Hindawi Publishing Corporation
Boundary Value Problems
Volume 2008, Article ID 859461, 9 pages
doi:10.1155/2008/859461
Research Article
The Existence and Uniqueness of
Solution of Duffing Equations with Non-C2
Perturbation Functional at Nonresonance
Zhou Ting and Huang Wenhua
School of Science, Jiangnan University, Wuxi Jiangsu 214122, China
Correspondence should be addressed to Huang Wenhua, hpjiangyue@163.com
Received 13 July 2007; Accepted 13 March 2008
Recommended by Martin Schechter
This paper deals with a boundary value problem for Duffing equation. The existence of unique
solution for the problem is studied by using the minimax theorem due to Huang Wenhua. The
existence and uniqueness result was presented under a generalized nonresonance condition.
Copyright q 2008 Z. Ting 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
In recent years, many authors are greatly attached to investigation for the existence and
uniqueness of solution of Duffing equations, for example, 1–11, and so forth. Some authors
8, 11, 12, etc. proved the existence and uniqueness of solution of Duffing equations under C2
perturbation functions and other conditions at nonresonance by employing minimax theorems.
In 1986, Tersian investigated the equation u ft, ut −pt using a minimax theorem
proved by himself and reaped a result of generalized solution 13. In 2005, Huang and Shen
generalized the minimax theorem of Tersian in 13. Using the generalized minimax theorem,
Huang and Shen proved a theorem of existence and uniqueness of solution for the equation
u ft, ut et 0 14 under the weaker conditions than those in 13.
Stimulated by the works in 13, 14, in the present paper, we investigate the solutions
of the boundary value problems of Duffing equations with non-C2 perturbation functions at
nonresonance using the minimax theorem proved by Huang in 15.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm · , respectively, X and Y
be two orthogonal closed subspaces of H such that H X ⊕ Y . Let Q : H → X, P : H → Y
2
Boundary Value Problems
denote the projections from H to X and from H to Y , respectively. The following theorem will
be employed to prove our main theorem.
Theorem 2.1 see 15. Let H be a real Hilbert space, let X and Y be orthogonal closed vector
subspace of H such that H X ⊕ Y , let f : H → R be an everywhere defined functional with
Gâteaux derivative, ∇f : H → H everywhere defined and hemicontinuous. Suppose that there exist
two continuous functions α : 0, ∞ → 0, ∞, β : 0, ∞ → 0, ∞ satisfying
αs −→ ∞, βs −→ ∞, as s −→ ∞,
∇fu − ∇fv, x1 − x2 ≤ −α u − v x1 − x2 ,
∇fu − ∇fv, y1 − y2 ≥ β u − v y1 − y2 ,
2.1
for u ∈ H, v ∈ H, x1 Qu ∈ X, x2 Qv ∈ X, y1 P u ∈ Y , y2 P v ∈ Y . Then, the following hold:
a f has a unique critical point v0 ∈ H such that ∇fv0 0;
b fv0 maxx∈X miny∈Y fx y miny∈Y maxx∈X fx y.
It is easy to prove the following corollary of the above theorem.
Corollary 2.2. Let H be a real Hilbert space, let X and Y be orthogonal closed vector subspace of H
such that H X ⊕ Y , and let f : H → R be an everywhere defined functional with second Gâteaux
differential. Suppose that there exist two continuous functions α : 0, ∞ → 0, ∞, β : 0, ∞ →
0, ∞ satisfying
αs −→ ∞, βs −→ ∞, as s −→ ∞,
2 ∇ f v tu − v u − v, x1 − x2 ≤ −α u − v x1 − x2 ,
2 ∇ f v tu − v u − v, y1 − y2 ≥ β u − v y1 − y2 ,
2.2
for u ∈ H, v ∈ H, x1 Qu ∈ X, x2 Qv ∈ X, y1 P u ∈ Y , y2 P v ∈ Y , 0 < t < 1. Then, the
following hold:
a f has a unique critical point v0 ∈ H such that ∇fv0 0;
b fv0 maxx∈X miny∈Y fx y miny∈Y maxx∈X fx y.
Proof. We note that f is a second Gâteaux differentiable functional, the mean-value theorem
ensures that there exists θ ∈ 0, 1 such that ∇fu−∇fv ∇2 fvθu−vu−v. Therefore,
for u ∈ H, v ∈ H, x1 Qu ∈ X, x2 Qv ∈ X, y1 P u ∈ Y , y2 P v ∈ Y , we have
∇fu − ∇fv, x1 − x2 ∇2 f v θu − v u − v, x1 − x2 ≤ −α u − v x1 − x2 ,
∇fu − ∇fv, y1 − y2 ∇2 f v θu − v u − v, y1 − y2 ≥ β u − v y1 − y2 .
2.3
The conclusion of the corollary follows immediately from Theorem 2.1.
Z. Ting and H. Wenhua
3
3. The main theorems
Consider the boundary value problem
u gt, u et,
u0 a,
3.1
u2π b,
where u : 0, 2π → R, g : 0, 2π × R → R is a potential Carathéodory function, e : 0, 2π → R
is a given function in L2 0, 2π.
Let ut vt ωt, ωt a2π − t bt/2π, t ∈ 0, 2π, then 3.1 may be written
in the form of
v g ∗ t, v et,
3.2
v0 v2π 0,
where g ∗ t, v gt, v ω. Clearly, g ∗ t, v is a potential Carathéodory function, and if v0 is a
solution of 3.2, then u0 v0 ω will be a solution of 3.1.
It is well known that L2 0, 2π is a Hilbert space with inner product:
u, v 2π
utvtdt
u, v ∈ L2 0, 2π ,
3.3
0
and norm u ctions,
2π
1/2
u, u 0 u2 tdt , respectively. The system of trigonometrical fun-
1
1
1
1
1
1
1
√ ; √ cosx, √ sinx; √ cos2x, √ sin2x; . . . ; √ cosnx, √ sinnx; . . . ,
π
π
π
π
π
π
2π
3.4
is a system of orthonormal functions in L2 0, 2π. Each v ∈ L2 0, 2π can be written as the
Fourier series
vt v, √
1
2π
√
1
2π
∞
n1
cosnt
v, √
π
cosnt
√
π
sinnt
v, √
π
sinnt
.
√
π
3.5
Define the linear operator L −d2 /dt2 : D∗ L ⊂ L2 0, 2π → L2 0, 2π,
∗
D L v ∈ L 0, 2π | vt v, √
1
1
∞
sin nt sin nt
cos nt cos nt
,
v, √
√ v, √
√
π
π
π
π
√ 2π
2π n1
2 2 ∞
cos nt sin nt 2
n 1 v, √
v, √
< ∞, v0 v2π 0 ⊂ L2 0, 2π,
π
π
n1
∞
sin nt sin nt
cos nt cos nt
2
v, √
,
Lv n
v, √
√
√
π
π
π
π
n1
σL n2 | n ∈ N .
3.6
2
4
Boundary Value Problems
Denote
DL a2π − t bt
u | ut vt ωt, v ∈ D L, ωt , t ∈ 0, 2π .
2π
∗
3.7
Clearly, L −d2 /dt2 is a self-adjoint operator, and D∗ L is a Hilbert space for the inner
product:
u, v 2π
u tv t utvt dt,
3.8
0
u, v ∈ L2 0, 2π, the norm induced by this inner product is
v 2
2π
2
v t v2 t dt.
3.9
0
Note that DL is not a space.
Since gt, u in 3.1, and hence g ∗ t, v in 3.2, is a potential Carathéodory function,
there exists a function Gt, u such that
∂Gt, u
,
∂u
3.10
∂Gt, v ω
,
∂u
3.11
gt, u and hence
g ∗ t, v and the mapping g, and hence g ∗ , generates a Nemytskii operator N : DL ⊂ L2 0, 2π →
L2 0, 2π by
Nu g t, ut ,
3.12
N ∗ v Nv ω g t, vt ωt g ∗ t, vt .
3.13
and hence
Define the functional f : DL ⊂ L2 0, 2π → R by
1
fu Lu, u − Gt, u etu,
2
3.14
where G satisfies 3.10 and et is in 3.1. We have
1
f ∗ v Lv, v − G∗ t, v etv etω.
2
3.15
It is easy to see that
∇fv Lu − Nu et,
∇f ∗ v Lv − N ∗ v et,
3.16
Z. Ting and H. Wenhua
5
where N ∗ v g ∗ t, vt. Clearly, v0 ∈ D∗ L is a critical point of f if and only if v0 is a
solution of the equation L − N ∗ v −et and hence a solution of 3.2 and thus u0 v0 ω v0 a2π − t bt/2π, t ∈ 0, 2π is a solution of 3.1.
Now, we suppose that there exists a real-bounded mapping bt, u u ∈ DL such that
g t, u2 − g t, u1 b t, u1 τ u2 − u1
u2 − u1 ,
τ ∈ 0, 1, u1 , u2 ∈ DL.
3.17
For u1 , u2 ∈ DL, v1 , v2 ∈ D∗ L, let
bt, u bt, v ω b∗ t, v.
3.18
g ∗ t, v2 − g ∗ t, v1 g t, v2 ω − g t, v1 ω
b t, v1 ω τ v2 − v1
v2 − v1
b t, v1 τ v2 − v1 ω v2 − v1
b∗ t, v1 τ v2 − v1
v2 − v1 ,
3.19
Since
equation 3.17 is equivalent to
g ∗ t, v2 − g ∗ t, v1 b∗ t, v1 τ v2 − v1
v2 − v1 ,
τ ∈ 0, 1, v1 , v2 ∈ D∗ L.
3.20
Suppose that for v ∈ D∗ L,
n2 < b∗ t, v < n 12
n ∈ N,
3.21
and for v1 , v2 , v ∈ D∗ L, define
α∗ v1 − v2 min
min n 12 − max b∗ t, v > 0, min b∗ t, v − n2 > 0 ,
3.22
min
min n 12 − max bt, v > 0, min bt, v − n2 > 0 .
3.23
v≤v1 −v2 n∈N
1≤t≤2π
0≤t≤2π
which is equivalent to
α u1 − u2 u≤u1 −u2 n∈N
0≤t≤2π
0≤t≤2π
Since L −d2 /dt2 is a self-adjoint operator, it possesses spectral resolution
L
∞
−∞
λdEλ ,
λ ∈ σL,
3.24
with a right continuous spectral family {Eλ : λ ∈ R}; and we let
Eα, β β
α
dEλ
3.25
6
Boundary Value Problems
for all α, β ∈ ρL∩{±∞} with α < β. Then, the operator L−b∗ t, vI has the spectral resolution:
L − b∗ t, vI ∞
−∞
λ − b∗ t, v dEλ ,
v ∈ D∗ L,
3.26
where I is an identity operator.
Define X and Y by
X E − ∞, b∗ t, v D∗ L,
Y E b∗ t, v, ∞ D∗ L.
3.27
By 3.21, we have
E − ∞, b∗ t, v I − E b∗ t, v, ∞ ,
v ∈ D∗ L,
3.28
and hence
D∗ L X ⊕ Y,
X and Y are orthogonal.
3.29
We need to prove a lemma before presenting our main theorem.
Lemma 3.1. Suppose that g ∗ : 0, 2π × R → R in 3.2 satisfies 3.20, b∗ t, v v ∈ D∗ L
commutes with the linear operator L −d2 /dt2 and satisfies 3.21, α∗ s is a continuous function
defined in 3.22. Then, for v1 x1 y1 ∈ D∗ L, x1 ∈ X, y1 ∈ Y, v2 x2 y2 ∈ D∗ L, x2 ∈
X, y2 ∈ Y, τ ∈ 0, 1, z v1 τv2 − v1 ∈ D∗ L,
2
x1 − x2 , x1 − x2 ≤ −α∗ v1 − v2 x1 − x2 ,
2
L − b∗ t, zI y1 − y2 , y1 − y2 ≥ α∗ v1 − v2 y1 − y2 .
L − b∗ t, zI
3.30
Proof. For z v1 τv2 − v1 ∈ D∗ L, v1 ∈ D∗ L, v2 ∈ D∗ L, v ∈ D∗ L, let
x E − ∞, b∗ t, z v,
y E b∗ t, z, ∞ v.
3.31
Note that b∗ t, z z ∈ D∗ L commutes with the linear operator L and
L − b∗ t, zI
x 1 − x2 L − b t, zI y1 − y2 ∗
∞
−∞
λ − b∗ t, z dEλ ◦ E − ∞, b∗ t, z v1 − v2
b∗ t,z
−∞
∞
−∞
λ − b∗ t, z dEλ v1 − v2 ,
λ − b∗ t, z dEλ ◦ E b∗ t, z, ∞ v1 − v2
∞
b∗ t,z
λ − b∗ t, z dEλ v1 − v2 .
3.32
Z. Ting and H. Wenhua
7
By 3.22,
L − b∗ t, zI
x 1 − x 2 , x1 − x 2 b∗ t,z
−∞
2
λ − b∗ t, z dEλ v1 − v2 ≤ −α∗ v1 − v2 b∗ t,z
−∞
2
dEλ v1 − v2 2
−α v1 − v2 x1 − x2 , v1 , v2 ∈ D∗ L, x1 , x2 ∈ X,
∞
2
L − b∗ t, zI y1 − y2 , y1 − y2 λ − b∗ t, z dEλ v1 − v2 ∗
b∗ t,z
≥ α∗ v1 − v2 ∞
b∗ t,z
2
dEλ v1 − v2 2
α∗ v1 − v2 y1 − y2 ,
v1 , v2 ∈ D∗ L, y1 , y2 ∈ Y.
3.33
Now, we show our main theorem dealing with 3.1.
Theorem 3.2. Let g : 0, 2π × R → R be a potential Carathéodory function satisfying 3.17. Suppose
that bt, u u ∈ DL, t ∈ 0, 2π commutes with the linear operator L −d2 /dt2 and satisfies
n2 < bt, u < n 12 ,
n ∈ N, u ∈ DL,
3.34
and the continuous function αs defined by 3.23 satisfies the conditions
α : 0, ∞ −→ 0, ∞,
s · αs −→ ∞ as s −→ ∞.
3.35
Then, 3.1 has a unique solution u0 ∈ DL such that
∇f u0 0,
f u0 max min fx y ω min max fx y ω,
x∈X y∈Y
y∈Y
x∈X
3.36
where f is a functional defined in 3.14 and ωt a2π − t bt/2π, t ∈ 0, 2π.
Proof. For v1 x1 y1 ∈ D∗ L, v2 x2 y2 ∈ D∗ L, x1 , x2 ∈ X, y1 , y2 ∈ Y , by Lemma 3.1, we
have
∇f ∗ v2 − ∇f ∗ v1 , x2 − x1 Lv2 − N ∗ v2 et − Lv1 N ∗ v1 − et, x2 − x1
L v2 − v1 − g ∗ t, v2 t − g ∗ t, v1 t , x2 − x1
v2 − v1 , x2 − x1
L v2 − v1 − b∗ t, v1 τ v2 − v1
L − b∗ t, v1 τ v2 − v1 I v2 − v1 , x2 − x1
L − b∗ t, v1 τ v2 − v1 I x2 − x1 , x2 − x1
2
≤ −α∗ v2 − v1 x1 − x2 , τ ∈ 0, 1,
8
Boundary Value Problems
∇f ∗ v2 − ∇f ∗ v1 , y2 − y1 Lv2 − N ∗ v2 et − Lv1 N ∗ v1 − et, y2 − y1
L v2 − v1 − g ∗ t, v2 t − g ∗ t, v1 t , y2 − y1
L − b∗ t, v1 τ v2 − v1 I y2 − y1 , y2 − y1
2
≥ α∗ v2 − v1 y1 − y2 ,
τ ∈ 0, 1.
3.37
Employing Theorem 2.1, we can know that there exists a unique v0 ∈ D∗ L such that
∇f ∗ v0 0, where v0 ∈ D∗ L is a solution of 3.2 and this means that 3.1 has a unique
solution u0 v0 ω ∈ DL such that
∇f ∗ v0 ∇f v0 ω ∇f u0 0,
f u0 f v0 ω f ∗ v0 max min f ∗ x y min max f ∗ x y
x∈X y∈Y
y∈Y
x∈X
3.38
max min fx y ω min max fx y ω,
x∈X y∈Y
y∈Y
x∈X
where f is a functional defined in 3.14 and ωt a2π − t bt/2π, t ∈ 0, 2π.
If the perturbation function Gt, u in 3.10 is a second Gâteaux differential, 3.17,
3.34, and 3.23 become
g t, u2 − g t, u1 gu t, u1 τ u2 − u1
u2 − u1 ,
α u1 − u2 τ ∈ 0, 1, u1 , u2 ∈ DL,
n2 < gu t, u < n 12 , n ∈ N, u ∈ DL,
min min n 12 − max gu t, u > 0, min gu t, u − n2 > 0 ,
u≤u1 −u2 n∈N
0≤t≤2π
0≤t≤2π
3.39
3.40
3.41
respectively. By 3.30 in Lemma 3.1, we have
2
x2 − x1 , x2 − x1 ≤ −α∗ v1 − v2 x1 − x2 ,
2
L − gu∗ t, vI y2 − y1 , y2 − y1 ≥ α∗ v1 − v2 y1 − y2 ,
L − gu∗ t, vI
3.42
where v v1 τv2 − v1 ∈ D∗ L, v1 , v2 ∈ D∗ L, τ ∈ 0, 1, x1 , x2 ∈ X, y1 , y2 ∈ Y .
We then have the following corollary of Theorem 3.2.
Corollary 3.3. Let g : 0, 2π × R → R be a potential Carathéodory function with first Gâteaux derivative gu satisfying 3.39 and 3.40 and gu t, u u ∈ DL, t ∈ 0, 2π commutes with the linear
operator L −d2 /dt2 . If the continuous function αs defined by 3.41 satisfies 3.35, then 3.1 has
a unique solution u0 ∈ DL and
f u0 max min fx y ω min max fx y ω,
x∈X y∈Y
y∈Y
x∈X
where f is a functional defined in 3.14 and ωt a2π − t bt/2π, t ∈ 0, 2π.
3.43
Z. Ting and H. Wenhua
9
Acknowledgment
The corresponding author is grateful to the referees for their helpful and valuable comments.
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