Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 176, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR A NONLINEAR
FRACTIONAL BOUNDARY VALUE PROBLEM VIA A LOCAL
MINIMUM THEOREM
CHUANZHI BAI
Abstract. This article concerns the existence of solutions to the nonlinear
fractional boundary-value problem
”
d “ α−1 c α
α−1 c α
(0 Dt u(t)) − t DT
(t DT u(t)) + λf (u(t)) = 0, a.e. t ∈ [0, T ],
0 Dt
dt
u(0) = u(T ) = 0,
where α ∈ (1/2, 1], and λ is a positive real parameter. The approach is based
on a local minimum theorem established by Bonanno.
1. Introduction
Fractional differential equations have been proved to be valuable tools in the
modeling of many phenomena in various fields of physic, chemistry, biology, engineering and economics. There has been significant development in fractional
differential equations, one can see the monographs of Miller and Ross [1], Samko
et al [2], Podlubny [3], Hilfer [4], Kilbas et al [5] and the papers [7, 8, 9, 10, 11, 12,
13, 6, 14, 15, 16] and the references therein.
Critical point theory has been very useful in determining the existence of solution
for integer order differential equations with some boundary conditions, for example
[21, 6, 19, 18, 17, 20]. But until now, there are few results on the solution to fractional BVP which were established by the critical point theory, since it is often very
difficult to establish a suitable space and variational functional for fractional BVP.
Recently, Jiao and Zhou [22] investigated the fractional boundary-value problem
d 1 −β 0
1 −β 0
a.e. t ∈ [0, T ],
0 Dt (u (t)) + t DT (u (t)) + ∇F (t, u(t)) = 0,
dt 2
2
u(0) = u(T ) = 0,
by using the critical point theory, where 0 Dt−β and t DT−β are the left and right
Riemann-Liouville fractional integrals of order 0 ≤ β < 1 respectively, F : [0, T ] ×
RN → R is a given function and ∇F (t, x) is the gradient of F at x.
2000 Mathematics Subject Classification. 58E05, 34B15, 26A33.
Key words and phrases. Critical points; fractional differential equations;
boundary-value problem.
c
2012
Texas State University - San Marcos.
Submitted July 30, 2012. Published October 12, 2012.
1
2
C. BAI
EJDE-2012/176
In this article, by using a local minimum theorem established by Bonanno in
[23], a new approach is provided to investigate the existence of solutions to the
following fractional boundary value problems
d α−1 c α
(0 Dt u(t)) − t DTα−1 (ct DTα u(t)) + λf (u(t)) = 0, a.e. t ∈ [0, T ],
0 Dt
dt
(1.1)
u(0) = u(T ) = 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 0 < α ≤ 1 respectively, λ is a positive real
parameter, and f : R → R is a continuous function.
2. Preliminaries
In this section, we introduce some definitions and properties of the fractional
calculus which are used in this article.
Definition 2.1 ([5]). Let f be a function defined
Riemann-Liouville fractional integrals of order α for
Z t
1
−α
(t − s)α−1 f (s)ds,
D
f
(t)
=
a t
Γ(α) a
Z b
1
−α
(s − t)α−1 f (s)ds,
t Db f (t) =
Γ(α) t
on [a, b]. The left and right
a function f are defined by
t ∈ [a, b], α > 0,
t ∈ [a, b], α > 0,
provided the right-hand sides are pointwise defined on [a, b], where Γ(α) is the
standard gamma function.
Definition 2.2 ([5]). Let γ ≥ 0 and n ∈ N.
(i) If γ ∈ (n − 1, n) and f ∈ AC n ([a, b], RN ), then the left and right Caputo
fractional derivatives of order γ for function f denoted by ca Dtγ f (t) and ct Dbγ f (t),
respectively, exist almost everywhere on [a, b], ca Dtγ f (t) and ct Dbγ f (t) are represented
by
Z t
1
c γ
(t − s)n−γ−1 f (n) (s)ds, t ∈ [a, b],
D
f
(t)
=
a t
Γ(n − γ) a
Z b
(−1)n
c γ
(s − t)n−γ−1 f (n) (s)ds, t ∈ [a, b],
t Db f (t) =
Γ(n − γ) t
respectively.
(ii) If γ = n − 1 and f ∈ AC n−1 ([a, b], RN ), then ca Dtn−1 f (t) and ct Dbn−1 f (t) are
represented by
c n−1
f (t)
a Dt
= f (n−1) (t),
and
c n−1
f (t)
t Db
= (−1)(n−1) f (n−1) (t),
t ∈ [a, b].
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 [5, 2].
Proposition 2.3 ([5, 2]). We have the following property of fractional integration
Z b
Z b
−γ
[a Dt f (t)]g(t)dt =
[t Db−γ g(t)]f (t)dt, γ > 0,
(2.1)
a
a
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EXISTENCE OF SOLUTIONS
3
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 6= 1, q =
6 1, 1/p + 1/q = 1 + γ.
Proposition 2.4 ([5]). Let n ∈ N and n − 1 < γ ≤ n. If f ∈ AC n ([a, b], RN ) or
f ∈ C n ([a, b], RN ), then
−γ c γ
a Dt (a Dt f (t)) = f (t) −
n−1
X
j=0
−γ c γ
t Db (t Db f (t))
= f (t) −
n−1
X
j=0
f (j) (a)
(t − a)j ,
j!
(−1)j f (j) (b)
(b − t)j ,
j!
for t ∈ [a, b]. In particular, if 0 < γ ≤ 1 and f ∈ AC([a, b], RN ) or f ∈
C 1 ([a, b], RN ), then
−γ c γ
a Dt (a Dt f (t))
= f (t) − f (a),
−γ c γ
t Db (t Db f (t))
and
= f (t) − f (b).
(2.2)
Remark 2.5. In view of (2.1) and Definition 2.2, it is obvious that u ∈ AC([0, T ])
is a solution of (1.1) if and only if u is a solution of the problem
d −β 0
−β 0
a.e. t ∈ [0, T ],
0 Dt (u (t)) + t DT (u (t)) + λf (u(t)) = 0,
dt
(2.3)
u(0) = u(T ) = 0,
where β = 2(1 − α) ∈ [0, 1).
To establish a variational structure for (1.1), it is necessary to construct appropriate function spaces. Denote by C0∞ [0, T ] the set of all functions g ∈ C ∞ [0, T ]
with g(0) = g(T ) = 0.
Definition 2.6 ([22]). Let 0 < α ≤ 1. The fractional derivative space E0α is defined
by the closure of C0∞ [0, T ] with respect to the norm
Z T
Z T
1/2
kukα =
|c0 Dtα u(t)|2 dt +
|u(t)|2 dt
, ∀u ∈ E α .
0
0
Remark 2.7. It is obvious that the fractional derivative space E0α is the space
of functions u ∈ L2 [0, T ] having an α-order Caputo fractional derivative c0 Dtα u ∈
L2 [0, T ] and u(0) = u(T ) = 0.
Proposition 2.8 ([22]). Let 0 < α ≤ 1. The fractional derivative space E0α is
reflexive and separable Banach space.
Lemma 2.9 ([22]). Let 0 < α ≤ 1. For all u ∈ E0α , we have
Tα
kc Dα ukL2 ,
Γ(α + 1) 0 t
(2.4)
T α−1/2
kc Dα ukL2 .
Γ(α)(2(α − 1) + 1)1/2 0 t
(2.5)
kukL2 ≤
kuk∞ ≤
By (2.4), we can consider E0α with respect to the norm
Z T
1/2
kukα =
|c0 Dtα u(t)|2 dt
= kc0 Dtα ukL2 ,
0
in the following analysis.
∀u ∈ E0α
(2.6)
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C. BAI
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Lemma 2.10 ([22]). Let 1/2 < α ≤ 1, then for all any u ∈ E0α , we have
Z T
1
c α
c α
| cos(πα)|kuk2α ≤ −
kuk2α .
0 Dt u(t) · t DT u(t)dt ≤
|
cos(πα)|
0
(2.7)
Our main tools is the local minimum theorem [23] which is recalled below. Given
a set X and two functionals Φ, Ψ : X → R, let
supu∈Φ−1 (]r1 ,r2 [) Ψ(u) − Ψ(v)
,
r2 − Φ(v)
1 ,r2 [)
Ψ(v) − supu∈Φ−1 (]−∞,r1 ]) Ψ(u)
ρ2 (r1 , r2 ) =
sup
,
Φ(v) − r1
v∈Φ−1 (]r1 ,r2 [)
β(r1 , r2 ) =
inf
v∈Φ−1 (]r
(2.8)
(2.9)
for all r1 , r2 ∈ R, with r1 < r2 .
Theorem 2.11 ([23]). Let X be a reflexive real Banach space; Φ : X → R be a
sequentially weakly lower semicontinuous, coercive and continuously Gateaux differential function whose Gateaux derivative admits a continuous inverse on X ∗ ;
Ψ : X → R be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Put Iλ = Φ − λΨ and assume that there are r1 , r2 ∈ R, with
r1 < r2 , such that
β(r1 , r2 ) < ρ2 (r1 , r2 ),
where β and ρ2 are given by (2.8) and (2.9). Then, for each λ ∈ ρ2 (r11 ,r2 ) , β(r11,r2 )
there is u0,λ ∈ Φ−1 (]r1 , r2 [) such that Iλ (u0,λ ) ≤ Iλ (u) for all u ∈ Φ−1 (]r1 , r2 [) and
Iλ0 (u0,λ ) = 0.
3. Main result
For given u ∈ E0α , we define functionals Φ, Ψ : E α → R as follows:
Z T
Z T
c α
c α
Φ(u) := −
Ψ(u) :=
F (u(t))dt,
0 Dt u(t) · t DT u(t)dt,
0
0
Ru
where F (u) = 0 f (s)ds. Clearly, Φ and Ψ are Gateaux differentiable functional
whose Gateaux derivative at the point u ∈ E0α are given by
Z T
Φ0 (u)v = −
(c0 Dtα u(t) · ct DTα v(t) + ct DTα u(t) · c0 Dtα v(t))dt,
0
Z
0
T
Z
T
Z
f (u(t))v(t)dt = −
Ψ (u)v =
0
0
t
f (u(s))ds · v 0 (t)dt,
0
for every v ∈ E0α . By Definition 2.2 and (2.2), we have
Z T
Φ0 (u)v =
(0 Dtα−1 (c0 Dtα u(t)) − t DTα−1 (ct DTα u(t))) · v 0 (t)dt.
0
Hence, Iλ = Φ − λΨ ∈ C 1 (E0α , R). If u∗ ∈ E0α is a critical point of Iλ , then
Z T
α−1 c α
0
0 = Iλ (u∗ )v =
(0 Dt u∗ (t)) − t DTα−1 (ct DTα u∗ (t))
0 Dt
0
Z t
f (u∗ (s))ds · v 0 (t)dt,
+λ
0
(3.1)
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EXISTENCE OF SOLUTIONS
5
for v ∈ E0α . We can choose v ∈ E0α such that
2kπt
2kπt
or v(t) = 1 − cos
, k = 1, 2, . . . .
T
T
The theory of Fourier series and (3.1) imply
Z t
α−1 c α
α−1 c α
(0 Dt u∗ (t)) − t DT (t DT u∗ (t)) + λ
f (u∗ (s))ds = C
0 Dt
v(t) = sin
(3.2)
0
a.e. on [0, T ] for some C ∈ R. By (3.2), it is easy to show that u∗ ∈ E0α is a solution
of (1.1).
By Lemma 2.9, when α > 1/2, for each u ∈ E0α we have
Z T
1/2
kuk∞ ≤ Ω
|c0 Dtα u(t)|2 dt
= Ωkukα ,
(3.3)
0
where
1
T α− 2
p
.
(3.4)
Γ(α) 2(α − 1) + 1
q
ωα,d
Given two constants c ≥ 0 and d 6= 0, with c 6= | cos(πα)|
· Ω, where Ω as in (3.4).
Put
4Γ2 (2 − α) 1−2α 2 2α−1
T
d (2
− 1).
ωα,d :=
Γ(4 − 2α)
Ω=
Theorem 3.1. Let f : R → R be a continuous function and 12 < α ≤ 1. Assume
that there exist a positive constant c and a constant d 6= 0 with
r
ωα,d
Ω < c,
(3.5)
| cos(πα)|
such that
max|η|≤c F (η)
0< 2
<
c | cos(πα)|
1
Γ(2−α)|d|
R Γ(2−α)|d|
0
F (x)dx
ωα,d Ω2
.
(3.6)
Then, for each
λ∈
ωα,d Γ(2 − α)|d|
c2 | cos(πα)|
,
,
R Γ(2−α)|d|
2
T 0
F (x)dx T Ω max|η|≤c F (η)
problem (1.1) admits at least one solution ū such that kūkα < c/Ω.
Proof. Let Φ, Ψ be the functionals defined above. It is well known that they satisfy
all regularity assumptions requested in Theorem 2.11 and that the critical point of
the functional Φ − λΨ in E0α is exactly the solution of (1.1). Put
| cos(πα)| 2
c ,
Ω2
(
2Γ(2−α)d
t,
t ∈ [0, T /2),
T
u0 (t) = 2Γ(2−α)d
(T
−
t),
t ∈ [T /2, T ]
T
r=
(3.7)
It is easy to check that u0 (0) = u0 (T ) = 0 and u0 ∈ L2 [0, T ]. The direct calculation
shows that
(
2d 1−α
,
t ∈ [0, T /2),
c α
T t
0 Dt u0 (t) =
2d 1−α
T 1−α
− 2(t − 2 )
), t ∈ [T /2, T ]
T (t
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C. BAI
EJDE-2012/176
and
ku0 k2α =
T
Z
(c0 Dtα u0 (t))2 dt =
0
4d2 h
= 2
T
T
Z
t
2(1−α)
(c0 Dtα u0 (t))2 dt
T /2
Z
T
dt − 4
0
1−α
T
(t − )1−α dt + 4
2
2
T
t
T /2
2α−1
That is,
T
Z
+
0
=
T
2
Z
2
4(1 + 2
)d 1−2α 16d
T
− 2
3 − 2α
T
c α
0 Dt u0
2
Z
t1−α (t −
T /2
Z
T
(t −
T /2
T 2(1−α) i
)
dt
2
T 1−α
)
dt < ∞.
2
E0α .
∈ L [0, T ]. Thus, u0 ∈
Moreover, the direct calculation shows
(
2d
1−α
− 2( T2 − t)1−α ), t ∈ [0, T /2),
c α
T ((T − t)
D
u
(t)
=
0
t T
2d
1−α
,
t ∈ [T /2, T ]
T (T − t)
and
Z
T
c α
0 Dt u0 (t)
Φ(u0 ) = −
· ct DTα u0 (t)dt
0
Z T
1−α 2d 2 h 2 1−α T
)
t
(T − t)1−α − 2
−t
dt
T
2
0
Z T
i
T
+
(T − t)1−α · t1−α − 2(t − )1−α dt
2
T /2
Z T2
Z
1−α i
T
2d h T 1−α
t
(T − t)1−α dt − 4
t1−α
−t
dt
= −( )2
T
2
0
0
2d h Γ2 (2 − α) 3−2α
Γ2 (2 − α) T 3−2α i
= −( )2
T
−4
( )
T
Γ(4 − 2α)
Γ(4 − 2α) 2
4Γ2 (2 − α) 1−2α 2α−1
=
T
(2
− 1)d2 = ωα,d ,
Γ(4 − 2α)
= −(
and
T
Z
Ψ(u0 ) =
F (u0 (t))dt =
0
T
Γ(2 − α)|d|
Z
Γ(2−α)|d|
F (x)dx.
0
Hence, from (3.5), one has 0 < ωα,d < | cos(πα)|
c2 ; that is, 0 < Φ(u0 ) < r. Moreover,
Ω2
α
−1
for all u ∈ E0 such that u ∈ Φ (] − ∞, r]), by (2.7) we have
| cos(πα)|kuk2α ≤ Φ(u) ≤ r,
which implies
1
r.
| cos(πα)|
kuk2α ≤
Thus, by (3.3), (3.8) and (3.7) we obtain
r
r
|u(t)| < Ωkukα ≤ Ω
= c,
| cos(πα)|
(3.8)
∀t ∈ [0, T ].
Hence,
Z
T
Z
F (u(t))dt ≤
Ψ(u) =
0
T
max F (η)dt = T max F (η),
0
|η|≤c
|η|≤c
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EXISTENCE OF SOLUTIONS
7
for all u ∈ E0α such that u ∈ Φ−1 (] − ∞, r]). Hence,
Ψ(u) ≤ T max F (η).
sup
|η|≤c
u∈Φ−1 (]−∞,r])
Hence, one has
supu∈Φ−1 (]−∞,r]) Ψ(u) − Ψ(u0 )
r − Φ(u0 )
R Γ(2−α)|d|
1
max|η|≤c F (η) − Γ(2−α)|d|
F (x)dx
0
2
≤Ω T
2
2
| cos(πα)|c − ωα,d Ω
β(0, r) ≤
< Ω2 T
= Ω2 T
ωα,d Ω2
| cos(πα)|c2 max|η|≤c
| cos(πα)|c2 − ωα,d Ω2
max|η|≤c F (η) −
(3.9)
F (η)
max|η|≤c F (η)
,
c2 | cos(πα)|
by condition (3.6). On the other hand, if u ∈ Φ−1 (] − ∞, 0]), then Φ(u) ≤ 0. Thus,
by (2.4) and (2.7) we have kukL2 = 0; that is, u(t) = 0, a.e. t ∈ [0, T ]. Hence,
Ψ(u0 ) − supu∈Φ−1 (]−∞,0]) Ψ(u)
Ψ(u0 )
=
Φ(u0 )
Φ(u0 )
R Γ(2−α)|d|
1
F (x)dx
Γ(2−α)|d| 0
=T
.
ωα,d
ρ2 (0, r) ≥
(3.10)
Thus, by (3.9), (3.10) and (3.6) it follows that β(0, r) < ρ2 (0, r). So, from Theorem
2.11 for each
ω Γ(2 − α)|d|
1
c2 | cos(πα)|
1 α,d
,
,
λ∈
⊂
,
R Γ(2−α)|d|
2
ρ2 (0, r) β(0, r)
F (x)dx T Ω max|η|≤c F (η)
T
0
the function Φ − λΨ admits at least one critical point ū such that 0 < Φ(ū) < r;
that is, kūkα < Ωc , and the conclusion is achieved.
We conclude with an example that illustrates the results obtained here. Let
α = 0.8, T = 1, and f (u) = cos(πu/3). Then (1.1) reduces to the boundary-value
problem
d −0.2 c 0.8
π
(0 Dt u(t)) − t D1−0.2 (ct D10.8 u(t)) + λ cos( u(t)) = 0, a.e. t ∈ [0, 1],
0 Dt
dt
3
u(0) = u(1) = 0.
(3.11)
Owing to Theorem 3.1, for each λ ∈ (3.2964, 4.30512), boundary-value problem
(3.11) admits at least one solution. In fact, put c = 2.5 and d = 1, it is easy to
calculate that Ω = 1.1089, ω0.8,1 = 1.4 and
r
ω0.8,1
Ω = 1.4588 < 2.5 = c.
| cos(0.8π)|
Moreover, we have
R Γ(2−α)|d|
1
Γ(2−α)|d|
0
ωα,d Ω2
F (x)dx
=
1
Γ(1.2)
R Γ(1.2)
0
3
π
sin(πx/3)dx
ω0.8,1 · Ω2
= 0.2467,
(3.12)
8
C. BAI
EJDE-2012/176
and
max|η|≤c F (η)
3/π
=
= 0.1889,
c2 | cos(πα)|
2.52 · | cos(0.8π)|
(3.13)
which implies that condition (3.6) holds. Thus, by Theorem 3.1, for each λ ∈
(3.2964, 4.3051), problem (3.11) admits at least one solution ū such that kūk0.8 <
2.2545.
Acknowledgements. This work is supported by Natural Science Foundation of
Jiangsu Province (grant BK2011407), and by the Natural Science Foundation of
China (grant 10771212).
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Chuanzhi Bai
Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China
E-mail address: czbai8@sohu.com