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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 247071, 11 pages
doi:10.1155/2008/247071
Research Article
Existence Theorems of Periodic Solutions for
Second-Order Nonlinear Difference Equations
Xiaochun Cai
1
and Jianshe Yu
2
1
College of Statistics, Hunan University, Changsha, Hunan 410079, China
2
College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China
Correspondence should be addressed to Xiaochun Cai,
Received 14 August 2007; Accepted 14 November 2007
Recommended by Patricia J. Y. Wong
The authors consider the second-order nonlinear difference equation of the type Δp
n
Δx
n−1
δ
q
n
x
δ
n
fn, x
n
,n∈ Z, using critical point theory, and they obtain some new results on the existence
of periodic solutions.
Copyright q 2008 X. Cai and J. Yu. This is an open access article distributed under the Creative
Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
We denote by
N, Z, R the set of all natural numbers, integers, and real numbers, respectively.
For a, b ∈
Z, define Za{a, a 1, }, Za, b{a, a 1, ,b} when a ≤ b.
Consider the nonlinear second-order difference equation
Δ
p
n
Δx
n−1
δ
q
n
x
δ
n
f
n, x
n
,n∈
Z, 1.1
where the forward difference operator Δ is defined by the equation Δx
n
x
n1
− x
n
and
Δ
2
x
n−1
ΔΔx
n−1
Δx
n
− Δx
n−1
. 1.2
In 1.1, the given real sequences {p
n
}, {q
n
} satisfy p
nT
p
n
> 0,q
nT
q
n
for any n ∈ Z,
f :
Z × R → R is continuous in the second variable, and fn T, zfn, z for a given positive
integer T and for all n, z ∈
Z × R. −1
δ
−1,δ>0, and δ is the ratio of odd positive integers.
By a solution of 1.1, we mean a real sequence x {x
n
},n∈ Z, satisfying 1.1.
In 1, 2, the qualitative behavior of linear difference equations of type
Δp
n
Δx
n
q
n
x
n
0 1.3
2 Advances in Difference Equations
has been investigated. In 3, the nonlinear difference equation
Δp
n
Δx
n−1
q
n
x
n
fn, x
n
1.4
has been considered. However, results on periodic solutions of nonlinear difference equations
are very scarce in the literature, see 4, 5. In particular, in 6, by critical point method, the
existence of periodic and subharmonic solutions of equation
Δ
2
x
n−1
f
n, x
n
0,n∈
Z, 1.5
has been studied. Other interesting contributions can be found in some recent papers 7–11
and in references contained therein. It is interesting to study second-order nonlinear difference
equations 1.1 because they are discrete analogues of differential equation
ptϕu
ft, u0. 1.6
In addition, they do have physical applications in the study of nuclear physics, gas aerody-
namics, infiltrating medium theory, and plasma physics as evidenced in 12, 13.
The main purpose here is to develop a new approach to the above problem by using
critical point method and to obtain some sufficient conditions for the existence of periodic
solutions of 1.1.
Let X be a real Hilbert space, I ∈ C
1
X, R, which implies that I is continuously Fr
´
echet
differentiable functional defined on X. I is said to be satisfying Palais-Smale condition P-S
condition if any sequence {Iu
n
} is bounded, and I
u
n
→ 0asn →∞possesses a conver-
gent subsequence in X.LetB
ρ
be the open ball in X with radius ρ andcenteredat0,andlet
∂B
ρ
denote its boundary.
Lemma 1.1 mountain pass lemma, see 14. Let X be a real Hilbert space, and assume that I ∈
C
1
X, R satisfies the P-S condition and the following conditions:
I
1
there exist constants ρ>0 and a>0 such that Ix ≥ a for all x ∈ ∂B
ρ
,whereB
ρ
{x ∈ X :
x
X
<ρ};
I
2
I0 ≤ 0 and there exists x
0
∈ B
ρ
such that Ix
0
≤ 0.
Then c inf
h∈Γ
sup
s∈0,1
Ihs is a positive critical value of I,where
Γ
h ∈ C
0, 1,X
: h00,h1x
0
. 1.7
Lemma 1.2 saddle point theorem, see 14, 15. Let X be a real Banach space, X X
1
⊕ X
2
, where
X
1
/
{0} and is finite dimensional. Suppose I ∈ C
1
X, R satisfies the P-S condition and
I
3
there exist constants σ, ρ>0 such that I|
∂B
ρ
X
1
≤ σ;
I
4
there is e ∈ B
p
X
1
and a constant ω>σsuch that I|
eX
2
≥ ω.
Then I possesses a critical value c ≥ ω and
c inf
h∈Γ
max
u∈B
ρ
X
1
Ihu, 1.8
where Γ{h ∈ C
B
ρ
X
1,
X|h|
∂B
ρ
X
1
id}.
X. Cai and J. Yu 3
2. Preliminaries
In this section, we are going to establish the corresponding variational framework for 1.1.
Let Ω be the set of sequences
x
x
n
n∈Z
,x
−n
, ,x
−1
,x
0
,x
1
, ,x
n
,
, 2.1
that is,
Ω
x
x
n
: x
n
∈ R,n∈ Z
. 2.2
For any x, y ∈ Ω,a,b∈
R, ax by is defined by
ax by :
ax
n
by
n
∞
n−∞
. 2.3
Then Ω is a vector space. For given positive integer T, E
T
is defined as a subspace of Ω by
E
T
x
x
n
∈ Ω : x
nT
x
n
,n∈ Z
. 2.4
Clearly, E
T
is isomorphic to R
T
, and can be equipped with inner product
x, y
T
i1
x
i
y
i
, ∀x, y ∈ E
T
, 2.5
by which the norm · can be induced by
x :
T
i1
x
2
i
1/2
, ∀x ∈ E
T
. 2.6
It is obvious that E
T
with the inner product defined by 2.5 is a finite-dimensional Hilbert
space and linearly homeomorphic to
R
T
. Define the functional J on E
T
as follows:
Jx
1
δ 1
T
n1
p
n
Δx
n−1
δ1
−
1
δ 1
T
n1
q
n
x
δ1
n
T
n1
Fn, x
n
, ∀x ∈ E
T
, 2.7
where Ft, z
z
0
ft, sds. Clearly, J ∈ C
1
E
T
, R, and for any x {x
n
}
n∈Z
∈ E
T
, by using
x
0
x
T
,x
1
x
T1
, we can compute the partial derivative as
∂J
∂x
n
−Δp
n
Δx
n−1
δ
− q
n
x
δ
n
fn, x
n
,n∈ Z1,T. 2.8
Thus x {x
n
}
n∈Z
is a critical point of J on E
T
i.e., J
x0 if and only if
Δp
n
Δx
n−1
δ
q
n
x
δ
n
fn, x
n
,n∈ Z1,T. 2.9
By the periodicity of x
n
and fn, z in the first variable n, we have reduced the existence
of periodic solutions of 1.1 to that of critical points of J on E
T
. In other words, the func-
tional J is just the variational framework of 1.1. For convenience, we identify x ∈ E
T
with
x x
1
,x
2
, ,x
T
T
.DenoteW {x
1
,x
2
, ,x
T
T
∈ E
T
: x
i
≡ v, v ∈ R,i∈ Z1,T} and
W
⊥
Y such that E
T
Y ⊕ W. Denote other norm ·
r
on E
T
as follows see, e.g., 16:
x
r
T
i1
|x
i
|
r
1/r
, for all x ∈ E
T
and r>1. Clearly, x
2
x.Dueto·
r
1
and ·
r
2
being equivalent when r
1
,r
2
> 1, there exist constants c
1
, c
2
, c
3
,andc
4
such that c
2
≥ c
1
> 0,
c
4
≥ c
3
> 0, and
c
1
x≤x
δ1
≤ c
2
x, 2.10
c
3
x≤x
β
≤ c
4
x, 2.11
for all x ∈ E
T
, δ>0andβ>1.
4 Advances in Difference Equations
3. Main results
In this section, we will prove our main results by using critical point theorem. First, we prove
two lemmas which are useful in the proof of theorems.
Lemma 3.1. Assume that the following conditions are satisfied:
F
1
there exist constants a
1
> 0, a
2
> 0,andβ>δ 1 such that
z
0
fn, sds ≤−a
1
|z|
β
a
2
, ∀z ∈ R; 3.1
F
2
q
n
≤ 0, ∀n ∈ Z. 3.2
Then the functional
Jx
1
δ 1
T
n1
p
n
Δx
n−1
δ1
−
1
δ 1
T
n1
q
n
x
δ1
n
T
n1
Fn, x
n
3.3
satisfies P-S condition.
Proof. For any sequence {x
l
}⊂E
T
, with Jx
l
being bounded and J
x
l
→ 0as
l → ∞, there exists a positive constant M such that |Jx
l
|≤M. Thus, by F
1
,
−M ≤ Jx
l
1
δ 1
T
n1
p
n
x
l
n
− x
l
n−1
δ1
− q
n
x
l
n
δ1
T
n1
F
n, x
l
n
≤
1
δ 1
T
n1
p
n
2
δ1
x
l
n
δ1
x
l
n−1
δ1
−
1
δ 1
T
n1
q
n
x
l
n
δ1
T
n1
F
n, x
l
n
≤
2
δ1
δ 1
T
n1
p
n
p
n1
x
l
n
δ1
−
1
δ 1
T
n1
q
n
x
l
n
δ1
− a
1
T
n1
x
l
n
β
a
2
T
1
δ 1
T
n1
2
δ1
p
n
p
n1
− q
n
x
l
n
δ1
− a
1
x
l
β
β
a
2
T.
3.4
Set
A
0
max
n∈Z1,T
2
δ1
p
n
p
n1
− q
n
. 3.5
Then A
0
> 0. Also, by the above inequality, we have
−M ≤ Jx
l
≤
A
0
δ 1
x
l
δ1
δ1
− a
1
x
l
β
β
a
2
T. 3.6
X. Cai and J. Yu 5
In view of
T
n1
x
l
n
δ1
≤ T
β−δ−1/β
T
n1
x
l
n
β
δ1/β
, 3.7
we have
x
l
β
β
≥ T
δ1−β/δ1
x
l
β
δ1
. 3.8
Then we get
−M ≤ Jx
l
≤
A
0
δ 1
x
l
δ1
δ1
− a
1
T
δ1−β/δ1
x
l
β
δ1
a
2
T. 3.9
Therefore, for any l ∈
N,
a
1
T
δ1−β/δ1
x
l
β
δ1
−
A
0
δ 1
x
l
δ1
δ1
≤ M a
2
T. 3.10
Since β>δ1, the above inequality implies that {x
l
} is a bounded sequence in E
T
. Thus {x
l
}
possesses convergent subsequences, and the proof is complete.
Theorem 3.2. Suppose that F
1
and following conditions hold:
F
3
for each n ∈ Z,
lim
z→0
fn, z
z
δ
0;
3.11
F
4
q
n
< 0, ∀n ∈ Z1,T.
3.12
Then there exist at least two nontrivial T-periodic solutions for 1.1.
Proof. We will use Lemma 1.1 to prove Theorem 3.2.First,byLemma 3.1, J satisfies P-S condi-
tion. Next, we will prove that conditions I
1
and I
2
hold. In fact, by F
3
, there exists ρ>0
such that for any |z| <ρand n ∈
Z1,T,
|Fn, z|≤ −
q
max
2δ 1
z
δ1
, 3.13
where q
max
max
n∈Z1,T
q
n
< 0. Thus for any x ∈ E
T
, x≤ρ for all n ∈ Z1,T, we have
Jx ≥−
q
max
δ 1
T
n1
x
δ1
n
q
max
2δ 1
T
n1
x
δ1
n
−
q
max
2δ 1
x
δ1
δ1
≥−
q
max
2δ 1
c
δ1
1
x
δ1
2
.
3.14
6 Advances in Difference Equations
Taking a −c
δ1
1
q
max
/2δ 1ρ
δ1
, we have
Jx|
∂B
ρ
≥ a>0, 3.15
and the assumption I
1
is verified. Clearly, J00. For any given w ∈ E
T
with w 1anda
constant α>0,
Jαw
1
δ 1
T
n1
p
n
αw
n
− αw
n−1
δ1
− q
n
αw
n
δ1
T
n1
Fn, αw
n
≤
1
δ 1
T
n1
p
n
2α
δ1
− q
n
α
δ1
− a
1
T
n1
|αw
n
|
β
a
2
T
≤
1
δ 1
T
n1
2
δ1
p
n
− q
n
α
δ1
α
δ1
− a
1
T
2−β/2
α
β
a
2
T
−→ − ∞ , α −→ ∞.
3.16
Thus we can easily choose a sufficiently large α such that α>ρand for
x αw ∈ E
T
, Jx < 0.
Therefore, by Lemma 1.1, there exists at least one critical value c ≥ a>0. We suppose that x is
a critical point corresponding to c,thatis,Jxc, and J
x0. By a similar argument to the
proof of Lemma 3.1, for any x ∈ E
T
, there exists x ∈ E
T
such that J
xc
max
. Clearly, x
/
0. If
x
/
x, and the proof is complete; otherwise, x x and c c
max
. By Lemma 1.1,
c inf
h∈Γ
sup
s∈0,1
J
hs
, 3.17
where Γ{h ∈ C0, 1,E
T
| h00,h1x}. Then for any h ∈ Γ,c
max
max
s∈0,1
Jhs.
By the continuity of Jhs in s, J0 ≤ 0andJ
x < 0 show that there exists some s
0
∈
0, 1 such that Jhs
0
c
max
. If we choose h
1
,h
2
∈ Γ such that the intersection {h
1
s |
s ∈ 0, 1}
{h
2
s | s ∈ 0, 1} is empty, then there exist s
1
,s
2
∈ 0, 1 such that Jh
1
s
1
Jh
2
s
2
c
max
. Thus we obtain two different critical points x
1
h
1
s
1
, x
2
h
2
s
2
of J in
E
T
. In this case, in fact, we may obtain at least two nontrivial critical points which correspond
to the critical value c
max
. The proof of Theorem 3.2 is complete. When fn, x
n
≡ h
n
,wehave
the following results.
Theorem 3.3. Assume that the following conditions hold:
G
1
qn < 0, ∀n ∈
Z1,T; 3.18
G
2
1
c
δ1
1
T
n1
h
2
n
δ1/2
T
n1
−q
n
<
p
min
λ
δ1/2
2
− q
max
T
n1
h
n
δ1
, 3.19
X. Cai and J. Yu 7
where p
min
min
n∈Z1,T
p
n
,q
max
max
n∈Z1,T
q
n
,c
1
is a constant in 2.10,andλ
2
is the minimal
positive eigenvalue of the matrix
A
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
2 −10··· 0 −1
−12−1 ··· 00
0 −12··· 00
··· ··· ··· ··· ··· ···
000··· 2 −1
−10 0··· −12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
T×T
. 3.20
Then equation
Δ
p
n
Δx
n−1
δ
q
n
x
δ
n
h
n
,n∈ Z, 3.21
possesses at least one T-periodic solution.
First, we proved the following lemma.
Lemma 3.4. Assume that G
1
holds, then the functional
Jx
1
δ 1
T
n1
p
n
Δx
n−1
δ1
−
1
δ 1
T
n1
q
n
x
δ1
n
T
n1
h
n
x
n
3.22
satisfies P-S condition on E
T
.
Proof. For any sequence {x
l
}⊂E
T
with Jx
l
being bounded and J
x
l
→ 0asn → ∞,
there exists a positive constant M such that |Jx
l
|≤M. In view of G
3
and
T
n1
|h
n
x
l
n
|≤
T
n1
h
2
n
1/2
T
n1
x
l
n
2
1/2
, 3.23
we have
M ≥ Jx
l
1
δ 1
T
n1
p
n
Δx
l
n−1
δ1
−
1
δ 1
T
n1
q
n
x
l
n
δ1
T
n1
h
n
x
l
n
≥−
1
δ 1
T
n1
q
n
x
l
n
δ1
−
T
n1
h
n
x
l
n
≥−
1
δ 1
q
max
T
n1
x
l
n
δ1
−
T
n1
h
2
n
1/2
T
n1
x
l
n
2
1/2
−
q
max
δ 1
x
l
δ1
δ1
−
T
n1
h
2
n
1/2
x
l
≥−
q
max
δ 1
c
δ1
1
x
l
δ1
−
T
n1
h
2
n
1/2
x
l
.
3.24
By δ 1 > 1, the above inequality implies that {x
l
} is a bounded sequence in E
T
.Thus{x
l
}
possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove
Theorem 3.3 by the saddle point theorem.
8 Advances in Difference Equations
Proof of Theorem 3.3. For any w z,z, ,z
T
∈ W, we have
Jw−
1
δ 1
T
n1
q
n
z
δ1
T
n1
h
n
z. 3.25
Take z
T
n1
h
n
/
T
n1
q
n
1/δ
and ρ w T
1/2
|
T
n1
h
n
/
T
n1
q
n
|
1/δ
, then
Jw
δ
δ 1
T
n1
h
n
δ1/δ
T
n1
q
n
1/δ
. 3.26
Set
σ
δ
δ 1
T
n1
h
n
δ1/δ
T
n1
q
n
1/δ
, 3.27
then we have
Jwσ, ∀w ∈ ∂B
ρ
Y. 3.28
On the other hand, for any x ∈ Y, we have
Jx
1
δ 1
T
n1
p
n
Δx
n−1
δ1
−
1
δ 1
T
n1
q
n
x
δ1
n
T
n1
h
n
x
n
≥
p
min
δ 1
T
n1
Δx
n−1
δ1
−
q
max
δ 1
T
n1
x
δ1
n
−
T
n1
|h
n
x
n
|
≥
p
min
δ 1
c
δ1
1
T
n1
Δx
n−1
2
δ1/2
−
q
max
δ 1
x
δ1
δ1
−
T
n1
h
2
n
1/2
x
p
min
δ 1
c
δ1
1
x
T
Ax
δ1/2
−
q
max
δ 1
x
δ1
δ1
−
T
n1
|h
n
x
n
|,
3.29
where x
T
x
1
,x
2
, ,x
T
.
Clearly, λ
1
0 is an eigenvalue of the matrix A and ξ v,v, ,v
T
∈ E
T
is an eigen-
vector of A corresponding to 0, where v
/
0,v∈ R.Letλ
2
,λ
3
, ,λ
T
be the other eigenvalues
of A. By matrix theory, we have λ
j
> 0 for all j ∈ Z2,T. Without loss of generality, we may
assume that 0 λ
1
<λ
2
≤ ··· ≤ λ
T
, then for any x ∈ Y,
Jx ≥
p
min
δ 1
c
δ1
1
λ
δ1/2
2
x
δ1
−
q
max
δ 1
x
δ1
δ1
−
T
n1
h
2
n
1/2
x
p
min
δ 1
c
δ1
1
λ
δ1/2
2
−
q
max
δ 1
c
δ1
1
x
δ1
−
T
n1
h
2
n
1/2
x
≥−
δ
δ 1
T
n1
h
2
n
1/2
T
n1
h
2
n
1/2
p
min
c
δ1
1
λ
δ1/2
2
− q
max
c
δ1
1
1/δ
,
3.30
X. Cai and J. Yu 9
as one finds by minimizing with respect to x. That is
Jx ≥−
δ
δ 1
T
n1
h
2
n
δ1/2δ
1/c
1
δ1/δ
p
min
λ
δ1/2
2
− q
max
1/δ
. 3.31
Set
w
0
−
δ
δ 1
T
n1
h
2
n
δ1/2δ
1/c
1
δ1/δ
p
min
λ
δ1/2
2
− q
max
1/δ
, 3.32
then by G
2
,wehave
Jx ≥ w
0
>σ, ∀x ∈ Y. 3.33
This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least
one critical point of J on E
T
, and the proof is complete. When q
n
> 0, we have the following
result.
Theorem 3.5. Assume that the following conditions are satisfied:
G
3
2
δ1
p
n
p
n1
<q
n
,q
n
> 0 for all n ∈ Z1,T;
G
4
T
n1
h
2
n
δ1/2δ
T
n1
q
n
1/δ
C
δ1
1
< −A
0
T
n1
h
n
δ1/δ
,
where A
0
max
n∈Z1,T
2
δ1
p
n
p
n1
− q
n
.
Then 3.21 possesses at least one T-periodic solution.
Before proving Theorem 3.5, first, we prove the following result.
Lemma 3.6. Assume that G
3
holds, then Jx defined by 3.22 satisfies P-S condition.
Proof. For any sequence {x
l
}∈E
T
with Jx
l
being bounded and J
x
l
→ 0as
n → ∞, there exists a positive constant M such that |Jx
l
|≤M.
Thus
−M ≤ J
x
l
≤
1
δ 1
T
n1
p
n
Δx
l
n−1
δ1
−
1
δ 1
T
n1
q
n
x
l
n
δ1
T
n1
h
n
x
l
n
≤
2
δ1
δ 1
T
n1
p
n
p
n1
x
l
n
δ1
−
1
δ 1
T
n1
q
n
x
l
n
δ1
T
n1
h
n
x
l
n
≤
1
δ 1
T
n1
2
δ1
p
n
p
n1
− q
n
x
l
n
δ1
T
n1
h
2
n
1/2
x
l
≤
1
δ 1
A
0
x
l
δ1
δ1
T
n1
h
2
n
1/2
x
l
≤
A
0
δ 1
c
δ1
2
x
l
δ1
T
n1
h
2
n
1/2
x
l
.
3.34
10 Advances in Difference Equations
That is,
−c
δ1
2
A
0
δ 1
x
l
δ1
−
T
n1
h
2
n
1/2
x
l
≤ M, ∀n ∈
N. 3.35
By δ 1 > 1, the above inequality implies that {x
l
} is a bounded sequence in E
T
. Thus {x
l
}
possesses convergent subsequences, and the proof is complete.
Proof of Theorem 3.5. For any w z,z, ,z
T
∈ W, we have
Jω−
1
δ 1
T
n1
q
n
z
δ1
T
n1
h
n
z. 3.36
Take z
T
n1
h
n
/
T
n1
q
n
,ρ w T
1/2
|
T
n1
h
n
/
T
n1
|
1/δ
, then
Jw
δ
δ 1
T
n1
h
n
δ1/δ
T
n1
q
n
1/δ
, ∀w ∈ ∂B
ρ
W. 3.37
Set
σ
δ
δ 1
T
n1
h
n
δ1/δ
|
T
n1
q
n
|
1/δ
, 3.38
then Jwσ for all w ∈ ∂B
ρ
W. On the other hand, for any x ∈ Y, we have
Jx ≤
1
δ 1
T
n1
2
δ1
p
n
p
n1
− q
n
x
δ1
n
T
n1
h
2
n
1/2
x
≤
A
0
δ 1
c
δ1
2
x
δ1
T
n1
h
2
n
1/2
x
≤−
δ
δ 1
1
A
0
1/δ
1
c
2
δ1/δ
T
n1
h
2
n
δ1/2δ
.
3.39
Set w
0
−δ/δ 11/A
0
1/δ
1/c
2
δ1/δ
T
n1
h
2
n
δ1/2δ
, then Jx ≤ w
0
<σ.Thus −Jx
satisfies the assumption of saddle point theorem, that is, there exists at least one critical point
of J on E
T
. This completes the proof of Theorem 3.5.
Acknowledgment
This project is supported by specialized research fund for the doctoral program of higher edu-
cation, Grant no. 20020532014.
X. Cai and J. Yu 11
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