Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 389624, 18 pages
doi:10.1155/2009/389624
Research Article
On the Nonexistence and Existence of Solutions for
a Fourth-Order Discrete Boundary Value Problem
Shenghuai Huang and Zhan Zhou
School of Mathematics and Information Science, Guangzhou University, Guangzhou,
Guangdong 510006, China
Correspondence should be addressed to Zhan Zhou, zzhou0321@hotmail.com
Received 16 July 2009; Accepted 16 October 2009
Recommended by Patricia J. Y. Wong
By using the critical point theory, we establish various sets of sufficient conditions on the
nonexistence and existence of solutions for the boundary value problems of a class of fourth-order
difference equations.
Copyright q 2009 S. Huang and Z. Zhou. 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.
1. Introduction
In this paper, 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 following boundary value problem BVP:
Δ2 pn − 1Δ2 un − 2 Δ qnΔun − 1 fn, un,
n ∈ Z1, k,
1.1
u−1 u0 0 uk 1 uk 2.
Here, k ∈ N, pn is nonzero and real valued for each n ∈ Z0, k 1, qn is real valued for
each n ∈ Z1, k 1. fn, u is real-valued for each n, u ∈ Z1, k × R and continuous in the
second variable u. Δ is the forward difference operator defined by Δun un 1 − un,
and Δ2 un ΔΔun.
We may think of 1.1 as being a discrete analogue of the following boundary value
problem:
ptx t
qtx t ft, xt,
xa x a 0,
t ∈ a, b
,
xb x b 0,
1.2
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which are used to describe the bending of an elastic beam; see, for example, 1–10
and
references therein. Owing to its importance in physics, many methods are applied to study
fourth-order boundary value problems by many authors. For example, fixed point theory
1, 3, 5–7
, the method of upper and lower solutions 8
, and critical point theory 9, 10
are widely used to deal with the existence of solutions for the boundary value problems of
fourth-order differential equations.
Because of applications in many areas for difference equations, in recent years, there
has been an increased interest in studying of fourth-order difference equation, which include
results on periodic solutions 11
, results on oscillation 12–14
, and results on boundary
value problems and other topics 15, 16
. Recently, a few authors have gradually paid
attention to applying critical point theory to deal with problems on discrete systems; for
example, Yu and Guo in 17
considered the existence of solutions for the following BVP:
Δ pnΔun − 1 qnun fn, un,
ua αua 1 A,
ub 2 βub 1 B.
1.3
The papers 17–20
show that the critical point theory is an effective approach to the study of
the boundary value problems of difference equations. In this paper, we will use critical point
theory to establish some sufficient conditions on the nonexistence and existence of solutions
for the BVP 1.1.
Let
an qn 1 − 2 pn pn 1 ,
bn pn − 1 4pn pn 1 − qn − qn 1.
1.4
Then the BVP 1.1 becomes
Lun fn, un,
n ∈ Z1, k,
u−1 u0 0 uk 1 uk 2,
1.5
where
Lun pn 1un 2 anun 1 bnun an − 1un − 1
pn − 1un − 2.
1.6
The remaining of this paper is organized as follows. First, in Section 2, we give some
preliminaries and establish the variational framework for BVP 1.5. Then, in Section 3, we
present a sufficient condition on the nonexistence of nontrivial solutions of BVP 1.5. Finally,
in Section 4, we provide various sets of sufficient conditions on the existence of solutions of
BVP 1.5 when f is superlinear, sublinear, and Lipschitz. Moreover, in a special case of f we
obtain a necessary and sufficient condition for the existence of unique solutions of BVP 1.5.
To conclude the introduction, we refer to 21, 22
for the general background on
difference equations.
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2. Preliminaries
In order to apply the critical point theory, we are going to establish the corresponding
variational framework of BVP 1.5. First we give some notations.
Let Rk be the real Euclidean space with dimension k. Define the inner product on Rk
as follows:
u, v k
u j v j ,
∀u, v ∈ Rk ,
2.1
∀u ∈ Rk ,
2.2
j1
by which the norm · can be induced by
⎞1/2
k
u ⎝ u2 j⎠ ,
⎛
j1
For BVP 1.5, consider the functional J defined on Rk as follows:
Ju 1
Mu, u − Fu,
2
∀u u1, u2, . . . , ukT ∈ Rk ,
2.3
where T is the transpose of a vector in Rk :
⎛
b1 a1 p2
⎜
⎜a1
⎜
⎜
⎜p2
⎜
⎜
⎜ 0
⎜
M⎜
⎜ ···
⎜
⎜
⎜ 0
⎜
⎜
⎜ 0
⎝
0
0
b2 a2 p3
0
···
0
···
a2 b3 a3 p4 · · ·
p3 a3 b4 a4 · · ·
0
0
0
0
0
0
⎟
⎟
⎟
⎟
⎟
0
0
0
⎟
⎟
⎟
0
0
0
⎟
⎟
···
···
··· ⎟
⎟
⎟
bk − 2 ak − 2 pk − 1⎟
⎟
⎟
ak − 2 bk − 1 ak − 1⎟
⎠
···
···
···
··· ···
0
0
0
0
···
0
0
0
0
···
0
0
0
0
· · · pk − 1 ak − 1
Fu ⎞
bk
k uj f j, s ds.
j1
,
2.4
k×k
2.5
0
After a careful computation, we find that the Fréchet derivative of J is
J u Mu − fu,
where fu is defined as fu f1, u1, f2, u2, . . . , fk, ukT .
2.6
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Expanding out J u, one can easily see that there is an one-to-one correspondence
between the critical point of functional J and the solution of BVP 1.5. Furthermore,
u u1, u2, . . . , ukT is a critical point of J if and only if {ut}k2
t−1
T
u−1, u0, u1, . . . , uk, uk 1, uk 2 is a solution of BVP 1.5, where u−1 u0 0 uk 1 uk 2.
Therefore, we have reduced the problem of finding a solution of 1.5 to that of seeking
a critical point of the functional J defined on Rk .
In order to obtain the existence of critical points of J on Rk , for the convenience of
readers, we cite some basic notations and some known results from critical point theory.
Let H be a real Banach space, J ∈ C1 H, R, that is, J is a continuously Fréchet
differentiable functional defined on H, and J is said to satisfy the Palais-Smale condition
P-S condition, if any sequence {xn } ⊂ H for which Jxn is bounded and J xn → 0 as
n → ∞ possesses a convergent subsequence in H.
Let Br denote the open ball in H about 0 of radius r and let ∂Br denote its boundary.
The following lemmas are taken from 23, 24
and will play an important role in the proofs
of our main results.
Lemma 2.1 Linking theorem. Let H be a real Banach space, H H1 H2 , where H1 is a finitedimensional subspace of H. Assume that J ∈ C1 H, R satisfies the P-S condition and the following.
F1 There exist constants σ, ρ > 0 such that J|∂Bρ ∩H2 ≥ σ.
F2 There is an e ∈ ∂B1 ∩ H2 and a constant R0 > ρ such that J|∂Q ≤ 0 and Q BR0 ∩
H1 {re | 0 < r < R0 }.
Then J possesses a critical value c ≥ σ, where
c inf maxJhu,
h∈Γ u∈Q
2.7
and Γ {h ∈ CQ, H|h|∂Q id}, where id denotes the identity operator.
Lemma 2.2 Saddle point theorem. Let H be a real Banach space, H H1 H2 , where H1 /
{0}
and is finite-dimensional. Suppose that J ∈ C1 H, R satisfies the P-S condition and the following.
F3 There exist constants σ, ρ > 0 such that J|∂Bρ ∩H1 ≤ σ.
F4 There is e ∈ Bρ ∩ H1 and a constant ω > σ such that J|eH2 ≥ ω.
Then J possesses a critical value c ≥ ω, where
c inf max Jhu,
h∈Γu∈Bρ ∩H1
2.8
and Γ {h ∈ CBρ ∩ H1 , H|h|∂Bρ ∩H1 id}, where id denotes the identity operator.
Lemma 2.3 Clark theorem. Let H be a real Banach space, J ∈ C1 H, R, with J being even,
bounded from below, and satisfying P-S condition. Suppose Jθ 0, there is a set K ⊂ H such that
K is homeomorphic to Sj−1 (j − 1 dimension unit sphere) by an odd map, and supK J < 0. Then J has
at least j distinct pairs of nonzero critical points.
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3. Nonexistence of Nontrivial Solutions
In this section, we give a result of nonexistence of nontrivial solutions to BVP 1.5.
Theorem 3.1. Suppose that matrix M is negative semidefinite and for n 1, 2, . . . , k,
zfn, z > 0,
for z /
0.
3.1
Then BVP 1.5 has no nontrivial solutions.
Proof. Assume, for the sake of contradiction, that BVP 1.5 has a nontrivial solution. Then J
has a nonzero critical point u∗ . Since
J u Mu − fu,
3.2
fu∗ , u∗ Mu∗ , u∗ ≤ 0.
3.3
we get
On the other hand, it follows from 3.1 that
k
u∗ nfn, u∗ n > 0.
fu∗ , u∗ 3.4
n1
This contradicts with 3.3 and hence the proof is complete.
In the existing literature, results on the nonexistence of solutions of discrete boundary
value problems are scarce. Hence Theorem 3.1 complements existing ones.
4. Existence of Solutions
Theorem 3.1 gives a set of sufficient conditions on the nonexistence of solutions of BVP
1.5. In this section, with part of the conditions being violated, we establish the existence
of solutions of BVP 1.5 by distinguishing three cases: f is superlinear, f is sublinear, and f
is Lipschitzian.
4.1. The Superlinear Case
In this subsection, we need the following conditions.
z
z
P1 For any n, z ∈ Z1, k × R, 0 fn, sds ≥ 0, and 0 fn, sds o|z|2 , as z → 0.
P2 There exist constants a1 > 0, a2 > 0 and β > 2 such that
z
0
fn, sds ≥ a1 |z|β − a2 ,
∀n, z ∈ Z1, k × R.
4.1
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P3 Matrix M exists at least one positive eigenvalue.
P4 fn, z is odd for the second variable z, namely,
fn, −z −fn, z,
∀n, z ∈ Z1, k × R.
4.2
Theorem 4.1. Suppose that fn, z satisfies P2 . Then BVP 1.5 possesses at least one solution.
Proof. For any u u1, u2, . . . , ukT ∈ Rk , we have
⎛
⎞
k uj
k β
f j, s ds ≥ a1 ⎝ u j ⎠ − a2 k
Fu j1
j1
0
⎛
⎞β/2
k 2
≥ a1 ⎝k2−β/β u j ⎠ − a2 k a1 k2−β/2 uβ − a2 k.
4.3
j1
Let A1 a1 k2−β/2 , A2 a2 k. We have, for any u u1, u2, . . . , ukT ∈ Rk ,
Fu ≥ A1 uβ − A2 .
4.4
Since matrix M is symmetric, its all eigenvalues are real. We denote by λ1 , λ2 , . . . , λk its
eigenvalues. Set λmax max{|λ1 |, |λ2 |, . . . , |λk |}. Thus for any u u1, u2, . . . , ukT ∈ Rk ,
Ju ≤
1
Mu, u − Fu
2
1
λmax u2 − A1 uβ A2
2
−→ −∞
4.5
as u −→ ∞.
The above inequality means that −Ju is coercive. By the continuity of Ju, J attains
its maximum at some point, and we denote it by u
, that is, J
u cmax , where cmax is a critical point of J. This completes the proof of Theorem 4.1.
supu∈Rk Ju. Clearly, u
Theorem 4.2. Suppose that fn, z satisfies the assumptions P1 , P2 , and P3 . Then BVP 1.5
possesses at least two nontrivial solutions.
To prove Theorem 4.2, we need the following lemma.
Lemma 4.3. Assume that P2 holds, then the functional J satisfies the P-S condition.
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Proof. Assume that {un } ⊂ Rk is a P-S sequence. Then there exists a constant c1 such that for
any n ∈ Z1, |Jun | ≤ c1 and J un → 0 as n → ∞. By 4.5 we have
1
Mun , un − F un
−c1 ≤ J un 2
2
β
1
≤ λmax un − A1 un A2 ,
2
4.6
β 1
2
A1 un − λmax un ≤ c1 A2 .
2
4.7
and so
Due to β > 2, the above inequality means {un } is bounded. Since Rk is a finite-dimensional
Hilbert space, there must exist a subsequence of {un } which is convergent in Rk . Therefore,
P-S condition is satisfied.
Proof of Theorem 4.2. Let λi , 1 ≤ i ≤ l, −μj , 1 ≤ j ≤ m be the positive eigenvalues and the
negative eigenvalues, where 0 < λ1 ≤ λ2 ≤ · · · ≤ λl , 0 > −μ1 ≥ −μ2 ≥ · · · ≥ −μm . Let ξi be an
eigenvector of M corresponding to the eigenvalue λi , 1 ≤ i ≤ l, and let ηj be an eigenvector of
M corresponding to the eigenvalue −μj , 1 ≤ j ≤ m, such that
ξi , ξj ⎧
⎨0,
as i /
j,
⎩1,
as i j,
ξi , ηj 0,
ηi , ηj ⎧
⎨0,
as i /
j,
⎩1,
as i j,
4.8
for any 1 ≤ i ≤ l, 1 ≤ j ≤ m.
Let E , E0 , and E− be subspaces of Rk defined as follows:
E− span ηj , 1 ≤ i ≤ m ,
E span{ξi , 1 ≤ i ≤ l},
⊥
E0 E
E− .
4.9
For any u ∈ Rk , u u u0 u− , where u ∈ E , u0 ∈ E0 , u− ∈ E− . Then
2
2
λ1 u ≤ Mu , u ≤ λl u ,
Let X1 E−
2 2
−μm u− ≤ Mu− , u− ≤ −μ1 u− .
4.10
E0 , X2 E , then Rk has the following decomposition of direct sum:
Rk X1
X2 .
4.11
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By assumption P1 , there exists a constant ρ > 0, such that for any n ∈ Z1, k, z ∈ Bρ ,
fn,
sds
≤ 1/4λ1 z2 . So for any u ∈ ∂Bρ ∩ X2 , n ∈ Z1, k,
0
z
Ju 1
Mu, u − Fu
2
1
1
1
≥ λ1 u2 − λ1 u2 λ1 ρ2 .
2
4
4
4.12
Denote σ 1/4λ1 ρ2 . Then
Ju ≥ σ,
∀u ∈ ∂Bρ ∩ X2 .
4.13
That is to say, J satisfies assumption F1 of Linking theorem.
Take e ∈ ∂B1 ∩ X2 . For any ω ∈ X1 , r ∈ R, let u re ω, because ω ω0 ω− , where
0
0
ω ∈ E , ω− ∈ E− . Then
1
Mre ω, re ω − Fre ω
2
k rejωj
1
1
f j, s ds
Mre, re Mω− , ω− −
2
2
j1 0
Ju ⎛
⎞
k 1 2 1 β
2
≤ λl r − μ1 ω− − a1 ⎝ re j ω j ⎠ a2 k
2
2
j1
⎛
⎞β/2
k 1 2 1 2
2
≤ λl r − μ1 ω− − a1 ⎝k2−β/β rej ωj ⎠ a2 k
2
2
j1
4.14
⎛
⎞β/2
k 2
1 2 1 2−β
/2
−
2
2
2
λl r − μ1 ω − a1 k ⎝
r e j ω j ⎠ a2 k
2
2
j1
≤
1 2
λl r − a1 k2−β/2 r β − a1 k2−β/2 ωβ a2 k.
2
Set g1 r 1/2λl r 2 − a1 k2−β/2 r β and g2 τ −a1 k2−β/2 τ β a2 k. Then
limr → ∞ g1 r −∞, limτ → ∞ g2 τ −∞. Furthermore, g1 r and g2 τ are bounded from
above. Accordingly, there is some R0 > ρ, such that for any u ∈ ∂Q, Ju ≤ 0, where
Q BR0 ∩ X1 {re | 0 < r < R0 }. By Linking theorem, J possesses a critical value c ≥ σ > 0,
where
c inf maxJhu,
h∈Γ u∈Q
Γ h ∈ C Q, Rk |h|∂Q id .
4.15
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Let u ∈ Rk be a critical point corresponding to the critical value c of J, that is, Ju c.
0 since c > 0. On the other hand, by Theorem 4.1, J has a critical point u
satisfying
Clearly, u /
u
, then Theorem 4.2 holds. Otherwise, u u
. Then cmax J
u supu∈Rk Ju ≥ c. If u /
J
u Ju c, which is the same as supu∈Rk Ju infh∈Γ supu∈Q Jhu.
Choosing h id, we have supu∈Q Ju cmax . Because the choice of e ∈ ∂B1 ∩ X2 ∈ Q BR0 ∩ X1 {re | 0 < r < R0 } is arbitrary, we can take −e ∈ ∂B1 ∩ X2 . Similarly, there exists a
positive number R1 > ρ, for any u ∈ ∂Q1 , Ju ≤ 0, where Q1 BR1 ∩X1 {−re | 0 < r < R1 }.
Again, by the Linking theorem, J possesses a critical value c0 ≥ σ > 0, where
c0 inf maxJhu,
h∈Γ1 u∈Q
Γ1 h ∈ C Q1 , Rk |h|∂Q1 id .
4.16
1
If c0 / cmax , then the proof is complete. Otherwise c0 cmax , supu∈Q1 Ju cmax . Because
J|∂Q ≤ 0 and J|∂Q1 ≤ 0, then J attains its maximum at some point in the interior of sets Q and
∈ Rk satisfying
Q1 . But Q ∩ Q1 ⊂ X1 , and Ju ≤ 0 for u ∈ X1 . Thus there is a critical point u
u
/
u
, J
u c0 cmax .
The proof of Theorem 4.2 is now complete.
Theorem 4.4. Suppose that fn, z satisfies the assumptions P1 , P2 , P3 , and P4 . Then BVP
1.5 possesses at least l distinct pairs of nontrivial solutions, where l is the dimension of the space
spanned by the eigenvectors corresponding to the positive eigenvalues of M.
Proof. From the proof of Theorem 4.2, it is easy to know that J is bounded from above and
satisfies the P-S condition. It is clear that J is even and J0 0, and we should find a set K
and an odd map such that K is homeomorphic to Sl−1 by an odd map.
We take K ∂Bρ ∩ X2 , where ρ and X2 are defined as in the proof of Theorem 4.2. It
is clear that K is homeomorphic to Sl−1 l − 1 dimension unit sphere by an odd map. With
4.13, we get supK −J < 0. Thus all the conditions of Lemma 2.3 are satisfied, and J has
at least l distinct pairs of nonzero critical points. Consequently, BVP 1.5 possesses at least l
distinct pairs nontrivial solutions. The proof of Theorem 4.4 is complete.
4.2. The Sublinear Case
In this subsection, we will consider the case where f is sublinear. First, we assume the
following.
P5 There exist constants a1 > 0, a2 > 0, R > 0 and 1 < α < 2 such that
Fu ≤ a1 uα a2 ,
∀u u1, u2, . . . , ukT ∈ Rk , u ≥ R.
4.17
The first result is as follows.
Theorem 4.5. Suppose that P5 is satisfied and that matrix M is positive definite. Then BVP 1.5
possesses at least one solution.
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Proof. The proof will be finished when the existence of one critical point of functional J
defined as in 2.3 is proved.
Assume that matrix M is positive definite. We denote by λ1 , λ2 , . . . , λk its eigenvalues,
where 0 < λ1 ≤ λ2 ≤ · · · ≤ λk . Then for any u u1, u2, . . . , ukT ∈ Rk , u ≥ R, followed
by P5 we have
Ju 1
Mu, u − Fu
2
≤ λ1 u2 − a1 uα − a2
−→ ∞
4.18
as u −→ ∞.
By the continuity of J on Rk , the above inequality means that there exists a lower
bound of values of functional J. Classical calculus shows that J attains its minimal value at
some point, and then there exist u such that Ju min{Ju | u ∈ Rk }. Clearly, u is a critical
point of the functional J.
Corollary 4.6. Suppose that matrix M is positive definite, and fn, z satisfies that there exist
constants a1 > 0, a2 > 0 and 1 < α < 2 such that
z
fn, sds ≤ a1 |z|α a2 ,
∀n, z ∈ Z1, k × R.
4.19
0
Then BVP 1.5 possesses at least one solution.
Corollary 4.7. Suppose that matrix M is positive definite, and fn, z satisfies the following.
P6 There exists a constant t0 > 0 such that for any n, z ∈ Z1, k × R, |fn, z |≤ t0 .
Then BVP 1.5 possesses at least one solution.
Proof. Assume that matrix M is positive definite. In this case, for any u
u1, u2, . . . , ukT ∈ Rk ,
k uj
k
f j, s ds ≤
t0 u j ≤ t0 ku.
|Fu| ≤
0
j1
j1
4.20
Since the rest of the proof is similar to Theorem 4.5, we do not repeat them here.
When M is neither positive definite nor negative definite, we now assume that M is
nonsingular, and we have the following result.
Theorem 4.8. Suppose that M is nonsingular, fn, z satisfies P6 . Then BVP 1.5 possesses at
least one solution.
Proof. We may assume that M is neither positive definite nor negative definite. Let
λ−l , λ−l1 , . . . , λ−1 , λ1 , λ2 , . . . , λm denote all eigenvalues of M, where λ−l ≤ λ−l1 ≤ · · · ≤ λ−1 <
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0 < λ1 ≤ λ2 ≤ · · · ≤ λm and l m k. For any j ∈ Z−l, −1 ∪ Z1, m, let ξj be an eigenvector
of M corresponding to the eigenvalue λj , j −l, −l 1, . . . , −1, 1, 2, . . . , m, such that
ξi , ξj ⎧
⎨0,
as i /
j,
⎩1,
as i j,
4.21
Let X1 and X2 be subspaces of Rk defined as follows:
X1 span{ξi , i ∈ Z1, m},
X2 span ξj , j ∈ Z−l, −1 .
4.22
Then Rk has the following decomposition of direct sum:
Rk X1
X2 .
4.23
Let Ju be defined as in 2.3. Then J ∈ C1 Rk , R. By 4.20,
|Fu| ≤ t0 ku,
∀u ∈ Rk .
4.24
Suppose that {un } ⊂ Rk is a P-S sequence. Then there exists a constant t1 such that for
any n ∈ Z1, |Jun | ≤ t1 and J un → 0 as n → ∞. Thus, for sufficiently large n and
for any u ∈ Rk , we have |J un , u| ≤ u.
Let un xn yn ∈ X1 X2 . We have, by 2.6, for any u u1, u2, . . . , ukT ∈
Rk ,
k
J un , u Mun , u − f j, un j · u j .
4.25
j1
Thus for sufficiently large n, we get
k
Mun , xn ≤
f j, un j · xn j xn j1
≤ t0
k n n j x x
j1
≤ t0 k 1 xn ,
2
Mun , xn Mxn , xn ≥ λ1 xn .
4.26
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Thus,
2 λ1 xn ≤ t0 k 1 xn ,
4.27
which implies that {xn } is bounded.
Now we are going to prove that {yn } is also bounded. By 4.25,
k
Mun , yn ≥
f j, un j · yn j − yn j1
≥ − t0 k 1 yn ,
4.28
2
Mun , yn Myn , yn ≤ λ−1 yn .
Thus we have
2
λ−1 yn ≥ − t0 k 1 yn .
4.29
2 −λ−1 yn − t0 k 1 yn ≤ 0.
4.30
And so
Due to λ−1 < 0, {yn } is bounded. Then {un } is bounded. Since Rk is a finitedimensional Hilbert space, there must exist a subsequence of {un } which is convergent in
Rk . Therefore, P-S condition is satisfied.
In order to apply the saddle point theorem to prove the conclusion, we consider the
functional −J and verify the conditions of Lemma 2.2.
For any y ∈ X2 , y y1, y2, . . . , ykT , we have
1
−J y − My, y F y
2
2
1
≥ − λ−1 y − t0 ky
2
2
1 ≥
t0 k .
2λ−1
This implies that F4 is true.
4.31
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For any x ∈ X1 , x x1, x2, . . . , xkT ,
1
−Jx − Mx, x Fx
2
4.32
1
≤ − λ1 x2 t0 kx
2
−→ −∞
as x −→ ∞.
This implies that F3 is true.
So far we have verified all the assumptions of Lemma 2.2 and hence −J has at least a
critical point in Rk . This completes the proof.
Consider the following special case
Δ4 un − 2 fn, un,
n ∈ Z1, k,
4.33
u−1 u0 0 uk 1 uk 2.
Here,
⎛
6 −4 1
⎜
⎜−4
⎜
⎜
⎜1
⎜
⎜
⎜0
⎜
M⎜
⎜· · ·
⎜
⎜
⎜0
⎜
⎜
⎜0
⎝
0
0 ··· 0
0
6 −4 1 · · · 0
0
−4 6 −4 · · · 0
0
1 −4 6 · · · 0
0
··· ··· ··· ··· ··· ···
0
⎞
⎟
0⎟
⎟
⎟
0⎟
⎟
⎟
0⎟
⎟
⎟
· · ·⎟
⎟
⎟
1⎟
⎟
⎟
−4⎟
⎠
0
0
0 · · · · · · −4
0
0
0 · · · −4 6
0
0
0 · · · 1 −4 6
.
4.34
k×k
It can be verified that M is positive definite, then we have the following corollaries.
Corollary 4.9. Suppose that there exist constants a1 > 0, a2 > 0 and 1 < α < 2 such that
z
fn, sds ≤ a1 |z|α a2 ,
∀n, z ∈ Z1, k × R.
4.35
0
Then BVP 4.33 possesses at least one solution.
Corollary 4.10. Suppose that fn, z satisfies P6 . Then BVP 4.33 possesses at least one solution.
14
Advances in Difference Equations
4.3. The Lipschitz Case
In this subsection, we suppose the following.
P7 Assume that there exist positive constants L, K such that for any n, z ∈ Z1, k ×
R,
fn, z ≤ L|z| K.
4.36
When fn, z is Lipschitzian in the second variable z, namely, there exists a constant
L > 0 such that for any n ∈ Z1, k, z1 , z2 ∈ R,
fn, z1 − fn, z2 ≤ L|z1 − z2 |,
4.37
then condition 4.36 is satisfied.
Theorem 4.11. Suppose that P7 is satisfied and M is nonsingular. If L < λmin min{λ1 , −λ−1 },
where λ1 and λ−1 are the minimal positive eigenvalue and maximal negative eigenvalue of M,
respectively, then BVP 1.5 possesses at least one solution.
Proof. Assume that {un } ⊂ Rk is a P-S sequence. Then J un → 0 as n → ∞. Thus for
sufficiently large n, we get J un ≤ 1. Since J un Mun − fun , then for sufficiently
large n,
Mun ≤ f un 1.
4.38
In view of 4.36, we have
k
k 2 2
Lun j K
f 2 j, un j ≤
f un j1
j1
k k n 2
L
u j 2LK un j K 2 k.
4.39
2
j1
j1
It follows, by using the inequality
Hölder’s inequality, that
√
abc ≤
√
a
√
b
√
c for a ≥ 0, b ≥ 0, c ≥ 0 and
1/2
f un ≤ Lun 2LK1/2 k1/4 un Kk1/2 .
4.40
By a similar argument to the proof of Theorem 4.8, we can decompose Rk into the
following form of direct sum:
Rk X1
X2 ,
4.41
Advances in Difference Equations
15
where X1 and X2 can be referred to 4.22. Thus un can be expressed by
un xn yn ,
2
2
4.42
2
and Mun Mxn Myn , where xn ∈ X1 , yn ∈ X2 . Therefore,
! 1/2
2
2 λ21 xn λ2−1 yn ≤ Mun ≤ 1 Lun 2LK1/2 k1/4 un Kk1/2 . 4.43
Hence,
1/2
λmin − Lun ≤ 1 2LK1/2 k1/4 un Kk1/2 .
4.44
By the fact that L < λmin , we know that the sequence {un } is bounded and therefore
the P-S condition is verified.
Now we are going to check conditions F3 and F4 for functional −J. In fact, by 4.36,
we have for any u ∈ Rk ,
k uj 1
f j, s ds ≤ Lu2 K ku.
|Fu| ≤
2
0
j1
4.45
Thus, for any y ∈ X2 , y y1, y2, . . . , ykT ,
1
−J y − My, y F y
2
2 1 2
1
≥ − λ−1 y − Ly − K ky
2
2
2
1
−λ−1 − Ly − K ky ≥ ω,
2
4.46
for some positive constant ω.
For any x ∈ X1 , x x1, x2, . . . , xkT , we have
1
−Jx − Mx, x Fx
2
1
1
≤ − λ1 x2 Lx2 K kx
2
2
4.47
1
− λ1 − Lx2 K kx −→ −∞ as x −→ ∞.
2
Until now, we have verified all the assumptions of Lemma 2.2 and hence −J has at least a
critical point in Rk . This completes the proof.
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Advances in Difference Equations
Finally, we consider the special case that fn, z is independent of the second variable
z; that is, fn, z ≡ gn for any n, z ∈ Z1, k × R, the BVP 1.1 becomes
Δ2 pn − 1Δ2 un − 2 Δ qnΔun − 1 gn,
n ∈ Z1, k,
4.48
u−1 u0 0 uk 1 uk 2.
As in Section 2, we reduce the existence of solutions of BVP 4.48 to the existence of
critical points of a functional J1 defined on Rk as follows:
J1 u 1
Mu, u − G, u,
2
∀u u1, u2, . . . , ukT ∈ Rk ,
4.49
where M is defined as in 2.4, and G g1, g2, . . . , gkT . Then we can see that the
critical point of J1 is just the solution to the following system of linear algebraic equations:
Mu − G 0.
4.50
By using the theory of linear algebra, we have the next necessary and sufficient
conditions.
Theorem 4.12. i BVP 4.48 has at least one solution if and only if rM rM, G, where
rM denotes the rank of matrix M and M, G is the augmented matrix defined as follows:
⎛
⎜b1
⎜
⎜
⎜a1
⎜
⎜
⎜
⎜p2
⎜
⎜
⎜
⎜
⎜ 0
M, G ⎜
⎜
⎜
⎜ ···
⎜
⎜
⎜
⎜ 0
⎜
⎜
⎜
⎜ 0
⎜
⎝
0
···
0
0
0
..
.
b2 a2 p3 · · ·
0
0
0
..
.
a2 b3 a3 · · ·
0
0
0
..
.
p3 a3 b4 · · ·
0
0
0
..
.
···
..
.
a1 p2
0
···
···
···
0
···
···
0
0
0
···
···
0
0
0
.
· · · ak − 2 bk − 1 ak − 1 ..
0
0
0
· · · pk − 1 ak − 1
.
ak − 2 pk − 1 ..
bk
..
.
⎞
g1 ⎟
⎟
⎟
g2 ⎟
⎟
⎟
⎟
g3 ⎟
⎟
⎟
⎟
⎟
··· ⎟
⎟
⎟
⎟
··· ⎟
⎟
⎟
⎟
gk − 2⎟
⎟
⎟
⎟
gk − 1⎟
⎟
⎠
gk
.
k×k1
4.51
ii BVP 4.48 has a unique solution if and only if rM k.
Advances in Difference Equations
17
Acknowledgment
This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction
no. 20071078001.
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