Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 963105, 13 pages
doi:10.1155/2012/963105
Research Article
Existence of Three Solutions for a Nonlinear
Fractional Boundary Value Problem via a Critical
Points Theorem
Chuanzhi Bai
Department of Mathematics, Huaiyin Normal University, Jiangsu, Huaian 223300, China
Correspondence should be addressed to Chuanzhi Bai, czbai8@sohu.com
Received 13 May 2012; Revised 8 July 2012; Accepted 9 July 2012
Academic Editor: Bashir Ahmad
Copyright q 2012 Chuanzhi Bai. 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.
This paper is concerned with the existence of three solutions to a nonlinear fractional boundary
α−1 C α
α
value problem as follows: d/dt1/20 Dtα−1 C
0 Dt ut − 1/2t DT t DT ut λatfut 0, a.e. t ∈ 0, T , u0 uT 0, where α ∈ 1/2, 1, and λ is a positive real parameter. The
approach is based on a critical-points theorem established by G. Bonanno.
1. Introduction
Differential equations with fractional order have recently proved to be strong tools in the
modeling of many physical phenomena in various fields of physical, chemical, biology,
engineering, and economics. There has been significant development in fractional differential
equations, one can see the monographs 1–5 and the papers 6–20 and the references
therein.
Critical-point theory, which proved to be very useful in determining the existence of
solution for integer-order differential equation with some boundary conditions, for example,
one can refer to 21–25. But till now, there are few results on the solution to fractional
boundary value problem which were established by the critical-point theory, since it is often
very difficult to establish a suitable space and variational functional for fractional boundary
value problem. Recently, Jiao and Zhou 26 investigated the following fractional boundary
value problem:
1
d 1
−β −β u t t DT u t ∇Ft, ut 0,
0 Dt
dt 2
2
u0 uT 0
a.e. t ∈ 0, T ,
1.1
2
Abstract and Applied Analysis
−β
−β
by using the critical point theory, where 0 Dt and t DT are the left and right RiemannLiouville fractional integrals of order 0 ≤ β < 1, respectively, F : 0, T × RN → R is a given
function and ∇Ft, x is the gradient of F at x.
In this paper, by using the critical-points theorem established by Bonanno in 27, a
new approach is provided to investigate the existence of three solutions to the following
fractional boundary value problems:
1
d 1
α−1 C a
α−1 C a
λatfut 0,
0 Dt
t DT
t D T ut
0 D t ut −
dt 2
2
a.e. t ∈ 0, T ,
1.2
u0 uT 0,
where α ∈ 1/2, 1, 0 Dtα−1 and t DTα−1 are the left and right Riemann-Liouville fractional
integrals of order 1 − α respectively, c0 Dtα and ct DTα are the left and right Caputo fractional
derivatives of order α respectively, λ is a positive real parameter, f : R → R is a continuous
function, and a : R → R is a nonnegative continuous function with at /
≡ 0.
2. Preliminaries
In this section, we first introduce some necessary definitions and properties of the fractional
calculus which are used in this paper.
Definition 2.1 see 5. Let f be a function defined on a, b. The left and right RiemannLiouville fractional integrals of order α for function f denoted by a Dt−α ft and t Db−α ft,
respectively, are defined by
−α
a Dt ft
1
Γα
−α
t Db ft
1
Γα
t
t − sα−1 fsds,
t ∈ a, b, α > 0,
a
b
2.1
s − t
α−1
fsds,
t ∈ a, b, α > 0,
t
provided the right-hand sides are pointwise defined on a, b, where Γα is the gamma
function.
Definition 2.2 see 5. Let γ ≥ 0 and n ∈ N.
i If γ ∈ n − 1, n and f ∈ ACn a, b, RN , then the left and right Caputo fractional
γ
C γ
derivatives of order γ for function f denoted by C
a Dt ft and t Db ft, respectively, exist
C γ
C γ
almost everywhere on a, b, a Dt ft and t Db ft are represented by
respectively.
C γ
a Dt ft
1
Γ n−γ
C γ
t Db ft
−1n
Γ n−γ
t
t − sn−γ−1 f n sds,
t ∈ a, b,
a
b
t
2.2
s − tn−γ−1 f n sds,
t ∈ a, b,
Abstract and Applied Analysis
3
ii If γ n − 1 and f ∈ ACn−1 a, b, RN , then
represented by
C n−1
a Dt ft
f n−1 t,
C n−1
t Db ft
C n−1
a Dt ft
−1n−1 f n−1 t,
and
C n−1
t Db ft
t ∈ a, b.
are
2.3
With these definitions, we have the rule for fractional integration by parts, and
the composition of the Riemann-Liouville fractional integration operator with the Caputo
fractional differentiation operator, which were proved in 2, 5.
Property 1 see 2, 5. we have the following property of fractional integration:
b
a
−γ
a Dt ft
b
−γ
gtdt t Db gt ftdt,
γ >0
2.4
a
provided that f ∈ Lp a, b, RN , g ∈ Lq a, b, RN , and p ≥ 1, q ≥ 1, 1/p 1/q ≤ 1 γ or
p/
1, q /
1, 1/p 1/q 1 γ.
Property 2 see 5. Let n ∈ N and n − 1 < γ ≤ n. If f ∈ ACn a, b, RN or f ∈ Cn a, b, RN ,
then
−γ
a Dt
C γ
a D t ft
ft −
n−1 j
f a
j!
j0
−γ
t Db
C γ
t D b ft
ft −
t − aj ,
n−1
−1j f j b
j!
j0
2.5
b − tj ,
for t ∈ a, b. In particular, if 0 < γ ≤ 1 and f ∈ ACa, b, RN or f ∈ C1 a, b, RN , then
−γ
a Dt
C γ
a D t ft
ft − fa,
−γ
t Db
C γ
t D b ft
ft − fb.
2.6
Remark 2.3. In view of Property 1 and Definition 2.2, it is obvious that u ∈ AC0, T is a
solution of BVP 1.2 if and only if u is a solution of the following problem:
d 1 −β 1 −β D
D
λatfut 0,
u
u
t
t
0
t
dt 2 t
2 T
a.e. t ∈ 0, T ,
2.7
u0 uT 0,
where β 21 − α ∈ 0, 1.
In order to establish a variational structure for BVP 1.2, it is necessary to construct
appropriate function spaces.
Denote by C0∞ 0, T the set of all functions g ∈ C∞ 0, T with g0 gT 0.
4
Abstract and Applied Analysis
Definition 2.4 see 26. Let 0 < α ≤ 1. The fractional derivative space E0α is defined by the
closure of C0∞ 0, T with respect to the norm
u
α T
0
1/2
T
2
C α
ut
dt
|ut|2 dt
0 Dt
∀u ∈ E0α .
,
0
2.8
Remark 2.5. It is obvious that the fractional derivative space E0α is the space of functions u ∈
α
2
L2 0, T having an α-order Caputo fractional derivative C
0 Dt u ∈ L 0, T and u0 uT 0.
Proposition 2.6 see 26. Let 0 < α ≤ 1. The fractional derivative space E0α is reflexive and
separable Banach space.
Lemma 2.7 see 26. Let 1/2 < α ≤ 1. For all u ∈ E0α , one has the following:
i
u
L2 ≤
Tα C α 0 Dt u 2 .
L
Γα 1
2.9
ii
u
∞ ≤
T α−1/2
Γα2α − 1 1
1/2
C α 0 Dt u 2 .
L
2.10
By 2.9, we can consider E0α with respect to the norm
u
α T
0
1/2
2
C α
0 Dt ut dt
α C
u
,
D
0
t 2
L
∀u ∈ E0α
2.11
in the following analysis.
Lemma 2.8 see 26. Let 1/2 < α ≤ 1, then for all any u ∈ E0α , one has
|cosπα|
u
2α ≤ −
T
0
C α
0 D t ut
α
·C
t DT utdt ≤
1
u
2α .
|cosπα|
2.12
Our main tool is the critical-points theorem 27 which is recalled below.
Theorem 2.9 see 27. Let X be a separable and reflexive real Banach space; Φ : X → R be
a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous
functional whose Gateaux derivative admits a continuous inverse on X ∗ ; Ψ : X → R be a
continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there
exists x0 ∈ X such that Φx0 Ψx0 0, and that
i lim
x
→ ∞ Φx − λΨx ∞, forallλ ∈ 0, ∞. Further, assume that there are
r > 0, x1 ∈ X such that
Abstract and Applied Analysis
5
ii r < Φx1 ;
iii supx∈Φ−1 −∞,r w Ψx < r/r Φx1 Ψx1 .
Then, for each
⎡
⎤
⎥
Φx1 λ ∈ Λ1 ⎥
⎦ Ψx1 − sup
x∈Φ−1 −∞,r w
⎢
r
⎢,
,
Ψx supx∈Φ−1 −∞,r w Ψx ⎣
2.13
the equation
Φ x − λΨ x 0
2.14
has at least three solutions in X and, moreover, for each h > 1, there exists an open interval
hr
Λ2 ⊂ 0,
rΨx1 /Φx1 − supx∈Φ−1 −∞,r w Ψx
2.15
and a positive real number σ such that, for each λ ∈ Λ2 , 2.14 has at least three solutions in X whose
norms are less than σ.
3. Main Result
For given u ∈ E0α , we define functionals Φ, Ψ : Eα → R as follows:
Φu : −
1
2
T
0
Ψu :
C α
0 D t ut
α
·C
t D T utdt,
3.1
T
atFutdt,
0
u
where Fu 0 fsds. Clearly, Φ and Ψ are Gateaux differentiable functional whose
Gateaux derivative at the point u ∈ E0α are given by
Φ uv −
Ψ uv T
0
1
2
T
0
C α
0 D t ut
α
C α
C α
dt,
·C
vt
ut
·
vt
D
D
D
t
t
0
T
T
t
atfutvtdt −
T t
0
0
3.2
asfusds · v tdt,
6
Abstract and Applied Analysis
for every v ∈ E0α . By Definition 2.2 and Property 2, we have
Φ uv T
0
1
1 α−1 C α
α−1 C α
−
· v tdt.
ut
D
ut
D
D
0 Dt
t
t
0
T
t
2
2 T
3.3
Hence, Iλ Φ − λΨ ∈ C1 E0α , R. If u∗ ∈ E0α is a critical point of Iλ , then
0 Iλ u∗ v
T
1
1 α−1 C α
α−1 C α
−
u
D
u
t
t
D
D
0 Dt
∗
t
∗
t
0
T
t
2
2 T
0
t
λ
3.4
· v tdt,
asfu∗ sds
0
for v ∈ E0α . We can choose v ∈ E0α such that
vt sin
2kπt
T
or vt 1 − cos
2kπt
,
T
k 1, 2, . . . .
3.5
The theory of Fourier series and 3.4 imply that
1
1 α−1 C α
α−1 C α
λ
0 Dt
t D T u∗ t
0 D t u∗ t − t D T
2
2
t
asfu∗ sds C
3.6
0
a.e. on 0, T for some C ∈ R. By 3.6, it is easy to know that u∗ ∈ E0α is a solution of BVP
1.2.
By Lemma 2.7, if α > 1/2, we have for each u ∈ E0α that
u
∞ ≤ Ω
T
0
1/2
2
C α
0 Dt ut dt
Ω
u
α ,
3.7
where
Ω
3.8.
T α−1/2
.
Γα 2α − 1 1
Given two constants c ≥ 0 and d /
0, with c /
3.8
2Aα/| cosπα|Ω · d, where Ω as in
For convenience, set
Aα :
8Γ2 2 − α 1−2α 1 33−2α 24α−5 − 22α−3 − 1 .
T
Γ4 − 2α
3.9
Abstract and Applied Analysis
7
Theorem 3.1. Let f : R → R be a continuous function,
x a : R → R be a nonnegative continuous
function with at /
≡ 0, and 1/2 < α ≤ 1. Put Fx 0 fsds for every x ∈ R, and assume that
there exist four positive constants c, d, μ, and p, with c < 2Aα/| cosπα|Ω · d and p < 2,
such that
H1 Fx ≤ μ1 |x|p , for all x ∈ R;
H2 Fx ≥ 0for all x ∈ 0, Γ2 − αd, and
Fx <
|cosπα|c2
|cosπα|c2 2Ω2 Aαd2 3T/4
× FΓ2 − αd
T
0
atdt
3.10
atdt
T/4
T
4Γ2 − αd
Γ2−αd
bsFsds ,
∀x ∈ −c, c,
0
where bs aT/4Γ2 − αds aT − T/4Γ2 − αds. Then, for each
λ ∈ Λ1
⎤
⎡
2
2
c
Aαd
|cosπα|
⎣,
,
⎦
T
Γ2−αd
T
a 0
bxFxdx − 0 atdt · max|x|≤c Fx 2Ω2 0 atdt · max|x|≤c Fx
3.11
3T/4
where a and denote FΓ2 − αd T/4 atdt and T/4Γ2 − αd respectively, the problem 1.2
admits at least three solutions in E0α and, moreover, for each h > 1, there exists an open interval
⎡
Λ2 ⊂ ⎣0,
⎤
a 2Γ2−α
0
hAαd2
bxFxdx − 2Ω2 Aαd2 /c2 |cosπα|
T
0
⎦
atdt · max|x|≤c Fx
3.12
such that, for each λ ∈ Λ2 , the problem 1.2 admits at least three solutions in E0α whose norms are
less that σ.
Proof. Let Φ, Ψ be the functionals defined in the above. By the Lemma 5.1 in 26, Φ is
continuous and convex, hence it is weakly sequentially lower semicontinuous. Moreover,
Φ is coercive, continuously Gateaux differentiable functional whose Gateaux derivative
admits a continuous inverse on E0α . The functional Ψ is well defined, continuously Gateaux
differentiable and with compact derivative. It is well known that the critical point of the
functional Φ − λΨ in E0α is exactly the solution of BVP 1.2.
8
Abstract and Applied Analysis
From H1 and 2.12, we get
lim Φu − λΨu ∞,
3.13
u
α → ∞
for all λ ∈ 0, ∞. Put
T
t ∈ 0,
,
4
⎧
4Γ2 − αd
⎪
⎪
t,
⎪
⎪
⎪
T
⎪
⎪
⎪
⎪
⎪
⎪
⎨
u1 t Γ2 − αd,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ 4Γ2 − αd
⎪
⎪
⎩
T − t,
T
t∈
t∈
T 3T
,
,
4 4
3.14
T
,T .
4
It is easy to check that u1 0 u1 T 0 and u1 ∈ L2 0, T . The direct calculation shows
⎧
4d 1−α
⎪
⎪
t ,
⎪
⎪
T
⎪
⎪
⎪
⎪
⎪
⎨ 4d
1−α
C α
t
−
t−
D
u
t
0 t 1
⎪T
⎪
⎪
⎪
⎪
⎪
4d 1−α
⎪
⎪
⎪
⎩ T t − t−
u1 2α
T 0
16d
T2
2
C α
u
t
dt
D
1
0
t
T
2
0
T
t ∈ 0,
,
4
T 3T
T 1−α
,
t∈
,
,
4
4 4
3T
T 1−α
3T 1−α
, t∈
,T ,
− t−
4
4
4
T/4 3T/4 T
0
T/4
3T/4
2
C α
u
dt
t
D
1
0
t
T T T 21−α
3T 21−α
21−α
t
dt dt dt
t−
t−
4
4
T/4
3T/4
T
T
T 1−α
3T 1−α
t1−α t −
dt − 2
t1−α t −
dt
4
4
T/4
3T/4
T T 1−α
3T 1−α
2
dt
t−
t−
4
4
3T/4
T
3−2α 3−2α 3−2α
T
3
1
T 1−α
16d2
1−α
1
−
2
t
dt
t
−
4
4
3 − 2α
4
T2
T/4
−2
−2
T
t
3T/4
1−α
3T
t−
4
1−α
dt 2
T
3T/4
T
t−
4
1−α 3T 1−α
dt < ∞
t−
4
.
3.15
Abstract and Applied Analysis
That is,
C α
0 Dt u1
9
∈ L2 0, T . Thus, u1 ∈ E0α . Moreover, the direct calculation shows
⎧ 1−α 1−α 3T
T
4d
⎪
1−α
⎪
⎪
−t
−t
−
,
T − t −
⎪
⎪
T
4
4
⎪
⎪
⎪ ⎪
1−α ⎪
⎨ 4d
3T
1−α
C α
,
−t
T − t −
t DT u1 t ⎪ T
4
⎪
⎪
⎪
⎪
4d
⎪
⎪
⎪
T − t1−α ,
⎪
⎪
⎩T
Φu1 −
1
2
T
0
8d2
− 2
T
8d2
− 2
T
C α
0 D t u1 t
T
,
t ∈ 0,
4
T 3T
t∈
,
,
4 4
3T
,T ,
t∈
4
α
·C
t D T u1 tdt
1−α 1−α 3T
T
dt
−t
−t
t
−
T − t −
4
4
0
3T/4 1−α T 1−α
3T
1−α
1−α
t − t−
dt
−t
T − t −
4
4
T/4
T 1−α 1−α T
3T
1−α
1−α
t − t−
− t−
T − t dt
4
4
3T/4
T/4
T
t
1−α
1−α
1−α
T − t
1−α
dt −
0
T/4
t1−α
0
T
−t
4
1−α
dt
3T/4 T 1−α
T 1−α 3T
3T 1−α
−t
t−
t−
dt −
T − t1−α dt
4
4
4
T/4
3T/4
3T/4
1−α T T 1−α
1−α
1−α 3T
−t
−
t
−
t−
T − t dt
4
4
0
T/4
8Γ2 2 − α 1−2α 2 1 33−2α 24α−5 − 22α−3 − 1 Aαd2 ,
T
d
Γ4 − 2α
T
atFu1 tdt
Ψu1 0
T/4
atF
0
3T/4
4Γ2 − αd
t dt atFΓ2 − αddt
T
T/4
T
atF
3T/4
FΓ2 − αd
4Γ2 − αd
T − t dt
T
3T/4
T/4
T
atdt 4Γ2 − αd
Let r | cosπα|/2Ω2 c2 . Since c <
Γ2−αd
bxFxdx.
0
3.16
2Aα/| cosπα|Ω · d, we obtain r < Φu1 .
10
Abstract and Applied Analysis
By 2.12 and 3.7, one has Φu ≤ r ⇒ u
∞ ≤ c. Thus,
sup
u∈Φ−1 −∞,r w
Ψu sup
u∈Φ−1 −∞,r Ψu ≤ maxFx
|x|≤c
T
3.17
atdt.
0
Moreover, we have
r
Ψu1 r Φu1 |cosπα|/2Ω2 c2
|cosπα|/2Ω2 c2 Aαd2
3T/4
× FΓ2 − αd
atdt T/4
|cosπα|c2
|cosπα|c2 2Ω2 Aαd2
3T/4
× FΓ2 − αd
atdt T/4
T
4Γ2 − αd
T
4Γ2 − αd
Γ2−αd
bxFxdx
3.18
0
Γ2−αd
bxFxdx .
0
Hence, from H2 one has
sup
u∈Φ−1 −∞,r w
Ψu <
r
Ψu1 .
r Φu1 3.19
Now, taking into account that
Φu1 Ψu1 − supu∈Φ−1 −∞,r w Ψu
≤
Aαd2
,
Γ2−αd
T
a 0
bxFxdx − 0 atdt · max|x|≤c Fx
r
supu∈Φ−1 −∞,r w Ψu
≥
2Ω2
T
0
c2 |cosπα|
atdt · max|x|≤c Fx
,
hr
rΨu1 /Φu1 − supu∈Φ−1 −∞,r w Ψu
≤
a 2Γ2−α
0
hAαd2
bxFxdx − 2Ω2 Aαd2 /c2 |cosπα|
T
0
atdt · max|x|≤c Fx
m.
3.20
Abstract and Applied Analysis
11
Thus, by Theorem 2.9 it follows that, for each λ ∈ Λ1 , BVP 1.2 admits at least three solutions,
and there exists an open interval Λ2 ⊂ 0, m and a real positive number σ such that, for each
λ ∈ Λ2 , BVP 1.2 admits at least three solutions in E0α whose norms are less than σ.
Finally, we give an example to show the effectiveness of the results obtained here.
√
Let α 0.8, T 1, at ≡ 1, and fu e−u u8 9 − u u. Then BVP 1.2 reduces to
the following boundary value problem:
1
√ d 1 −0.2 C 0.8
−0.2 C 0.8
λ e−u u8 9 − u u
0 Dt
t D 1 ut
0 D t ut − t D 1
dt 2
2
0,
3.21
a.e. t ∈ 0, 1,
u0 u1 0.
Example 3.2. Owing to Theorem 3.1, for each λ ∈0.291, 0.318, BVP 3.21 admits at least
three solutions. In fact, put c 1 and d 2, it is easy to calculate that Ω 1.1089, A0.8 1.3313, and
2A0.8
Ω · d 4.0235 > 1 c.
|cos0.8π|
3.22
Since
Fx x
0
2
fsds e−x x9 x3/2 ,
3
3.23
we have that condition H1 holds. Moreover, Fx ≥ 0 for each x ∈ 0, 2Γ1.2, and
2Γ1.2
1
1
|cos0.8π|
F2Γ1.2 Fsds
4Γ1.2 0
|cos0.8π| 2Ω2 A0.8 · 22 2
2
> 1.064 > 1.0345 e−1 ≥ Fx,
3
3.24
|x| ≤ 1,
which implies that condition H2 holds. Thus, by Theorem 3.1, for each λ ∈ 0.291, 0.318,
the problem 3.21 admits at least three nontrivial solutions in E00.8 . Moreover, for each h > 1,
there exists an open interval Λ ⊂0, 3.4674h and a real positive number σ such that, for each
λ ∈ Λ, the problem 3.21 admits at least three solutions in E00.8 whose norms are less than σ.
Acknowledgments
The author thanks the reviewers for their suggestions and comments which improved the
presentation of this paper. This work is supported by Natural Science Foundation of Jiangsu
Province BK2011407 and Natural Science Foundation of China 10771212.
12
Abstract and Applied Analysis
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