Acta Mathematica Scientia 2011,31B(4):1337–1346
http://actams.wipm.ac.cn
THREE POINT BOUNDARY VALUE PROBLEMS
FOR NONLINEAR FRACTIONAL
DIFFERENTIAL EQUATIONS∗
Mujeeb ur Rehman1 Rahmat Ali Khan2 Naseer Ahmad Asif 2
1. National University of Sciences and Technology (NUST),
Centre for Advanced Mathematics and Physics, Sector H-12 Islamabad, Pakistan
2. University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhawa, Pakistan
E-mail: [email protected]; rahmat [email protected]; [email protected]
Abstract In this paper, we study existence and uniqueness of solutions to nonlinear
three point boundary value problems for fractional differential equation of the type
cDδ
0+u(t) = f(t, u(t), cDσ
0+u(t)), t ∈ [0, T ],
u(0) = αu(η), u(T ) = βu(η),
where 1 < δ < 2, 0 < σ < 1, α, β ∈ R, η ∈ (0, T ), αη(1 − β) + (1 − α)(T − βη) 6= 0 andcDδ
0+, cDσ0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem
and contraction mapping principle to obtain existence and uniqueness results. Examples
are also included to show the applicability of our results.
Key words fractional differential equations; three point boundary conditions; existence
and uniqueness results
2000 MR Subject Classification 34A08
1 Introduction
Fractional differential equations have been the focus of many studies due to their frequent
applications in various fields of physics, mathematics, engineering, biology, control theory, fi-
nance and dynamical processes in self-similar structures, see for example, [10, 13, 18, 21]. There
are many excellent books and papers that provide broad and deep understanding of the theory
and applications of fractional calculus and fractional differential equations. In particular, the
book by Oldham and Spanier [18] had a chronological listing on major works in the study of
fractional calculus. There was a significant development in fractional differential equations in
recent years; see the monographs of Hilfer [10], Machando [14], Miller and Ross [15], Nishimoto
[16], Podlubny [19], Samko, Kilbas and Marichev [22] and the references therin. The physical
and geometric interpretations of fractional integrals and derivatives was discussed in [20].
∗Received September 27, 2009; revised November 20, 2010.
1338 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
There were several approaches to fractional derivatives such as Riemann-Liouville, Caputo,
Weyl, Hadamard, Grunwald-Letnikov and Erdelyi-Kober. Applied problems require those defi-
nitions of fractional derivatives that allow the utilization of physically interpretable initial and
boundary conditions. The Caputo fractional derivative fulfills these requirements.
Recently, existence and uniqueness of solutions to boundary value problems for fractional
differential equations had attracted the attention of many authors, see for example, [1, 5–7, 11,
12, 24, 28] and the references therein.
M. Shahed [23] studied existence and nonexistence of positive solution of nonlinear frac-
tional two-point boundary value problem derivative
Dδ0+u(t) + λa(t)f(u(t)) = 0, 0 < t < 1, 2 < δ < 3,
u(0) = u′(0) = u′(1) = 0,
where λ is a positive parameter and a : (0, 1) → [0,∞) is continuous. Z. Bai and L. Haishen
[4] studied existence and multiplicity of positive solutions for nonlinear fractional differential
equation
Dα0+u(t) + f(t, u(t)) = 0, 0 < t < 1, 2 < δ < 3
subject to the two-point boundary conditions u(0) = 0, u(1) = 0. Existence and uniqueness of
positive solution for the above fractional differential equation under the three-point boundary
conditions u(0) = 0, βu(η) = u(1) was recently been studied by Z. Bai [3].
B. Ahmad and J.J. Nieto [1] studied existence and uniqueness results for the following gen-
eral three point fractional boundary value problem involving a nonlinear fractional differential
equation of order q,
cDq0+u(t) = f(t, u(t)), 0 < t < 1, q ∈ (m − 1, m], m ∈ N, m ≥ 2,
u(0) = 0, u′(0) = 0, u′′(0) = 0, · · · , u(m−2)(0) = 0, u(1) = αu(η),
where cDq0+ is the Caputo fractional derivative.
However, very little work have been done on the case when the nonlinearity f depends on
the fractional derivative cDβ0+u(t) of the unknown function. X. Su and S. Zhang [26] studied
the existence and uniqueness of solutions for following nonlinear two-point fractional boundary
value problem
cDα0+u(t) = f(t, u(t), cD
β0+u(t)), t ∈ (0, 1),
a1u(0) − a2u′(0) = A, b1u(1) + b2u
′(1) = B,
where α, β, ai, bi (i = 1, 2) satisfy certain conditions.
We study existence and uniqueness of solutions to the nonlinear fractional differential
equations under the more general three point boundary conditions of the type
cDδ0+u(t) = f(t, u(t), cDσ
0+u(t)), t ∈ [0, T ],
u(0) = αu(η), u(T ) = βu(η),(1.1)
where 1 < δ < 2, 0 < σ < 1, α, β ∈ R, η ∈ (0, T ), αη(1 − β) + (1 − α)(T − βη) 6= 0 and cDδ0+,
cDσ0+ are the Caputo fractional derivatives.
No.4 M. Rehman et al: THREE POINT BOUNDARY VALUE PROBLEMS 1339
This paper is organized as fallows: In Section 2, we recall some basic definitions and
preliminary results. In Section 3, we study existence and uniqueness of solutions to the boundary
value problem (1.1). In Section 4, we give examples to illustrate the applicability of our results.
2 Preliminaries
Riemann’s modified form of Liouville’s fractional integral operator is a generalization of
Cauchy’s integral formula [15]∫ t
a
dt1
∫ t1
a
dt2 · · ·∫ tn−1
a
g(tn)dtn =1
Γ(n)
∫ t
a
g(s)
(t − s)1−nds, (2.1)
where Γ is the Euler’s gamma function. Clearly, the right-hand side of equation (2.1) is mean-
ingful for any positive real values of n. Hence, it is natural to define fractional integral as
follows:
Definition 2.1 If g ∈ C([a, b]) and α > 0, then the Riemann-Liouville fractional integral
is defined by
Iαa+g(t) =
1
Γ(α)
∫ t
a
g(s)
(t − s)1−αds. (2.2)
For a = 0, the fractional integral (2.2) can be written as Iα0+h(t) = h(t) ∗ ϕα(t), where ϕα(t) =
tα−1
Γ(α) for t > 0 and ϕα(t) = 0 for t ≤ 0 .
Definition 2.2 The Caputo fractional derivative of order α > 0 of a continuous function
g : (a, b) → R is defined by
cDαa+g(t) =
1
Γ(n − α)
∫ t
a
g(n)(s)
(t − s)α−n+1ds,
where n = [α] + 1 (The notation [α] stands for the largest integer not greater than α).
Remark 2.3 Under the natural conditions on g(t), the Caputo fractional derivative
becomes the conventional integer order derivative of the function g(t) as α → n.
We state the following known results in the sequel [25, 27].
Lemma 2.4 For α > 0, g(t) ∈ C(0, 1) ∩ L(0, 1), the homogenous fractional differential
equation cDα0+g(t) = 0 has a solution g(t) = c1 + c2t + c3t
2 + · · · + cntn−1, where ci ∈ R,
i = 0, · · · , n and n = [α] + 1.
Lemma 2.5 Assume that g(t) ∈ C(0, 1)∩ L(0, 1) with derivative of order n that belongs
to C(0, 1) ∩ L(0, 1), then Iα0+
cDα0+g(t) = g(t) + c1 + c2t + c3t
2 + · · · + cntn−1, where ci ∈ R,
i = 0, · · · , n and n = [α] + 1.
3 Main Results
For convenience, we define X ={
u : u ∈ C([0, T ]), cDσ0+u ∈ C([0, T ]), 0 < σ < 1
}
equipped with the norm ‖u‖ = max0≤t≤1
|u(t)| + max0≤t≤1
|cDσ0+u(t)|. The space X is a Banach space
[26].
Lemma 3.1 Let h ∈ C[0, 1], then the unique solution of the linear problem
cDδ0+u(t) = h(t), 0 < t < 1, 1 < δ < 2, t ∈ [0, T ], (3.1)
u(0) = αu(η), u(T ) = βu(η) (3.2)
1340 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
is given by
u(t) = Iδ0+h(t) +
1
p{(αT + (β − α)t)Iδ
0+h(η) − (αη + (1 − α)t)Iδ0+h(T )}, (3.3)
which is equivalent to u(t) =∫ T
0 G(t, s)h(s)ds, where p = αη(1 − β) + (1−α)(T − βη) 6= 0 and
G(t, s) is the kernel of Hammerstien integral operator (Green’s function) and is given by
G(t, s) =
(t − s)δ−1
Γ(δ)+
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ),
if 0 ≤ s ≤ t, η ≥ s,
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ),
if 0 ≤ t ≤ s ≤ η,
(t − s)δ−1
Γ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ), if η ≤ s ≤ t ≤ 1,
− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ), if 0 ≤ t ≤ s, s ≥ η.
(3.4)
Proof In view of Lemma 2.5 the general solution of (3.1) is given by
u(t) = Iδ0+h(t) + c1 + c2t, (3.5)
where c1, c2 ∈ R. Using boundary conditions (3.2) and (3.5), we have
(1 − α)c1 − αηc2 = αIδ0+h(η), (1 − β)c1 + (T − βη)c2 = βIδ
0+h(η) − Iδ0+h(T ).
Thus, we have c1 = αp
(TIδ0+h(η)−ηIδ
0+h(T )), c2 = 1p((β−α)Iδ
0+h(η)+(α−1)Iδ0+h(T ). Therefore
the unique solution of boundary value problem (3.1), (3.2) is
u(t) = Iδ0+h(t) +
1
p{(αT + (β − α)t)Iδ
0+h(η) − (αη + (1 − α)t)Iδ0+h(T )}.
Now, for t ≤ η, we have
u(t) =
∫ t
0
(t − s)δ−1
Γ(δ)h(s)ds +
(∫ t
0
+
∫ η
t
)
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)h(s)ds
−(
∫ t
0
+
∫ η
t
+
∫ T
η
)
(αη + (1 − α)t)(T − s)δ−1
pΓ(δ)h(s)ds
=
∫ t
0
(
(t − s)δ−1
Γ(δ)+
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ)
)
h(s)ds
+
∫ η
t
(
(t − s)δ−1
Γ(δ)+
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ)
)
h(s)ds
−∫ T
η
(αη + (1 − α)t)(T − s)δ−1
pΓ(δ)h(s)ds =
∫ T
0
G(t, s)h(s)ds.
For t ≥ η, we have
u(t) =
(∫ η
0
+
∫ t
η
)
(t − s)δ−1
Γ(δ)h(s)ds +
∫ η
0
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)h(s)ds
No.4 M. Rehman et al: THREE POINT BOUNDARY VALUE PROBLEMS 1341
−(
∫ η
0
+
∫ t
η
+
∫ T
t
)
(αη + (1 − α)t)(T − s)δ−1
pΓ(δ)h(s)ds
=
∫ η
0
(
(t − s)δ−1
Γ(δ)+
(αT + (β − α)t)(η − s)δ−1
pΓ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ)
)
h(s)ds
+
∫ t
η
(
(t − s)δ−1
Γ(δ)− (αη + (1 − α)t)(T − s)δ−1
pΓ(δ)
)
h(s)ds
−∫ T
t
(αη + (1 − α)t)(T − s)δ−1
pΓ(δ)h(s)ds =
∫ T
0
G(t, s)h(s)ds.
The proof is completed.
Define l∗ =: max(
∫ T
0| ∂∂t
G(t, s)φ(s)|ds)
, l = max(
∫ T
0|G(t, s)φ(s)|ds
)
, N = |β − α|ηδ +
|1 − α|T δ and M =[
(p + |α|η)T δ + |α|Tηδ + T σ−1
Γ(2−σ) (pT δ−1 + N)]
.
Define an operator A : x → R by
Au(t) =
∫ T
0
G(t, s)f(s, u(s), cDσ0+u(s))ds.
Our first result is based on the Schauder’s fixed point theorem.
Theorem 3.2 Let f : [0, T ] × R × R → R be continuous. Suppose that one of the
following condition is satisfied:
(H1) There exists a nonnegative function φ ∈ L[0, T ] such that
|f(t, u, v)| ≤ φ(t) + c1|u|θ1 + c2|v|θ2 ,
where c1, c2 ∈ R are nonnegative constants and 0 < θ1, θ2 < 1.
(H2) |f(t, u, v)| ≤ φ(t) + c1|u|θ1 + c2|v|θ2 , where c1, c2 ∈ R are nonnegative constants and
θ1, θ2 > 1.
Then, the boundary value problem (1.1) has a solution.
Proof Suppose (H1) holds. Choose R ≥ max{
3(l + l∗T 1−σ
Γ(2−σ) ), (3Mc1)1
1−θ1 , (3Mc2)1
1−θ2
}
and define U = {u ∈ X : ‖u‖ ≤ R}. For any u ∈ U , using (H1), we have
|Au(t)| =
∣
∣
∣
∣
∣
∫ T
0
G(t, s)f(s, u(s), cDσ0+u(s))ds
∣
∣
∣
∣
∣
≤∫ T
0
|G(t, s)φ(s)|ds + (c1Rθ1 + c2R
θ2)
[∫ t
0
(t − s)δ−1
Γ(δ)ds
+1
p(|α|T + |β − α|t)
∫ η
0
(η − s)δ−1
Γ(δ)ds +
1
p(|α|η + |1 − α|t)
∫ T
0
(T − s)δ−1
Γ(δ)ds
]
≤ l + (c1Rθ1 + c2R
θ2)
(
tδ
δΓ(δ)+
1
p(|α|T + |β − α|t) ηδ
δΓ(δ)+
1
p(|α|η + |1 − α|t) T δ
Γ(δ)
)
,
which implies that |Au(t)| ≤ l + (c1Rθ1+c2Rθ2 )δpΓ(δ)
(
pT δ + |α|(Tηδ + ηT δ) + NT)
. Also,
|(Au)′(t)| ≤∫ T
0
∣
∣
∣
∣
∂
∂tG(t, s)
∣
∣
∣
∣
|f(s, u(s), cDσ0+u(s))|ds
≤∫ T
0
∣
∣
∣
∣
∂
∂tG(t, s)φ(s)
∣
∣
∣
∣
ds + (c1Rθ1 + c2R
θ2)
(∫ t
0
(t − s)δ−2
Γ(δ − 1)ds
1342 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
+|β − α|
p
∫ η
0
(η − s)δ−1
Γ(δ)ds +
|1 − α|p
∫ T
0
(T − s)δ−1
Γ(δ)ds
)
≤ l∗ + (c1Rθ1 + c2R
θ2)
(
tδ−1
(δ − 1)Γ(δ − 1)+
N
δpΓ(δ)
)
.
Hence,
|Dσ0+A(u)| =
∣
∣
∣
∣
1
Γ(1 − σ)
∫ t
0
(t − s)−σ(Au)′(s)ds
∣
∣
∣
∣
≤ 1
Γ(1 − σ)
∫ t
0
(t − s)−σ|(Au)′(s)|ds
≤ l∗t1−σ
(1 − σ)Γ(1 − σ)+
(c1Rθ1 + c2R
θ2)t1−σ
(1 − σ)Γ(1 − σ)
(
tδ−1
Γ(δ)+
|β − α|ηδ
δpΓ(δ)+
|1 − α|T δ
δpΓ(δ)
)
,
which implies that
|Dσ0+A(u)| ≤ l∗T 1−σ
Γ(2 − σ)+
(c1Rθ1 + c2R
θ2)
δpΓ(δ)Γ(2 − σ)
(
pT δ−1 + N)
T 1−σ.
By the definition of the norm ‖.‖ = max0≤t≤1
|.(t)| + max0≤t≤1
|cDσ0+.(t)| and the above relations, we
obtain
‖Au(t)‖ ≤ l +l∗T 1−σ
Γ(2 − σ)+
c1Rθ1 + c2R
θ2
δpΓδ
[
(p + |α|η)T δ + |α|Tηδ +T σ−1
Γ(2 − σ)(pT δ−1 + N)
]
≤ R
3+(
c1Rθ1 + c2R
θ2
)
M ≤ R
3+
R
3+
R
3= R,
which implies that A : U → U . The continuity of the operator A follows from the continuity of
f and G.
Now, if (H2) holds, we choose
0 < R ≤ min
{
3
(
l +l∗T 1−σ
Γ(2 − σ)
)
,
(
1
3Mc1
)1
1−θ1
,
(
1
3Mc1
)1
1−θ1
}
and by the same process as above, we obtain
‖Au(t)‖ ≤ R
3+(
c1Rθ1 + c2R
θ2
)
M ≤ R
3+
R
3+
R
3= R,
which implies that A : U → U .
Now, we show that A is a completely continuous operator. Let
K = max{|f(t, u(t), cDσ0+u(t))| : t ∈ [0, T ], u ∈ U}.
For t1, t2 ∈ [0, T ] such that t1 < t2, we have
|Au(t1) − Au(t2)| =
∣
∣
∣
∣
∣
∫ T
0
(G(t1, s) − G(t2, s))f(s, u(s), cDσ0+u(s))ds
∣
∣
∣
∣
∣
≤ K
[∫ t1
0
|G(t1, s) − G(t2, s)|ds +
∫ t2
t1
|G(t1, s) − G(t2, s)|ds
+
∫ T
t2
|G(t1, s) − G(t2, s)|ds
]
,
No.4 M. Rehman et al: THREE POINT BOUNDARY VALUE PROBLEMS 1343
which, in view of the definition of G, implies that
|Au(t1) − Au(t2)| ≤ K
[∫ t2
0
(t2 − s)δ−1
Γ(δ)ds −
∫ t1
0
(t1 − s)δ−1
Γ(δ)ds
+|β − α|(t2 − t1)
pΓ(δ)
∫ η
0
(η − s)δ−1ds +|1 − α|(t2 − t1)
pΓ(δ)
∫ T
0
(η − s)δ−1ds
]
≤ K
δpΓ(δ)[p(tδ2 − tδ1) + (|β − α|ηδ + |1 − α|T δ)(t2 − t1)]
=K
δpΓ(δ)
[
p(tδ2 − tδ1) + N(t2 − t1)]
and
|cDσ0+A(u)(t1) − cDσ
0+A(u)(t2)|
=1
Γ(1 − σ)
∣
∣
∣
∣
∫ t2
0
(t2 − s)−σ(Au)′(s)ds −∫ t1
0
(t1 − s)−σ(Au)′(s)ds
∣
∣
∣
∣
≤ 1
Γ(1 − σ)
∣
∣
∣
∣
∫ t2
0
(t2 − s)−σ(Au)′(s)ds −∫ t1
0
(t2 − s)−σ(Au)′(s)ds
∣
∣
∣
∣
+1
Γ(1 − σ)
∣
∣
∣
∣
∫ t1
0
(t2 − s)−σ(Au)′(s)ds −∫ t1
0
(t1 − s)−σ(Au)′(s)ds
∣
∣
∣
∣
≤ 1
Γ(1 − σ)
(∫ t2
t1
(t2 − s)−σ|(Au)′(s)|ds +
∫ t1
0
((t2 − s)−σ − (t1 − s)−σ)|(Au)′(s)|ds
)
≤ 1
Γ(1 − σ)
[
∫ t2
t1
(t2 − s)−σ
(
∫ T
0
∣
∣
∣
∣
∂
∂sG(s, z)
∣
∣
∣
∣
|f(z, u(z), cDσ0+u(z))|dz
)
ds
+
∫ t1
0
((t2 − s)−σ − (t1 − s)−σ)
(
∫ T
0
∣
∣
∣
∣
∂
∂sG(s, z)
∣
∣
∣
∣
|f(z, u(z), cDσ0+u(z))|dz
)
ds
]
≤ K
δpΓ(δ)Γ(1 − σ)(pT δ−1 + N)
[∫ t2
t1
(t2 − s)−σds +
∫ t1
0
((t2 − s)−σ − (t1 − s)−σ)ds
]
≤ K(pT δ−1 + N)
δpΓ(δ)Γ(2 − σ)(t1−σ
2 − t1−σ1 ).
Hence,
‖Au(t1) − Au(t2)‖ ≤ K
δpΓ(δ)
[
p(tδ2 − tδ1) + N(t2 − t1) +pT δ−1 + N
Γ(2 − σ)(t1−σ
2 − t1−σ1 )
]
,
which implies that ||Au(t1)−Au(t2)|| → 0 as t1 → t2. By Arzela-Ascoli Theorem, it follows that
A : X → X is completely continuous. As a consequence of the Schauder’s fixed point theorem,
A has a fixed point which implies that the boundary value problem (1.1) has a solution.
Uniqueness of solutions is based on application of the Banach contraction principle. Define
kδα,β =
2|α| + |β|pδΓ(δ)
, kδα =
1 + |α|pδΓ(δ)
and kδ =T δ
pδΓ(δ).
Theorem 3.3 Assume that there exists a constant k > 0 such that
|f(t, u, v) − f(t, u, v))| ≤ k(|u − u| + |v − v|) for each t ∈ [0, T ] and all u, u ∈ R, v, v ∈ R.
1344 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
If k <(
kδα,β
(
T δ + ηδT 1−σ
Γ(2−σ)
)
+ (kδαT δ+1 + kδ)
(
1 + T−σ
Γ(2−σ)
))−1, then the boundary value problem
(1.1) has a unique solution.
Proof By the definition of A, we have
|A(u)(t) − A(u)(t)| ≤∫ T
0
|G(t, s)||f(s, u(s), cDσ0+u(s)) − f(s, u, cDσ
0+u(s))|ds
≤ k‖u − u‖pΓ(δ)
[∫ t
0
(t − s)δ−1ds + (|α|T + |β − α|t)∫ η
0
(η − s)δ−1ds
+(|α|η + |1 − α|t)∫ T
0
(T − s)δ−1ds
]
≤ k‖u − u‖pδΓ(δ)
(
tδ + (|α|T + |β − α|t)ηδ + (|α|η + |1 − α|t)T δ)
≤ k‖u − u‖(
(1 + T + (η + T )|α|)T δ
pδΓ(δ)+
(2|α| + |β|)Tηδ
pδΓ(δ)
)
≤ k‖u − u‖(
T δ
pδΓ(δ)+
(2|α| + |β|)ηT δ
pδΓ(δ)+
(1 + |α|)T δ+1
pδΓ(δ)
)
= k‖u − u‖(
kδ + kδα,βηT δ + kδ
αT δ+1)
and
|cDσ0+(Au)(t) − cDσ
0+(Au)(t)| =
∣
∣
∣
∣
1
Γ(1 − σ)
∫ t
0
(t − s)−σ((Au)′(s) − (Au)′(s))ds
∣
∣
∣
∣
≤ 1
Γ(1 − σ)
∫ t
0
(t − s)−σ
(∫ 1
0
∣
∣
∣
∣
∂
∂sG(s, z)
∣
∣
∣
∣
|f(z, u(z), cDσ0+u(z)
−f(z, u(z), cDσ0+u(z)|dz
)
ds
≤ k‖u − u‖Γ(1 − σ)
∫ t
0
(t − s)−σ
(∫ 1
0
∣
∣
∣
∣
∂
∂sG(s, z)
∣
∣
∣
∣
dz
)
ds.
Using the definition of G(t, s), we obtain∫ 1
0
∣
∣
∣
∣
∂
∂tG(t, s)
∣
∣
∣
∣
ds ≤∫ t
0
(t − s)δ−2
Γ(δ − 1)ds +
|β − α|pΓ(δ)
∫ η
0
(η − s)δ−1ds +|1 − α|pΓ(δ)
∫ T
0
(T − s)δ−1ds
≤ tδ−1
δΓ(δ)+
|β − α|ηδ + |1 − α|T δ
pδΓ(δ)≤ T δ−1
δΓ(δ)+
|β − α|ηδ + |1 − α|T δ
pδΓ(δ).
Consequently,
|cDσ0+(Au)(t) − cDσ
0+(Au)(t)| ≤ k‖u − u‖t1−σ
(1 − σ)Γ(1 − σ)
(
T δ−1
δΓ(δ)+
|β − α|ηδ + |1 − α|T δ
pδΓ(δ)
)
≤ k‖u − u‖T 1−σ
δΓ(δ)Γ(2 − σ)
(
T δ−1 +|β − α|ηδ + |1 − α|T δ
p
)
≤ k‖u − u‖T 1−σ
Γ(2 − σ)
(
T δ−1
pδΓ(δ)+
(2|α| + |β|)ηδ
pδΓ(δ)+
(1 + |α|)T δ
pδΓ(δ)
)
=k‖u − u‖Γ(2 − σ)
(
kδT−σ + kδα,βηδT 1−σ + kδ
αT δ−σ+1)
.
Hence, it follows that ‖Au − Au‖ ≤ L‖u − u‖, where
L = k
(
kδα,β
(
T δ +ηδT 1−σ
Γ(2 − σ)
)
+ (kδαT δ+1 + kδ)
(
1 +T−σ
Γ(2 − σ)
))
.
No.4 M. Rehman et al: THREE POINT BOUNDARY VALUE PROBLEMS 1345
Clearly L < 1. Hence, by the contraction mapping principle, BVP (1.1) has a unique solution.
4 Examples
Example 4.1 Consider the following boundary value problem
cDδ0+u(t) =
Γ(δ + 1)
64√
πet +
Γ(δ + 1)
120e−κt(|u(t)|)θ1 +
e−πt cos t
187(|cDσ
0+u(t)|)θ2 , t ∈ [0, 2],
u(0) =3
7u(3
2
)
, u(1) =9
11u(3
2
)
,
(4.1)
where 1 < δ < 2, 0 < σ < 1 and κ > 0.
Choose α = 37 , β = 9
11 , η = 32 , φ(t) = Γ(δ+1)
64√
πet, c1 = Γ(δ+1)
120 , c2 = 1187 and
f(t, u, v) =Γ(δ + 1)
64√
πet +
Γ(δ + 1)
120e−κt(|u(t)|)θ1 +
e−πt cos t
187(|v(t)|)θ2 ,
then p = 58471078 , and for t ∈ [0, 2], we have
|f(t, u, v)| ≤ φ(t) + c1|u|θ1 + c1|v|θ2 .
For 0 < θ1, θ1 < 1, condition (H1) of Theorem 3.2 is satisfied and for θ1, θ1 > 1, condition (H2)
of Theorem 3.2 is satisfied. Therefore the boundary value problem (4.1) has a solution.
Example 4.2 Consider the following fractional BVP:
cD3
2
0+u(t) =e−πt(|u| + |cD
1
2
0+u(t)|)(24
√π + e−πt)(1 + |u| + |v|) , t ∈ [0, 1],
u(0) =5
7u(
1
4), u(1) =
9
7u(
1
4).
(4.2)
Set f(t, u, v) = e−πt(|u(t)|+|v(t)|)(24
√π+e−πt)(1+|u|+|v|) , t ∈ [0, 1], u, v ∈ [0,∞). For t ∈ [0, 1] and u, u, v, v ∈
[0,∞), we have
|f(t, u, v) − f(t, u, v)| =e−πt
(24√
π + e−πt)
∣
∣
∣
∣
u(t) + v(t)
1 + |u| + |v| −u(t) + v(t)
1 + |u| + |v|
∣
∣
∣
∣
≤ e−πt(|u(t) − u(t)| + |v(t) − v(t)|)(24
√π + e−πt)(1 + |u| + |v|)(1 + |u| + |v|)
≤ e−πt(|u(t) − u(t)| + |v(t) − v(t)|)24
√π + e−ct
≤ 1
24√
π(|u(t) − u(t)| + |v(t) − v(t)|).
For α = 57 , β = 9
7 , η = 14 , δ = 3
2 and σ = 12 , we have p = 3, kδ
α, β = 193√
π, kδ
α = 4√π
and kδα = 4√
π.
Therefore
L =
(
1
24√
π
)(
3(8√
π + 9)
π
)
=8√
π + 9
8π√
π< 1.
Hence by Theorem 3.3 the fractional BVP (4.2) has a unique solution.
1346 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
References
[1] Ahmad B, Nieto J J. Existence of solutions for nonlocal boundary value problems of higher order nonlinear
fractional differential equations. Abs Appl Anal, 2009, Article ID 494720
[2] Ahmad B, Nieto J J. Existence results for a coupled system of nonlinear fractional differential equations
with three-point boundary conditions. Comp and Math Appl, 2009, 58: 1838–1843
[3] Bai Z. On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Analysis, 2009,
72: 916–924
[4] Bai Z, Haishen L. Positive solutions for boundary-value problem of nonlinear fractional differential equa-
tion. J Math Anal Appl, 2005, 311: 495–505
[5] Bai C, Fang J. The existence of positive solution for singular coupled system of nonlinear fractional
equations. Appl Math Comput, 2004, 150: 611–621
[6] Benchohra M, Henderson J, Ntoyuas S K, Ouahab A. Existence results for fractional order functional
differential equations with infinite delay. J Math Anal Appl, 2008, 338: 1340–1350
[7] Delbosco D, Rodino I. Existence and uniqueness for a nonlinear fractional differential equation. J Math
Anal Appl, 1996, 204: 609–625
[8] El-Shahed M, Nieto J J. Nontrivial solutions for a nonlinear multi-point boundary value problem of frac-
tional order Comput Math Appl, 2010, 59: 3438–3443
[9] Eloe P W, Bashir Ahmad. Positive solutions of a nonlinear nth boundary value problems with nonlocal
conditions. Applied Math Let, 2005, 18: 521527
[10] Hilfer R, Ed. Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000
[11] Ibrahim R W, Momani S. On existence and uniqueness of solutions of a class of fractional differential
equations. J Math Anal Appl, 2007, 3334: 1–10
[12] Lakshmikantham V, Leela S. Nagumo-type uniqueness result for fractional differential equations. Nonlinear
Analysis, TMA, 2009, 71(7/8): 2886–2889
[13] Machando J T. Disrete time fractional-order controllers. Frac Cal App Anal, 2001, 4: 47–66
[14] Sabatier J, Agrawal O P, Machado Tenreiro J A. Advances in Fractional Calculus. Springer, 2007
[15] Miller K S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential equations. New
York: Wiley, 1993
[16] Nishimoto K. An Essence of Neshimoto’s Fractional Calculus. Koriyama: Descartes Press, 1991
[17] Nieto J J. Maximum principles for fractional differential equations derived from MittagLeffler functions.
Appl Math Lett, 2010, 23: 1248–1251
[18] Oldham K B, Spanier J. The Fractional Calculus. New York: Academic Press, 1974
[19] Podlubny I. Fractional Differential Equations. San Diedo: Academic Press, 1999
[20] Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation.
Frac Cal and App Anal, 2002, 4: 366–386
[21] Sabatier J, Agarwal O P, Ttenreiro Machado J A. Advances in Fractional Calculus, Theoretical Develop-
ments and Applications in Physics and Engineering. Springer, 2007
[22] Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives, Theory and Applications.
Amsterdam: Gordon and Breach, 1993
[23] El-Shahed M. Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation.
Abst Appl Analy, 2007, Article ID 10368
[24] Stojanovic M. Existence-uniqueness results for a nonlinear n-term fractional equation. J Math Anal Appl,
2009, 353: 244–255
[25] Su X, Zhang S. Solutions to boundary-value problems for nonlinear differential equations of fractional
order. Elec J Diff Equat, 2009, 26: 1–15
[26] Su X. Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl
Math Letters, 2009, 22: 64–69
[27] Zang S. Positive solutions for boundary-valve problems of nonlinear fractional differential equations. Elec
J Diff Equat, 2006, 36: 1–12
[28] Zhang S. Existence of solution for boundary value problem of fractional order. Acta Math Sci, 2006,
26B(2): 220–228
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