that
Ft, x − Ft, 0 −→ ∞ as |x| −→ ∞ in H0 .
3.29
Discrete Dynamics in Nature and Society
11
Next, for all u ∈ H0⊥ , from F2 with α 1 and 2.6, we have
1
T
T
∇Ft, sut, utds
Ft, ut − Ft, 0 t1 0
t1
≤
T
T
1
pt|ut|2 qt|ut|
2 t1
t1
≤
1
M2 u2∞ M3 u∞
2
≤
1
M2 u2 M3 u.
2
3.30
By 2.7, 2.12, and 3.30, we obtain
ϕu −
T
T
1
|Δut|2 Ft, ut − Ft, 0
2 t1
t1
3.31
M2
1
≤ − λ1 u2 u2 M2 u.
2
2
Since λ1 > M2 , we have ϕu → −∞ as u → ∞ in H0⊥ . The proof of Theorem 3.2 is
complete.
4. Examples
In this section, we give two examples to illustrate our results.
Example 4.1. Let
2πt 7/4
2πt
Ft, x sin
1 |x|3/2 ht, x,
|x| sin
T
T
4.1
where h : Z1, T → RN and ht T ht. It is easy to see that
2πt 3/4 3 2πt
7 1|x|1/2 |ht|
|x| sin
|∇Ft, x| ≤ sin
4
T
2
T
2πt 7 sin
ε |x|3/4 aε |ht|, ∀t, x ∈ Z1, T × RN ,
≤
4 T 4.2
where ε > 0 and aε is a positive constant and is dependent on ε. It is easy to see that Ft, x
satisfies F1. From 4.2, we can let p, q, and α be
2πt 7 sin
ε ,
pt 4 T qt aε |ht|,
α
3
,
4
4.3
12
Discrete Dynamics in Nature and Society
which shows that F2 is satisfied. Moreover, if we let T 2, then we have
lim inf |x|−2α
|x| → ∞
T
Ft, x 2,
t1
2
T 147 sin 2πt ε 147 ε2 .
p2 t 128 t1 T 128
t1
T
λT/2 2λ1 λ1 λT/2 4,
2λ21
4.4
If we let ε2 < 256/147, then we obtain
lim inf |x|−2α
|x| → ∞
T
Ft, x 2 >
t1
T
147 2 λT/2 2λ1 ε p2 t,
128
2λ21
t1
4.5
which shows that F3’ holds. Then from Theorem 3.1, problem 1.1 has at least one periodic
solution with period T .
Example 4.2. Let
Ft, x 2πt 1
2πt
1
sin
|x|2 sin
|x|3/2 ht, x,
4
T
2
T
4.6
where h : Z1, T → RN and ht T ht. It is easy to see that Ft, x satisfies F1 and
2πt 1 3 2πt 1/2
1 sin
sin
|x|
|x| |ht|
2
T
2
2
T 2πt 1 1 ε
sin
≤
|x| bε |ht|, ∀t, x ∈ Z1, T × RN ,
2 T
2
|∇Ft, x| ≤
4.7
where ε > 0 and bε is a positive constant and is dependent on ε. The above shows that F2
holds with α 1 and
pt 2πt 1 1 sin
ε
,
2 T
2
qt bε |ht|.
4.8
Let T 2, then λ0 0, λ1 λT/2 4. Observe that
|x|
−2
T
−2
Ft, x |x|
t1
T 1
t1
1
4
2πt 1
2πt
2
3/2
sin
|x| sin
|x| ht, x
4
T
2
T
T
ht, |x|−2 x .
t1
4.9
Discrete Dynamics in Nature and Society
13
On the other hand, we have
T
T
2πt 1 1
1 sin
ε
ε,
pt 2
T
2
2
t1
t1
2
2
T
T
2πt 1 1 1 1
2
sin
ε ε .
p t 4 T
2
2 2
t1
t1
We can choose ε sufficiently small such that
lim inf |x|−2
|x| → ∞
T
Ft, x t1
T
t1
4.10
pt < 4 and
2
27 − 2ε 1
1
>
ε
4 167 − ε 2
T
λ1 − Tt1 pt pt λ1/2
1
p2 t,
2λ1 λ1 − Tt1 pt
t1
λT/2 λ1/2
1 T
4.11
t1
which shows that F4 holds. Then from Theorem 3.2, problem 1.1 has at least one periodic
solution with period T .
Acknowledgment
X. H. Tang is supported by the NNSF no. 10771215 of China.
References
1 R. P. Agarwal, Difference Equations and Inequalities, vol. 228 of Monographs and Textbooks in Pure and
Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
2 C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems, vol. 16 of Kluwer Texts in the
Mathematical Sciences, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
3 S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer,
New York, NY, USA, 2nd edition, 1999.
4 R. P. Agarwal and J. Popenda, “Periodic solutions of first order linear difference equations,”
Mathematical and Computer Modelling, vol. 22, no. 1, pp. 11–19, 1995.
5 R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular discrete -Laplacian
problems via variational methods,” Advances in Difference Equations, no. 2, pp. 93–99, 2005.
6 Z. M. Guo and J. S. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear
difference equations,” Science in China A, vol. 46, no. 4, pp. 506–515, 2003.
7 Z. M. Guo and J. S. Yu, “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian
systems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 55, no. 7-8, pp. 969–983, 2003.
8 Z. M. Guo and J. S. Yu, “The existence of periodic and subharmonic solutions of subquadratic second
order difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419–430, 2003.
9 H. H. Liang and P. X. Weng, “Existence and multiple solutions for a second-order difference boundary
value problem via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 326, no.
1, pp. 511–520, 2007.
10 J. Rodriguez and D. L. Etheridge, “Periodic solutions of nonlinear second-order difference equations,”
Advances in Difference Equations, no. 2, pp. 173–192, 2005.
11 Y.-F. Xue and C.-L. Tang, “Existence of a periodic solution for subquadratic second-order discrete
Hamiltonian system,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 67, no. 7, pp. 2072–
2080, 2007.
14
Discrete Dynamics in Nature and Society
12 J. S. Yu, Z. M. Guo, and X. Zou, “Periodic solutions of second order self-adjoint difference equations,”
Journal of the London Mathematical Society, vol. 71, no. 1, pp. 146–160, 2005.
13 J. S. Yu, Y. H. Long, and Z. M. Guo, “Subharmonic solutions with prescribed minimal period of a
discrete forced pendulum equation,” Journal of Dynamics and Differential Equations, vol. 16, no. 2, pp.
575–586, 2004.
14 J. S. Yu, X. Q. Deng, and Z. M. Guo, “Periodic solutions of a discrete Hamiltonian system with a
change of sign in the potential,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp.
1140–1151, 2006.
15 Z. Zhou, J. S. Yu, and Z. M. Guo, “Periodic solutions of higher-dimensional discrete systems,”
Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 5, pp. 1013–1022, 2004.
16 P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, vol.
65 of CBMS Regional Conference Series in Mathematics, Conference Board of the Mathematical Sciences,
Washington, DC, USA, 1986.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014