Volume 2009, Article ID 287834, 12 pagesdoi:10.1155/2009/287834 Research Article The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Jiang Zhengxian
Trang 1Volume 2009, Article ID 287834, 12 pages
doi:10.1155/2009/287834
Research Article
The Solution of Two-Point Boundary Value
Problem of a Class of Duffing-Type Systems with
Jiang Zhengxian and Huang Wenhua
School of Sciences, Jiangnan University, 1800 Lihu Dadao, Wuxi Jiangsu 214122, China
Correspondence should be addressed to Huang Wenhua,hpjiangyue@163.com
Received 14 June 2009; Accepted 10 August 2009
Recommended by Veli Shakhmurov
This paper deals with a two-point boundary value problem of a class of Duffing-type systems with
non-C1perturbation term Several existence and uniqueness theorems were presented
Copyrightq 2009 J Zhengxian and H Wenhua 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
Minimax theorems are one of powerful tools for investigation on the solution of differential equations and differential systems The investigation on the solution of differential equations and differential systems with non-C1 perturbation term using minimax theorems came into being in the paper of Stepan A.Tersian in 19861 Tersian proved that the equation Lut
f t, ut L −d2/dt2 exists exactly one generalized solution under the operators B j j
1, 2 related to the perturbation term ft, ut being selfadjoint and commuting with the operator L −d2/dt2 and some other conditions in 1 Huang Wenhua extended Tersian’s theorems in1 in 2005 and 2006, respectively, and studied the existence and uniqueness of solutions of some differential equations and differential systems with non-C1 perturbation term 2 4, the conditions attached to the non-C1 perturbation term are that the operator
B u related to the term is self-adjoint and commutes with the operator A where A is a selfadjoint operator in the equation Au ft, u Recently, by further research, we observe that the conditions imposed upon Bu can be weakened, the self-adjointness of Bu can be removed and Bu is not necessarily commuting with the operator A.
In this note, we consider a two-point boundary value problem of a class of
Duffing-type systems with non-C1perturbation term and present a result as the operator Bu related
to the perturbation term is not necessarily a selfadjoint and commuting with the operator
L We obtain several valuable results in the present paper under the weaker conditions than
those in2 4
Trang 22 Boundary Value Problems
2 Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , respectively, let X and
Y be two orthogonal closed subspaces of H such that H X ⊕ Y Let P : H → X, Q : H → Y denote the projections from H to X and from H to Y , respectively The following theorem
will be employed to prove our main theorem
with Gˆateaux derivative ∇f : H → H everywhere defined and hemicontinuous Suppose that
there exist two closed subspaces X and Y such that H X ⊕ Yand two nonincreasing functions α :
0, ∞ → 0, ∞, β : 0, ∞ → 0, ∞ satisfying
and
∇fh1 y− ∇fh2 y, h1− h2
≤ −αh1− h2h1− h22, 2.2
for all h1, h2∈ X, y ∈ Y, and
∇fx k1 − ∇fx k2, k1− k2
≥ βk1− k2k1− k22, 2.3
for all x ∈ X, k1, k2∈ Y Then
a f has a unique critical point v0 ∈ H such that ∇fv0 0;
b fv0 maxx ∈X miny ∈Y f x y min y ∈Y maxx ∈X f x y.
We also need the following lemma in the present work To the best of our knowledge, the lemma seems to be new
λ2 ≤ · · · ≤ λ n be the eigenvalues of A and B, respectively, where each eigenvalue is repeated according
to its multiplicity If A commutes with B, that is, AB BA, then A B is a diagonalization matrix
and μ1 λ1≤ μ2 λ2 ≤ · · · ≤ μ n λ n are the eigenvalues of A B.
Proof Since A is a diagonalization n × n matrix, there exists an inverse matrix P such that
eigenvalues of A, Ei i 1, 2, , s are the r i × r i r1 r2 · · · r s n identity matrices And
since AB BA, that is,
μ1E1, μ2E2, , μ sEs
P−1, 2.4
we have
diag
μ1E1, μ2E2, , μ sEs
μ1E1, μ2E2, , μ sEs
Trang 3
Denote P−1BP Cij, where Cij are the submatrices such that EiCijand CijEi i 1, 2, , s
are defined, then, by2.5,
μ iCij μ jCij
Noticed that μ i / μ j i / j, we have C ij O i / j, and hence
where Cii and Ei i 1, 2, , s are the same order square matrices Since B is a
diagonalization n × n matrix, there exists an invertible matrix Q diag Q1, Q2, , Q s such that
1 , Q−12 , , Q−1s
· diag C11, C22, , C ss · diag Q1, Q2, , Q s
diagQ−1
1 C11Q1, Q−12 C22Q2, , Q−1s CssQs
diag λ1, λ2, , λ n ,
2.8
where λ1≤ λ2≤ · · · ≤ λ nare the eigenvalues of B.
Let R PQ, then R is an invertible matrix such that R−1BR diag λ1, λ2, , λ n and
R−1A BR R−1AR R−1BR Q−1
diagQ−1
1 , Q−12 , , Q−1s
· diagμ1E1, μ2E2, , μ sEs
· diag Q1, Q2, , Q s
diag λ1, λ2, , λ n
diagμ1E1, μ2E2, , μ sEs
diag λ1, λ2, , λ n
diagμ1, μ2, , μ n
diag λ1, λ2, , λ n
diagμ1 λ1, μ2 λ2, , μ n λ n
.
2.9
A B.
The proof ofLemma 2.2is fulfilled
Let·, · denote the usual inner product on R n and denote the corresponding norm
by |u| {n
i1u2
i}1/2
, where u u1, u2, , u nT Let ·, · denote the inner product on
L20, π, R n It is known very well that L20, π, R n is a Hilbert space with inner product
u, v
π 0
ut, vtdt, u, v ∈ L20, π, R n 2.10
and normu u, u π
0ut, utdt 1/2
, respectively
Trang 44 Boundary Value Problems Now, we consider the boundary value problem
⎧
⎨
⎩
u Au gt, u ht, t ∈ 0, π,
where u :0, π → R n , A is a real constant diagonalization n× n matrix with real eigenvalues
μ1 ≤ μ2 ≤ · · · ≤ μ n each eigenvalue is repeated according to its multiplicity, g : 0, π ×
Rn → Rnis a potential Carath´eodory vector-valued function , h :0, π → R nis continuous,
Let ut vt ωt, ωt 1 − t/πa t/πb, t ∈ 0, π , then 2.11 may be
written in the form
⎧
⎨
⎩
v Av g∗t, v h∗t,
where g∗t, v gt, v ω, h∗t ht − Aωt Clearly, g∗t, v is a potential Carath´eodory
vector-valued function, h∗:0, π → R n Clearly, if v0is a solution of2.12, u0 v0 ω will
be a solution of2.11
Assume that there exists a real bounded diagonalization n × n matrix Bt, u t ∈
0, π, u ∈ R n such that for a.e t ∈ 0, π and ξ, η ∈ L20, π, R n
where τ diagτ1 , τ2, , τ n , τ i ∈ 0, 1 i 1, 2, , n, Bt, u commutes with A and is
possessed of real eigenvalues λ1t, u ≤ λ2t, u ≤ · · · ≤ λ n t, u In the light ofLemma 2.2,
· · · ≤ μ n λ n t, u each eigenvalue is repeated according to its multiplicity Assume that
there exist positive integers N i i 1, 2, , n such that for u ∈ L20, π, R n
N i2− μ i < λ i t, u < N i 12− μ i i 1, 2, , n. 2.14
Letξ i i 1, 2, , n be n linearly independent eigenvectors associated with the eigenvalues
μ i λ i t, u i 1, 2, , n and let γ i i 1, 2, , n be the orthonormal vectors obtained by
orthonormalizing to the eigenvectorsξ i i 1, 2, , n of μ i λ i t, u i 1, 2, , n Then
for every u∈ Rn
A Bt, uγ iμ i λ i t, uγ i i 1, 2, , n. 2.15 And let the set{γ1, γ2, , γ n} be a basis for the space Rn, then for every u∈ Rn,
Trang 5It is well known that each v ∈ L20, π, R n can be represented by the absolutely convergent Fourier series
v
2
π
n
i1
∞
k1
C ki sin ktγ i , C ki
2
π
π 0
v i t sin ktdt i 1, 2, , n; k 1, 2, 2.17
Define the linear operator L −d2/dt2 : DL ⊂ L20, π, R n → L20, π, R n ,
DL
⎧
⎨
2
π
n
i1
∞
k1
C ki sin ktγi,
C ki
2
π
π 0
v i t sin ktdt, i 1, 2, , n,n
i1
∞
k1
C ki2k4< ∞
⎫
⎬
⎭,
Lv
2
π
n
i1
∞
k1
k2C ki sin ktγi, σ L n2| n ∈ N.
2.18
Clearly, L −d2/dt2 is a selfadjoint operator and DL is a Hilbert space for the inner
product
u, v
π 0
u t, v t ut, vtdt, u, v ∈ DL, 2.19 and the norm induced by the inner product is
v2
π 0
v t, v t vt, vtdt, v ∈ DL. 2.20 Define
X
⎧
⎨
2
π
n
i1
N i
k1
C ki sin ktγ i , t ∈ 0, π,
C ki
2
π
π 0
x i t sin ktdt
⎫
⎬
⎭,
2.21
Y
⎧
⎨
2
π
n
i1
∞
k N i 1
C ki sin ktγ i , t ∈ 0, π,
C ki
2
π
π 0
y i t sin ktdt,n
i1
∞
k N i 1
C2ki k4 < ∞
⎫
⎬
⎭.
2.22
Clearly, X and Y are orthogonal closed subspaces of DL and DL X ⊕ Y.
Trang 66 Boundary Value Problems
Define two projective mappings P : DL → X and Q : DL → Y by Pv x ∈ X and
Qv y ∈ Y, v x y ∈ DL, then S P − Q is a selfadjoint operator.
Using the Riesz representation theorem , we can define a mapping T : L20, π, R n →
L20, π, R n by
Tu, v
π
0
− Au, v − gt, u, v ht, vdt, ∀v ∈ L20, π, R n . 2.23
We observe that T in2.23 is defined implicity Let Tu ∇Fu in 2.23, we have
∇Fu, v
π
0
− Au, v − gt, u, v ht, vdt, ∀v ∈ DL ⊂ L20, π, R n .
2.24
Clearly,∇F and hence F is defined implicity by 2.24 It can be proved that u is a solution of
2.11 if and only if u satisfies the operator equation
3 The Main Theorems
Now, we state and prove the following theorem concerning the solution of problem2.11
Theorem 3.1 Assume that there exists a real diagonalization n × n matrix Bt, u u ∈
L20, π, R n with real eigenvalues λ1t, u ≤ λ2t, u ≤ · · · ≤ λ n t, u satisfying 2.14 and
commuting with A Denote
αu min
u≤umin 1≤i≤n min 0≤t≤π
λ i t, u μ i − N2
i > 0
βu min
u≤umin 1≤i≤n min 0≤t≤π
N i 12− μ i − λ i t, u > 0. 3.2
If
α : 0, ∞ −→ 0, ∞, β : 0, ∞ −→ 0, ∞,
problem2.11 has a unique solution u0, and u0satisfies ∇Fu0 0, and
Fu0 max
x∈X min
y∈Y F x y ω min
y∈Y max
where F is a functional defined in2.24 and ω 1 − t/πa t/πb, t ∈ 0, π
Trang 7Proof First, by virtue of2.21 and 2.22, we have
π
0
dt
π 0
−x , x
dt
≤ π 0
⎛
⎝
2
π
n
i1
N i2
N i
k1
C ki sin ktγ i ,
2
π
n
i1
N i
k1
C ki sin ktγ i
⎞
⎠dt
≤
max 1≤i≤nN i
2 π 0
x, xdt,
3.5
π
0
dt
π 0
−y , y
dt
π 0
⎛
⎝
2
π
n
i1
∞
k N i 1
k2C ki sin ktγ i ,
2
π
n
i1
∞
k N i 1
C ki sin ktγ i
⎞
⎠dt,
3.6
1
max1≤i≤nN i 12
π 0
dt
π 0
⎛
⎝
2
π
n
i1
∞
k N i 1
k2 max1≤i≤nN i 12C ki sin ktγ i , y
⎞
⎠dt
≥
π 0
y, ydt.
3.7
Denote∇Fu ∇Fv ω ∇F∗v.
By2.24, 2.13, 3.5, 3.6, 3.7, 3.1, and 3.2, for all x1, x2 ∈ X, y ∈ Y, let v1
x 1 y ∈ DL, v2 x 2 y ∈ DL, v v1− v2 x1− x2 x ∈ X, x1 Pv1 ∈ X, x2 Pv2∈ X,
∇F∗v1 − ∇F∗v2, x1− x2 ∇Fu1 − ∇Fu2, x1− x2 ∇Fu1, x − ∇Fu2, x
π 0
u 1, x
− Au1, x − gt, u1, x ht, x dt
− π 0
u 2, x
− Au2, x − gt, u2, x ht, x dt
π 0
u1− u2 , x
− Au1− u2, x − gt, u1 − gt, u2, xdt
π 0
−v , x
− Av, x − Bt, v2 ω τvv, xdt
π 0
−x , x
− Ax, x − Bt, v2 ω τvx, xdt
Trang 88 Boundary Value Problems
≤ π 0
⎡
⎣
⎛
⎝n
i1
N i2·
2
π
N i
k1
C ki sin ktγ i , x
⎞
⎠
−
⎛
⎝n
i1
2
π
N i
k1
C ki sin ktA Bt, vγ i , x
⎞
⎠
⎤
⎦dt
≤
π 0
⎛
⎝n
i1
N i2− μ i − λ i t, v
2
π
N i
k1
C ki sin ktγ i , x
⎞
⎠dt
≤ −αv
π 0
x, xdt
−αv1− v2 1
max1≤i≤nN i2 1
×
π 0
max1≤i≤nN i2
x, x x, x dt
≤ −α∗v1− v2x1− x22,
%
α∗v1− v2 αv1− v2
max1≤i≤nN i2 1
&
,
3.8
for all x∈ X, y1, y2∈ Y, let v1 x y 1∈ DL, v2 x y 2∈ DL, v v1− v2 y1− y2 y ∈ Y,
y1 Qv1∈ Y, y2 Qv2∈ Y, x Pv1 Pv2∈ X, we have
∇F∗v1 − ∇F∗v2, y1− y2
∇Fu1 − ∇Fu2, y1− y2
π
0
u1− u2 , y
− Au1− u2, y − gt, u1 − gt, u2, ydt
π
0
− Av, y − Bt, vv, ydt
π
0
− A Bt, vy, ydt
≥
π
0
⎡
⎣y , y
−
⎛
⎝
2
π
n
i1
∞
k N i 1
k2 max1≤i≤nN i 12C ki sin ktμ i λ i t, vγ i , y
⎞
⎠
⎤
⎦dt
π
0
⎡
⎢
⎣−y , y
−
⎛
max 1≤i≤nN i 12
2
π
n
i1
∞
k N i 1
k2C ki sin ktμ i λ i t, vγ i , y
⎞
⎟
⎤
⎥
⎦dt
Trang 9π
0
⎡
⎣
⎛
⎝
2
π
n
i1
∞
k N i 1
k2C ki sin ktγ i , y
⎞
⎠
−
⎛
max1≤i≤nN i 12
2
π
n
i1
∞
k N i 1
k2C ki sin ktμ i λ i t, vγ i , y
⎞
⎠
⎤
⎦dt
max1≤i≤nN i 12
π 0
⎛
⎝
2
π
n
i1
∞
k N i 1
k2C ki sin ktN i 12−μ i λ i t, vγ i , y
⎞
⎠dt
≥ minv≤vmin1≤i≤nmint ∈0,π
N i 12− μ i − λ i t, v > 0
max1≤i≤nN i 12 1 ·
%
max1≤i≤nN i 12
& π 0
dt
max1≤i≤nN i 12 1
π 0
y, ydt
β∗vy2 β∗v1− v2y1− y22,
%
β∗v1− v2 βv1− v2
max1≤i≤nN i 12 1
&
.
3.9
By 3.3, s · α∗s → ∞, s · β∗s → ∞, as s → ∞ Clearly, α∗ and β∗
are nonincreasing Now, all the conditions in the Theorem 2.1 are satisfied By virtue of Theorem 2.1, there exists a unique v0 ∈ DL such that ∇F∗v0 ∇Fv0 ω ∇Fu0 0
and F∗v0 Fv0 ω Fu0 maxx∈Xminy∈Y F x y ω min y∈Y maxx∈X F x y ω,
where F is a functional defined implicity in2.24 and ωt 1 − t/πa t/πb, t ∈ 0, π
v0t is just a unique solution of 2.12 and u0t v0t ωt is exactly a unique solution of
2.11 The proof ofTheorem 3.1is completed
Now, we assume that there exists a positive integer N such that
N2− μ i < λ i t, u < N 12− μ i i 1, 2, , n 3.10
for u∈ L20, π, R n , t ∈ 0, π Define
X
⎧
⎨
2
π
n
i1
N
k1
C ki sin ktγ i , t ∈ 0, π,
C ki
2
π
π 0
x i t sin ktdt
⎫
⎬
⎭,
3.11
Trang 1010 Boundary Value Problems
Y
⎧
⎨
2
π
n
i1
∞
k N 1
C ki sin ktγ i , t ∈ 0, π,
C ki
2
π
π 0
y i t sin ktdt,n
i1
∞
k N 1
C2
ki k4< ∞
⎫
⎬
⎭,
3.12
αu min
u≤umin 1≤i≤nmin 0≤t≤π
λ i t, u μ i − N2 > 0
βu min
u≤umin 1≤i≤nmin 0≤t≤π
N 12− μ i − λ i t, u > 0. 3.14
Replace the condition 2.14 by 3.10 and replace 2.21, 2.22, 3.1, and 3.2 by
3.11, 3.12, 3.11, and 3.14, respectively Using the similar proving techniques in the Theorem 3.1, we can prove the following theorem
Theorem 3.2 Assume that there exists a real diagonalization n × n matrix Bt, u t ∈ 0, π, u ∈
Rn with real eigenvalues λ1t, u ≤ λ2t, u ≤ · · · ≤ λ n t, u satisfying 2.13 and 3.10 and
commuting with A If the functions α and β defined in (3.11) and3.14 satisfy 3.3, problem 2.11
has a unique solution u0, and u0satisfies ∇Fu0 0 and 3.4.
It is also of interest to the case of A O
diagonalization n × n matrix Bt, u t ∈ 0, π, u ∈ R n with real eigenvalues λ1t, u ≤ λ2t, u ≤
· · · ≤ λ n t, u satisfying 2.13 and N2
i < λ i t, u < N i 12 N i∈ Z , i 1, 2, , n Denote
αu min
u≤umin 1≤i≤nmin 0≤t≤π
λ i t, u − N2
i > 0
,
βu min
u≤umin 1≤i≤nmin 0≤t≤π
N i 12− λ i t, u > 0.
3.15
If α and β satisfy3.3, the problem
⎧
⎨
⎩
u gt, u ht, t ∈ 0, π,
has a unique solution u0, and u0satisfies ∇Fu0 0 and 3.4, where F is a functional defined in
∇Fu, v
π 0
− gt, u, v ht, vdt, v ∈ DL. 3.17
...Clearly, X and Y are orthogonal closed subspaces of DL and DL X ⊕ Y.
Trang 66 Boundary Value. .. v2 ω τvx, xdt
Trang 88 Boundary Value Problems
≤...
Theorem 3.1 Assume that there exists a real diagonalization n × n matrix Bt, u u ∈
L20, π, R n with real eigenvalues λ1t,