Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2008, Article ID 469815, 11 pages
doi:10.1155/2008/469815
Research Article
On Nonresonance Problems of Second-Order
Difference Systems
Ruyun Ma, Hua Luo, and Chenghua Gao
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma, mary@nwnu.edu.cn
Received 8 March 2007; Revised 14 November 2007; Accepted 24 January 2008
Recommended by Alberto Cabada
Let T be an integer with T ≥ 3, and let T : {1, . . . , T }. We study the existence and uniqueness of
solutions for the following two-point boundary value problems of second-order difference systems:
Δ2 ut − 1 ft, ut et, t ∈ T, u0 uT 1 0, where e : T → Rn and f : T × Rn → Rn is
a potential function satisfying ft, · ∈ C1 Rn and some nonresonance conditions. The proof of the
main result is based upon a mini-max theorem.
Copyright q 2008 Ruyun Ma 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.
1. Introduction
The existence and uniqueness of solutions of nonresonance problems of differential equations
have been studied extensively see 1–5, and the references therein. However, very few results have been established for nonresonance problems of differential equations. Although
we have seen some results of the existence of solutions of discrete equations subjected to
diverse boundary conditions, such as in 6–13, none of them addresses the nonresonance
problems.
In this paper, we consider nonlinear boundary value problems of second-order difference systems of the form
Δ2 ut − 1 f t, ut et,
u0 uT 1 0,
t ∈ T,
1.1
where e : T → Rn and f : T × Rn → Rn is a potential vector-valued function for t ∈ T.
Why do we pay attention to the discrete problem 1.1? Note that the continuous eigenvalue problem
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Advances in Difference Equations
y t ηut 0,
t ∈ 0, 1,
u0 u1 0
1.2
has a sequence of eigenvalues
η1 < η2 < · · · < ηn < · · · −→ ∞,
1.3
while the discrete eigenvalue problem
Δ2 yt − 1 μyt 0,
t ∈ T,
y0 yT 1 0
1.4
has exactly T real eigenvalues
μ 1 < μ2 < · · · < μT ,
1.5
: {0, 1, . . . , T 1}. Thus, the study of
where T is an integer with T ≥ 3, T : {1, . . . , T }, and T
nonresonance problems near the largest eigenvalue μT is new and interesting.
Furthermore, the eigenspace corresponding to any eigenvalue in 1.5 is one-dimensional; see 14 for more extensive discussion of these topics. For every e1 : T → R1 , the corresponding nonhomogeneous problem
Δ2 yt − 1 μyt e1 t,
t ∈ T,
y0 yT 1 0
1.6
has a unique solution if μ ∈
/ {μ1 , . . . , μT }. The purpose of this paper is to provide some nonresonance conditions which guarantee the existence and uniqueness of solutions of 1.1. Especially, we allow that the nonlinearity may be superlinear at ∞. The main tool in this paper is a
mini-max theorem due to Lazer 1.
2. Statement of the main result
In this section, we state our main result. First, we need to introduce some notations and preliminary results.
Let ·, ·n be the usual scalar product in Rn . Let LS Rn be the set of all symmetric n × n
real matrices. For A, B ∈ LS Rn , we say that A B if
B − Aξ, ξ
n
≥ 0 ∀ ξ ∈ Rn .
2.1
A
Lemma 2.1. Let A, B ∈ LS Rn be two commutative matrices with A B. Let λA
1 < λ2 < · · · <
A
A
λn be the eigenvalues of A, and let wk be the eigenvector corresponding to λk . Then, there exists γk
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3
such that
Bwk γk wk ,
k 1, . . . , n.
2.2
Proof. Since A and B are commutative, we have that
A
A Bwk B Awk B λA
k wk λk Bwk .
2.3
Then, there exists γk such that Bwk γk wk .
Remark 2.2. It is worth remarking that the conditions of Lemma 2.1 cannot guarantee that
γ 1 ≤ γ2 ≤ · · · ≤ γ n .
2.4
Therefore, 4, Assumption H2.2 is not suitable.
A
A
Lemma 2.3. Let A, B ∈ LS Rn be two commutative matrices with A B. Let λA
1 < λ2 < · · · < λn
and γ1 , γ2 , . . . , γn be the eigenvalues of A and B, respectively. Let wk be the eigenvector corresponding
to both λA
and γk . Then,
k
λA
k ≤ γk ,
k 1, . . . , n.
2.5
Proof. From the fact that
Awk λA
k wk ,
Bwk γk wk ,
2.6
we have
B − Awk γk − λA
k wk .
2.7
A
0 ≤ B − Awk , wk n γk − λA
wk , wk n .
k wk , wk n γk − λk
2.8
Subsequently,
This implies that γk ≥ λA
for k 1, . . . , n.
k
Definition 2.4. One says that f : T × Rn → Rn , ft, ξ f1 t, ξ1 , . . . , ξn , . . . , fn t, ξ1 , . . . , ξn , is a
potential vector-valued function for t ∈ T if there exists a function G : T × Rn → R1 such that
∂ G t, ξ1 , . . . , ξn ,
fi t, ξ1 , . . . , ξn ∂ξi
for all t, ξ1 , . . . , ξn ∈ T × Rn .
Let one suppose that
H1 ft, · ∈ C1 Rn for t ∈ T;
H2 ft, ξ is a potential function for t ∈ T.
i 1, . . . , n,
2.9
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Denote
⎛
Hft,ξ
f11 t, ξ1 , . . . , ξn ⎜
⎜
..
⎜
.
⎝
⎞
· · · f1n t, ξ1 , . . . , ξn ⎟
⎟
..
..
⎟,
.
.
⎠
2.10
fn1 t, ξ1 , . . . , ξn · · · fnn t, ξ1 , . . . , ξn where ξ ξ1 , . . . , ξn , fij t, ξ1 , . . . , ξn ∂/∂ξj fi t, ξ1 , . . . , ξn .
The following theorem is our main result.
Theorem 2.5. Let (H1) and (H2) hold. Assume that
H3 there exist two diagonal matrices A and B:
⎛ A
⎞
λ1 0 · · · 0
⎜
⎟
⎜ 0 λA · · · 0 ⎟
⎜
⎟
2
⎟,
A⎜
⎜ .. ..
⎟
.
.
⎜ . .
⎟
.
⎝
⎠
A
0 0 · · · λn
⎛
γ1 0 · · · 0
⎜
⎜ 0 γ2 · · ·
⎜
B⎜
⎜ .. ..
⎜. .
⎝
0 0 ···
⎞
⎟
0⎟
⎟
⎟,
... ⎟
⎟
⎠
γn
2.11
A
A
where λA
1 < λ2 < · · · < λn such that one of the following conditions holds:
n
A
a A Hft,ξ B, λA
1 , max{γk | k 1, . . . , n} ⊂ μ1 , μT , and ∪k1 λk , γk ∩
{μ1 , . . . , μT } ∅;
b Hft,ξ A and λA
n < μ1 ;
c B Hft,ξ and min{γk | k 1, . . . , n} > μT .
Then, the boundary value problem 1.1 has exact one solution for every e : T → Rn .
, γk to replace the interval λ1k , λ2k Remark 2.6. In a in H3, we use the revised interval λA
k
which was used in 4, Assumption H2.2.
Remark 2.7. c in H3 allows that the nonlinearity f may be superlinear at ∞ and −∞.
3. The main tools
Lemma 3.1 see 1. Let X and Y be two closed subspaces of a real Hilbert space H such that X is
finite-dimensional and H X ⊕ Y . Let f : H → R be a functional and let ∇f and D2 f denote the
gradient and Hessian of f, respectively. Suppose that there exist two positive constants m1 and m2 such
that
2
D fuh, h ≤ −m1 h2 ,
3.1
2
D fuk, k ≥ m2 k2
for all u ∈ H, h ∈ X, k ∈ Y . Then, f has a unique critical point. Moreover, this critical point of f is
characterized by the equality
fv max minfx y.
x∈X
y∈Y
3.2
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5
In order to introduce the other tool, we give firstly some notations. Let E denote real
Banach space with norm · E . If E∗ is the topological dual of E, then the symbol ·, · will
denote the duality pair between E and E∗ .
Let {un } be a sequence in E. We say that un converges weakly to u, written as un u, if
ψ, un − u → 0 as n → ∞ for all ψ ∈ E∗ .
Let g : E → R be a functional. We say that g is weakly continuous if for every {un } ⊂ E
with un u, we have that
gu lim g un .
3.3
n→∞
We say that g is weakly lower semicontinuous if {un } ⊂ E and un u imply that
gu ≤ lim infg un .
3.4
We say that g is weakly upper semicontinuous if {un } ⊂ E and un u imply that
gu ≥ lim supg un .
3.5
n→∞
n→∞
Lemma 3.2 see 15, Theorem 1.7, page 417. Let E be a real reflexive Banach space. Let f : E → R
be weakly upper semicontinuous (resp., weakly lower semicontinuous) and let it satisfy
lim fx ∞
resp., lim fx −∞ .
3.6
x→∞
x→∞
Then, there exists x0 ∈ E such that
f x0 minfx
resp., f x0 maxfx .
x∈E
x∈E
3.7
4. Preliminary lemmas
In this section, we give and prove some preliminary lemmas which are necessary for the proof
of the main result, Theorem 2.5.
Let
u | u:T
−→ R, u0 uT 1 0 ,
D
4.1
D u | u : T −→ R .
Let
i 1, . . . , n ,
y:T
−→ Rn | y y1 , . . . , yn T , yi ∈ D,
H
T
H z : T −→ Rn | z z1 , . . . , zn , zi ∈ D, i 1, . . . , n .
4.2
with
For u
, v ∈ H
⎞
u
1 t
⎟
⎜
⎟
⎜
u
t ⎜ ... ⎟ ,
⎠
⎝
u
n t
⎛
⎛
⎞
v1 t
⎜
⎟
⎜
⎟
v t ⎜ ... ⎟ ,
⎝
⎠
vn t
,
t∈T
4.3
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Advances in Difference Equations
let us define the inner product
u
, v
T 1
u
t, v t
t0
n
n
T u
j t
vj t.
4.4
t1 j1
Similarly for u, v ∈ H with
⎞
u1 t
⎟
⎜
⎟
⎜
ut ⎜ ... ⎟ ,
⎠
⎝
un t
⎛
⎛
⎞
v1 t
⎜
⎟
⎜
⎟
vt ⎜ ... ⎟ ,
⎝
⎠
vn t
t ∈ T,
4.5
we also define the inner product
n
T
T u, v u
j t
vj t.
ut, vt n t1
4.6
t1 j1
· and H, · are Hilbert spaces.
Then, both H,
Then,
Lemma 4.1. Let u, w ∈ D.
T
wkΔ2 uk − 1 −
k1
T
ΔukΔwk.
4.7
k0
Proof. Using “summation by parts”see 14, Theorem 2.8, we have
T
k1
T
T 1 wkΔ2 uk − 1 wkΔuk − 1 1 − ΔwkΔuk
k1
−w1Δu0 −
T
ΔwkΔuk
k1
−Δw0Δu0 −
T
4.8
ΔwkΔuk
k1
−
T
ΔukΔwk.
k0
→ R by
Define a functional J : H
Ju T
T
T
1
Δut, Δut n − G t, ut ut, et n ,
2 t0
t1
t1
4.9
where G satisfies
T
grad Gt, ξ ft, ξ f1 t, ξ1 , . . . , ξn , . . . , fn t, ξ1 , . . . , ξn
.
4.10
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→ Rn is weakly semicontinuous and J ∈ C2 .
Lemma 4.2. Let (H1) and (H2) hold. Then, J : H
Proof. The proof is standard; so we omit it.
is a critical point of J if and only if u is a solution of 1.1.
Lemma 4.3. u ∈ H
Proof. It is an immediate consequence of Lemma 4.1 and the definition of the Gâteaux-differentiation.
From now on we assume that the eigenfunction ϕk corresponding to the eigenvalue μk
satisfies
T
ϕk tϕk t 1.
4.11
t1
The following result is a special case of 14, Theorem 7.2.
Lemma 4.4. Tt1 ϕk tϕj t 0 for k, j ∈ T with k / j.
5. Proof of the main result
Now, we give the proof of Theorem 2.5. We divide the proof into three cases.
n
A
Case 1 λA
1 , max{γk | k 1, . . . , n} ⊂ μ1 , μT and ∪k1 λk , γk ∩ {μ1 , . . . , μT } ∅. For k ∈
{1, . . . , n}, we define two sets Zk and Yk as
⎧
⎪
, γk ⊂ − ∞, μ1 ,
{0}
as λA
⎪
k
⎪
⎨
Zk span ϕ1 , . . . , ϕmk
, γk ⊂ μmk , μmk 1 ,
as λA
k
⎪
⎪
⎪
⎩
, γk ⊂ μ T , ∞ ,
span ϕ1 , . . . , ϕT
as λA
k
⎧
⎪
as λA
, γk ⊂ −∞, μ1 ,
span {ϕ1 , . . . , ϕT }
⎪
k
⎪
⎨
A
Yk span {ϕmk 1 , . . . , ϕT } as λk , γk ⊂ μmk , μmk 1 ,
⎪
⎪
⎪
⎩
, γk ⊂ μT , ∞.
{0}
as λA
k
5.1
⎞
c11 ϕ1 · · · c1T ϕT
⎟
⎜
⎟
⎜
...
ut ⎜
⎟,
⎠
⎝
cn1 ϕ1 · · · cnT ϕT
5.2
with
For u ∈ H
⎛
→ Z1 × · · · × Zn and Q : H
→ Y1 × · · · × Yn by
we define the orthogonal projectors P : H
⎛
⎞
c11 ϕ1 · · · c1m1 ϕm1
⎜
⎟
⎜
⎟
..
Pu ⎜
⎟,
.
⎝
⎠
cn1 ϕ1 · · · cnmn ϕmn
⎛
⎞
c1m1 1 ϕm1 1 · · · c1T ϕT
⎜
⎟
⎜
⎟
..
Qu ⎜
⎟.
.
⎝
⎠
cnmn 1 ϕmn 1 · · · cnT ϕT
5.3
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Advances in Difference Equations
Let
| x Pu ,
X x∈H
| y Qu .
Y y∈H
5.4
By a in H3,
X ⊕ Y,
H
X ⊥ Y.
5.5
→ R which is defined in 4.9:
Let us consider the functional J : H
Ju T
T
T
1
Δut, Δut n − G t, ut ut, et n .
2 t0
t1
t1
5.6
It is easy to check that for h, k ∈ H,
T
T
T
∇Jut, ht − Δ2 ut − 1, ht n −
ft, u, ht n et, ht n ,
t1
t1
t1
T
2
D futkt, ht − Δ2 ht − 1 − Hft,ut ht, kt n .
5.7
t1
and x ∈ X with
Now, from a in H3 and Lemma 4.4, for u ∈ H
⎛
⎞
c11 ϕ1 · · · c1T ϕT
⎜
⎟
⎜
⎟
..
ut ⎜
⎟,
.
⎝
⎠
cn1 ϕ1 · · · cnT ϕT
⎛
⎞
c11 ϕ1 · · · c1m1 ϕm1
⎜
⎟
⎜
⎟
..
x⎜
⎟,
.
⎝
⎠
cn1 ϕ1 · · · cnmn ϕmn
5.8
we have that
⎛ m1
⎞ ⎛ m1
⎞
⎛ A
⎞ A
λ1 0 · · · 0
c1j ϕj ⎟
⎜ c1j ϕj ⎟ ⎜ λ1
⎜
⎟⎜
⎟ ⎜ j1
⎟
⎜ 0 λA · · · 0 ⎟ ⎜ j1
⎟ ⎜
⎟
⎜
⎜
⎟⎜
⎟
⎟
2
.
.
..
..
⎟⎜
⎟⎜
⎟,
Axt ⎜
⎜
⎜
⎜ .. ..
⎟
⎟
⎟
.
.. ⎟ ⎜
⎜ . .
⎟ ⎜ m
⎟
m
⎟ ⎜ n
⎟
⎝
⎠ ⎜n
⎝
⎝
⎠
A
λn cnj ϕj ⎠
cnj ϕj
0 0 · · · λA
n
j1
T
t1
mk
T n 2 A
ckj
λk .
Axt, xt n t1 k1 j1
j1
5.9
Ruyun Ma et al.
Thus,
9
2 D J ut xt, xt
T
− Δ2 xt − 1 − Hft,ut xt, xt
t1
T
− Δ2 xt − 1, xt
t1
≤
T
− Δ2 xt − 1, xt
t1
−
n
−
n
T
n
Hft,ut xt, xt
t1
T
Axt, xt
t1
n
n
⎞ ⎛ m1
⎞
⎛ m1
⎜ c1j μj ϕj t ⎟ ⎜ c1j ϕj t ⎟
⎟ ⎜ j1
⎟
⎜
⎟ ⎜
⎟
⎜ j1
T
T
⎜
⎟
⎟
⎜
.
.
..
..
⎟,⎜
⎟
⎜
≤
−
Axt, xt n
⎟ ⎜
⎟
⎜
⎟ ⎜m
⎟ n t1
⎜m
t1
⎟ ⎜n
⎟
⎜n
⎝ c μ ϕ t⎠ ⎝ c ϕ t⎠
nj j j
nj j
j1
5.10
j1
mk
mk
n n T T 2
2 A
ckj
μj −
ckj
λk
t1 k1 j1
≤ −δ1
mk
n T t1 k1 j1
t1 k1 j1
2
ckj
−δ1 xt, xt ,
and y ∈ Y , it follows from a in
− μk | k 1, . . . , T }. Similarly, for u ∈ H
where δ1 min{λA
k
H3 and Lemma 4.4 that
2
D Juy, y ≥ δ2 y, y ,
5.11
where δ2 min{μk1 − γk | k 1, . . . , T }. Now, applying Lemma 3.1, J has a unique critical
such that
point v ∈ H
Jv max minJx y.
x∈X
y∈Y
Case 2 Hft,ξ A and λA
n < μ1 . In this case, it is easy to verify that
2
h, h .
D Juh, h ≥ μ1 − λA
n
5.12
5.13
such that
Applying Lemma 3.2, we obtain that J has a unique critical point v ∈ H
Jv minJh.
h∈H
Case 3 B Hft,ξ and min{γk | k 1, . . . , n} > μT . In this case, we have that
2
D Juh, h ≤ − min γk | k 1, . . . , n − μT h, h .
5.14
5.15
such that
Applying Lemma 3.2, J has a unique critical point v ∈ H
Jv maxJh.
h∈H
This completes the proof of Theorem 2.5.
5.16
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Advances in Difference Equations
6. An example
Example 6.1. Let us consider the following boundary value problem of second-order difference
system:
Δ2 ut − 1 f t, ut et,
t ∈ {1, 2, 3},
u0 u4 0,
6.1
where u u1 , u2 T , ft, u u1 , 17/12u2 . Clearly, the conditions H1 and H2 hold, and
⎛
Hft,ξ ⎝
1 0
0
17
12
⎞
⎠.
6.2
Since the linear eigenvalue problem of difference equation
Δ2 yt − 1 μyt 0,
t ∈ {1, 2, 3},
y0 y4 0
6.3
has exactly 3 eigenvalues
2−
√
2,
2,
2
√
2,
6.4
now choose
⎛
A⎝
1 0
0
4
3
⎞
⎠,
⎛
⎞
5
0
⎜
⎟
B ⎝3 ⎠ .
3
0
2
6.5
Then, it is easy to show that the condition a in H3 holds. According to Theorem 2.5, the
boundary value problem 6.1 has a unique solution for every e : {1, 2, 3} → R2 .
Remark 6.2. Note that γ1 > γ2 in B; this case cannot be handled in 4.
Acknowledgments
The work is supported by the NSFC no. 10671158, the NSF of Gansu Province no. 3ZS051A25-016, NWNU-KJCXGC-03-17, the Spring-Sun Program no. Z2004-1-62033, SRFDP no.
20060736001, and the SRF for ROCS, SEM 2006 311.
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