Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 387629, 15 pages
doi:10.1155/2012/387629
Research Article
Some Existence Results for Impulsive
Nonlinear Fractional Differential Equations with
Closed Boundary Conditions
Hilmi Ergören1 and Adem Kiliçman2
1
2
Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia,
43400 Serdang, Malaysia
Correspondence should be addressed to Adem Kiliçman, akilicman@putra.upm.edu.my
Received 30 September 2012; Accepted 22 October 2012
Academic Editor: Beata Rzepka
Copyright q 2012 H. Ergören and A. Kiliçman. 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.
We investigate some existence results for the solutions to impulsive fractional differential
equations having closed boundary conditions. Our results are based on contracting mapping
principle and Burton-Kirk fixed point theorem.
1. Introduction
This paper considers the existence and uniqueness of the solutions to the closed boundary
value problem BVP, for the following impulsive fractional differential equation:
Dα xt ft, xt, t ∈ J : 0, T , t /
tk , 1 < α ≤ 2,
− Δxtk Ik x tk , Δx tk Ik∗ x t−k , k 1, 2, . . . , p,
C
xT ax0 bT x 0,
1.1
T x T cx0 dT x 0,
α
where C D is Caputo fractional derivative, f ∈ CJ × R, R, Ik , Ik∗ ∈ CR, R,
Δxtk x tk − x t−k ,
1.2
2
Abstract and Applied Analysis
with
x tk lim xtk h,
h→0
x t−k lim− xtk h,
h→0
1.3
and Δx tk has a similar meaning for x t, where
0 t0 < t1 < t2 < · · · < tp < tp1 T,
1.4
a, b, c, and d are real constants with Δ : c1 − b 1 − a1 − d /
0.
The boundary value problems for nonlinear fractional differential equations have been
addressed by several researchers during last decades. That is why, the fractional derivatives
serve an excellent tool for the description of hereditary properties of various materials and
processes. Actually, fractional differential equations arise in many engineering and scientific
disciplines such as, physics, chemistry, biology, electrochemistry, electromagnetic, control
theory, economics, signal and image processing, aerodynamics, and porous media see 1–
7. For some recent development, see, for example, 8–14.
On the other hand, theory of impulsive differential equations for integer order
has become important and found its extensive applications in mathematical modeling of
phenomena and practical situations in both physical and social sciences in recent years. One
can see a noticeable development in impulsive theory. For instance, for the general theory
and applications of impulsive differential equations we refer the readers to 15–17.
Moreover, boundary value problems for impulsive fractional differential equations
have been studied by some authors see 18–20 and references therein. However, to the
best of our knowledge, there is no study considering closed boundary value problems for
impulsive fractional differential equations.
Here, we notice that the closed boundary conditions in 1.1 include quasi-periodic
boundary conditions b c 0 and interpolate between periodic a d 1, b c 0 and
antiperiodic a d −1, b c 0 boundary conditions.
Motivated by the mentioned recent work above, in this study, we investigate the
existence and uniqueness of solutions to the closed boundary value problem for impulsive
fractional differential equation 1.1. Throughout this paper, in Section 2, we present some
notations and preliminary results about fractional calculus and differential equations to be
used in the following sections. In Section 3, we discuss some existence and uniqueness results
for solutions of BVP 1.1, that is, the first one is based on Banach’s fixed point theorem,
the second one is based on the Burton-Kirk fixed point theorem. At the end, we give an
illustrative example for our results.
2. Preliminaries
Let us set J0 0, t1 , J1 t1 , t2 , . . . , Jk−1 tk−1 , tk , Jk tk , tk1 , J : 0, T \ {t1 , t2 , . . . , tp }
and introduce the set of functions:
P CJ, R {x : J → R : x ∈ Ctk , tk1 , R, k 0, 1, 2, . . . , p and there exist xtk and
−
xtk , k 1, 2, . . . , p with xt−k xtk } and
P C1 J, R {x ∈ P CJ, R, x ∈ Ctk , tk1 , R, k 0, 1, 2, . . . , p and there exist x tk −
and x tk , k 1, 2, . . . , p with x t−k x tk } which is a Banach space with the norm x supt∈J {xP C , x P C } where xP C : sup{|xt| : t ∈ J}.
The following definitions and lemmas were given in 4.
Abstract and Applied Analysis
3
Definition 2.1. The fractional arbitrary order integral of the function h ∈ L1 J, R of order
α ∈ R is defined by
I0α ht
1
Γα
t
2.1
t − sα−1 hsds,
0
where Γ· is the Euler gamma function.
Definition 2.2. For a function h given on the interval J, Caputo fractional derivative of order
α > 0 is defined by
C
α
D0 ht
1
Γn − α
t
t − sn−α−1 hn sds,
n α 1,
2.2
0
where the function ht has absolutely continuous derivatives up to order n − 1.
Lemma 2.3. Let α > 0, then the differential equation
C
Dα ht 0
2.3
has solutions
ht c0 c1 t c2 t2 · · · cn−1 tn−1 ,
ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1.
2.4
The following lemma was given in 4, 10.
Lemma 2.4. Let α > 0, then
α
I αC D ht ht c0 c1 t c2 t2 · · · cn−1 tn−1 ,
2.5
for some ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1.
The following theorem is known as Burton-Kirk fixed point theorem and proved in
21.
Theorem 2.5. Let X be a Banach space and A, D : X → X two operators satisfying:
a A is a contraction, and
b D is completely continuous.
Then either
i the operator equation x Ax Dx has a solution, or
ii the set ε {x ∈ X : x λAx/λ λDx} is unbounded for λ ∈ 0, 1.
Theorem 2.6 see 22, Banach’s fixed point theorem. Let S be a nonempty closed subset of a
Banach space X, then any contraction mapping T of S into itself has a unique fixed point.
4
Abstract and Applied Analysis
Next we prove the following lemma.
Lemma 2.7. Let 1 < α ≤ 2 and let h : J → R be continuous. A function xt is a solution of the
fractional integral equation:
⎧ t
t − sα−1
⎪
⎪
hsds Ω1 tΛ1 Ω2 tΛ2 , t ∈ J0
⎪
⎪
⎪
Γα
⎪
0
⎪
⎪
⎪ t
⎪
α−1
k ti
⎪
ti − sα−1
⎪
⎪ t − s hsds 1 Ω1 t
hsds
⎪
⎪
⎪
Γα
Γα
⎪
tk
i1 ti−1
⎪
⎪
⎪
⎪
k ti
⎪
ti − sα−2
⎪
⎪
⎪
hsds
T
−
t
Ω
−
t
Ω
t
t
t
k
1
2
k
⎪
⎪
⎪
ti−1 Γα − 1
i1
⎪
⎪
⎪
ti
T
⎨
k−1
ti − sα−2
T − sα−1
xt hsds Ω1 t
hsds
1 Ω1 t
t
k − ti ⎪
Γα
⎪
tk
ti−1 Γα − 1
i1
⎪
⎪
⎪
T
⎪
k
⎪
− − ⎪
T − sα−2
∗
⎪
⎪
hsds 1 Ω1 t
Ω2 t
I i x ti I k x tk
⎪
⎪
⎪
tk Γα − 1
⎪
i1
⎪
⎪
⎪
k−1
k−1
⎪
− ⎪
⎪
∗
⎪ Ω1 t
Ii∗ x t−i
−
t
x
t
Ω
T
I
t
⎪
i
2
i
i
⎪
⎪
⎪
i1
i1
⎪
⎪
k−1
⎪
⎪
− ⎪
∗
⎪ t − ti Ii x ti , t ∈ Jk , k 1, 2, . . . , p
⎩
i1
2.6
if and only if xt is a solution of the fractional BVP
Dα xt ht, t ∈ J ,
Δxtk Ik x t−k ,
Δx tk Ik∗ x t−k ,
C
xT ax0 bT x 0,
T x T cx0 dT x 0,
where
Λ1 :
T
tk
k
T − sα−1
hsds
Γα
i1
k
i1
T − tk ti
ti−1
ti
ti−1
ti − sα−1
hsds
Γα
k−1
ti − sα−2
hsds
tk − ti Γα − 1
i1
ti
ti−1
k
k−1
Ii x t−i T − ti Ii∗ x t−i Ik∗ x t−k ,
i1
i1
ti − sα−2
hsds
Γα − 1
2.7
Abstract and Applied Analysis
Λ2 :
T
tk
5
k
T − sα−2
hsds
Γα − 1
i1
ti
Ω1 t : −
Ω2 t :
ti−1
k−1
ti − sα−2
hsds
Ii∗ x t−i ,
Γα − 1
i1
ct
1 − d
−
,
Δ
TΔ
1 − bT 1 − at
−
.
Δ
Δ
2.8
Proof. Let x be the solution of 2.7. If t ∈ J0 , then Lemma 2.4 implies that
xt I ht − c0 − c1 t α
x t t
0
t
0
t − sα−1
hsds − c0 − c1 t,
Γα
t − sα−2
hsds − c1 ,
Γα − 1
2.9
for some c0 , c1 ∈ R.
If t ∈ J1 , then Lemma 2.4 implies that
t
t − sα−1
hsds − d0 − d1 t − t1 ,
Γα
t1
t
t − sα−2
hsds − d1 ,
x t t1 Γα − 1
xt 2.10
for some d0 , d1 ∈ R. Thus we have
t1
t1 − sα−1
hsds − c0 − c1 t1 ,
x t1 −d0 ,
Γα
t0
t1
t1 − sα−2
hsds − c1 ,
x t1 −d1 .
x t−1 Γα
−
1
t0
x t−1 2.11
Observing that
Δxt1 x t1 − x t−1 I1 x t−1 ,
Δx t1 x t1 − x t−1 I1∗ x t−1 ,
2.12
6
Abstract and Applied Analysis
then we have
t1
t1 − sα−1
hsds − c0 − c1 t1 I1 x t−1 ,
Γα
t0
t1
t1 − sα−2
−d1 hsds − c1 I1∗ x t−1 ,
Γα
−
1
t0
−d0 2.13
hence, for t ∈ t1 , t2 ,
t1
t − sα−1
t1 − sα−1
hsds hsds
Γα
Γα
t1
t0
t1
t1 − sα−2
hsds I1 x t−1 t − t1 I1∗ x t−1 − c0 − c1 t,
t − t1 t0 Γα − 1
t1
t
t − sα−2
t1 − sα−2
hsds hsds I1∗ x t−1 − c1 .
x t t1 Γα − 1
t0 Γα − 1
xt t
2.14
If t ∈ J2 , then Lemma 2.4 implies that
t
t − sα−1
hsds − e0 − e1 t − t2 ,
Γα
t2
t
t − sα−2
hsds − e1 ,
x t t2 Γα − 1
xt 2.15
for some e0 , e1 ∈ R. Thus we have
x t−2 t2
t1
t2 − sα−1
t1 − sα−1
hsds hsds
Γα
Γα
t1
t0
t1
t1 − sα−2
hsds I1 x t−1 t2 − t1 I1∗ x t−1 − c0 − c1 t2 ,
t2 − t1 t0 Γα − 1
x t2 − e0 ,
t1
t2
t2 − sα−2
t1 − sα−2
−
hsds hsds I1∗ x t−1 − c1 ,
x t2 Γα
−
1
Γα
−
1
t1
t0
x t2 − e 1 .
2.16
Similarly we observe that
Δxt2 x t2 − x t−2 I2 x t−2 ,
Δx t2 x t2 − x t−2 I2∗ x t−2 ,
2.17
Abstract and Applied Analysis
7
thus we have
−e0 x t−2 I2 x t−2 ,
−e1 x t−2 I2∗ x t−2 .
2.18
Hence, for t ∈ t2 , t3 ,
xt t1
t2
t − sα−1
t1 − sα−1
t2 − sα−1
hsds hsds hsds
Γα
Γα
Γα
t2
t0
t1
t1
t1 − sα−2
hsds
t2 − t1 t0 Γα − 1
t
t2
1
t1 − sα−1
t2 − sα−1
t − t2 hsds hsds
t0 Γα − 1
t1 Γα − 1
t
2.19
I1 x t−1 I2 x t−2 t − t1 I1∗ x t−1 I2∗ x t−2 − c0 − c1 t.
By a similar process, if t ∈ Jk , then again from Lemma 2.4 we get
⎧
t
k ti
⎪
t − sα−1
ti − sα−1
⎪
⎪
hsds
hsds
⎪
⎪
⎪
Γα
Γα
⎪
tk
i1 ti−1
⎪
⎪
⎪
ti
ti
⎨
k
k−1
ti − sα−2
ti − sα−2
xt hsds
hsds
t − tk tk − ti ⎪
⎪
ti−1 Γα − 1
ti−1 Γα − 1
i1
i1
⎪
⎪
⎪
⎪
k
k−1
⎪
⎪
⎪
⎪ Ii x t−i t − ti Ii∗ x t−i Ik∗ x t−k − c0 − c1 t,
⎩
x t i1
t
tk
2.20
i1
k
t − sα−2
hsds
Γα − 1
i1
ti
ti−1
k−1
ti − sα−2
hsds
Ii∗ x t−i − c1 .
Γα − 1
i1
Now if we apply the conditions:
xT ax0 bT x 0,
T x T cx0 dT x 0,
2.21
we have
Λ1 1 − ac0 T 1 − bc1 ,
c
Λ2 − c0 1 − dc1 ,
T
−c0 −
1 − dΛ1 1 − bT Λ2
,
Δ
Δ
−c1 −
cΛ1 1 − aΛ2
−
.
TΔ
Δ
2.22
8
Abstract and Applied Analysis
In view of the relations 2.8, when the values of −c0 and −c1 are replaced in 2.9 and 2.20,
the integral equation 2.7 is obtained.
Conversely, assume that x satisfies the impulsive fractional integral equation 2.6,
then by direct computation, it can be seen that the solution given by 2.6 satisfies 2.7. The
proof is complete.
3. Main Results
Definition 3.1. A function x ∈ P C1 J, R with its α-derivative existing on J is said to be
α
a solution of 1.1, if x satisfies the equation C D xt ft, xt on J and satisfies the
conditions:
Δxtk Ik x t−k ,
Δx tk Ik∗ x t−k ,
xT ax0 bT x 0,
3.1
T x T cx0 dT x 0.
For the sake of convenience, we define
Ω∗1 sup|Ω1 t|,
Ω∗2 sup|Ω2 t|,
t∈J
t∈J
Ω∗∗
1 sup Ω1 t ,
t∈J
Ω∗∗
2 sup Ω2 t .
t∈J
3.2
The followings are main results of this paper.
Theorem 3.2. Assume that
A1 the function f : J × R → R is continuous and there exists a constant L1 > 0 such that
ft, u − ft, v ≤ L1 u − v, for all t ∈ J, and u, v ∈ R,
A2 Ik , Ik∗ : R → R are continuous, and there exist constants L2 > 0 and L3 > 0 such
that Ik u − Ik v ≤ L2 u − v, Ik∗ u − Ik∗ v ≤ L3 u − v for each u, v ∈ R and
k 1, 2, . . . , p.
Moreover, consider the following:
L1 T α−1 ∗ L1 T α Ω 1p
1 Ω∗1 1 p 2pα Γα 1
Γα 2
∗
∗
∗
1 Ω1 pL2 L3 Ω1 T Ω2 T pL3 < 1.
Then, BVP 1.1 has a unique solution on J.
Proof. Define an operator F : P C1 J, R → P C1 J, R by
Fxt t
tk
k
t − sα−1
fs, xsds 1 Ω1 t
Γα
i1
T − tk Ω1 t Ω2 t t − tk ti
ti−1
ti − sα−1
fs, xsds
Γα
k ti
ti − sα−2
fs, xsds
Γα − 1
i1 ti−1
3.3
Abstract and Applied Analysis
1 Ω1 t
k−1
9
tk − ti ti
ti−1
i1
Ω1 t
T
tk
Ω2 t
T
tk
Ω1 t
k−1
ti − sα−2
fs, xsds
Γα − 1
T − sα−1
fs, xsds
Γα
k
− − T − sα−2
∗
fs, xsds 1 Ω1 t
I i x ti I k x tk
Γα − 1
i1
k−1
k−1
Ii∗ x t−i T − ti Ii∗ x t−i Ω2 t
t − ti Ii∗ x t−i .
i1
i1
i1
3.4
Now, for x, y ∈ P CJ, R and for each t ∈ J, we obtain
Fxt − Fy t ≤
t
tk
t − sα−1 fs, xs − f s, ys ds
Γα
|1 Ω1 t|
k ti
ti − sα−1 fs, xs − f s, ys ds
Γα
i1 ti−1
|T − tk Ω1 t Ω2 t t − tk |
k ti
ti − sα−2 fs, xs − f s, ys ds
×
Γα − 1
i1 ti−1
|1 Ω1 t|
k−1
tk − ti ti
ti−1
i1
|Ω1 t|
T
tk
ti − sα−2 fs, xs − f s, ys ds
Γα − 1
T − sα−1 fs, xs − f s, ys ds
Γα
T
T − sα−2 fs, xs − f s, ys ds
tk Γα − 1
k − − ∗ − − ∗
Ii x t
|1 Ω1 t|
− Ii y t I x t
−I y t |Ω2 t|
i
i
i1
|Ω1 t|
k−1
T − ti Ii∗ x t−i − Ii∗ y t−i i1
|Ω2 t|
k−1 I ∗ x t− − I ∗ y t− i
i
i
i
i1
k−1
i1
|t − ti |Ii∗ x t−i − Ii∗ y t−i ,
k
k
k
k
10
Abstract and Applied Analysis
Fxt − Fy t ≤
L1 T α−1 ∗ L1 T α 1 Ω∗1 1 p 2pα Ω 1p
Γα 1
Γα 2
∗
∗
∗
1 Ω pL2 L3 Ω T Ω T pL3 xs − ys.
1
2
1
3.5
Therefore, by 3.3, the operator F is a contraction mapping. In a consequence of Banach’s
fixed theorem, the BVP 1.1 has a unique solution. Now, our second result relies on the
Burton-Kirk fixed point theorem.
Theorem 3.3. Assume that (A1)-(A2) hold, and
A3 there exist constants M1 > 0, M2 > 0, M3 > 0 such that ft, u ≤ M1 , Ik u ≤ M2 ,
Ik∗ u ≤ M3 for each u, v ∈ R and k 1, 2, . . . , p.
Then the BVP 1.1 has at least one solution on J.
Proof. We define the operators A, D : P C1 J, R → P C1 J, R by
k
− − ∗
I i x ti I k x tk
Axt 1 Ω1 t
i1
Ω1 t
k−1
k−1
k−1
Ii∗ x t−i T − ti Ii∗ x t−i Ω2 t
t − ti Ii∗ x t−i ,
i1
Dxt t
tk
i1
t − sα−1
fs, xsds 1 Ω1 t
Γα
T − tk Ω1 t Ω2 t t − tk k−1
tk − ti ti−1
i1
Ω1 t
T
tk
ti
ti−1
i1
k ti
i1
1 Ω1 t
i1
k ti
ti−1
ti − sα−1
fs, xsds
Γα
3.6
ti − sα−2
fs, xsds
Γα − 1
ti − sα−2
fs, xsds
Γα − 1
T − sα−1
fs, xsds Ω2 t
Γα
T
tk
T − sα−2
fs, xsds.
Γα − 1
It is obvious that A is contraction mapping for
1 Ω∗1
pL2 L3 Ω∗1 T Ω∗2 T pL3 < 1.
3.7
Now, in order to check that D is completely continuous, let us follow the sequence of the
following steps.
Abstract and Applied Analysis
11
Step 1 D is continuous. Let {xn } be a sequence such that xn → x in P CJ, R. Then for t ∈ J,
we have
|Dxn t − Dxt| ≤
t
tk
t − sα−1 fs, xn s − fs, xsds
Γα
k ti
ti − sα−1 |1 Ω1 t|
fs, xn s − fs, xsds
Γα
i1 ti−1
|T − tk Ω1 t Ω2 t t − tk |
k ti
ti − sα−2 fs, xn s − fs, xsds
×
Γα
−
1
i1 ti−1
k−1
|1 Ω1 t|
tk − ti ti−1
i1
|Ω1 t|
T
tk
|Ω2 t|
T
tk
ti
ti − sα−2 fs, xn s − fs, xsds
Γα − 1
T − sα−1 fs, xn s − fs, xsds
Γα
T − sα−2 fs, xn s − fs, xsds.
Γα − 1
3.8
Since f is continuous function, we get
Dxn t − Dxt −→ 0 as n −→ ∞.
3.9
Step 2 D maps bounded sets into bounded sets in P CJ, R. Indeed, it is enough to show
that for any r > 0, there exists a positive constant l such that for each x ∈ Br {x ∈ P CJ, R :
x ≤ r}, we have Dx ≤ l. By A3, we have for each t ∈ J,
|Dxt| ≤
t
tk
t − sα−1 fs, xsds
Γα
|1 Ω1 t|
k ti
ti − sα−1 fs, xsds
Γα
i1 ti−1
|T − tk Ω1 t Ω2 t t − tk |
|1 Ω1 t|
k−1
i1
tk − ti ti
ti−1
k ti
ti − sα−2 fs, xsds
Γα − 1
i1 ti−1
ti − sα−2 fs, xsds
Γα − 1
12
Abstract and Applied Analysis
|Ω1 t|
T
tk
Dx ≤
T − sα−1 fs, xsds |Ω2 t|
Γα
T
tk
T − sα−2 fs, xsds,
Γα − 1
M1 T α−1 ∗ M1 T α Ω2 1 p : l.
1 Ω∗1 1 p 2pα Γα 1
Γα
3.10
Step 3 D maps bounded sets into equicontinuous sets in P C1 J, R. Let τ1 , τ2 ∈ Jk , 0 ≤ k ≤ p
with τ1 < τ2 and let Br be a bounded set of P C1 J, R as in Step 2, and let x ∈ Br . Then
Dy τ2 − Dτ1 ≤
τ2 Dy sds ≤ Lτ2 − τ1 ,
3.11
τ1
where
Dx t ≤
t
tk
k
t − sα−2 fs, xsds Ω1 t
Γα − 1
i1
k
T − tk Ω1 t Ω2 t 1
i1
k−1
Ω1 t
tk − ti i1
Ω1 t
≤
T
tk
ti
ti−1
ti
ti−1
ti
ti−1
ti − sα−1 fs, xsds
Γα
ti − sα−2 fs, xsds
Γα − 1
ti − sα−2 fs, xsds
Γα − 1
T − sα−1 fs, xsds Ω2 t
Γα
T
tk
T − sα−2 fs, xsds,
Γα − 1
M1 T α−1 M1 T α ∗ Ω1 1 p 2pα 1 Ω∗2 1 p : L.
Γα 1
Γα
3.12
This implies that A is equicontinuous on all the subintervals Jk , k 0, 1, 2, . . . , p. Therefore,
by the Arzela-Ascoli Theorem, the operator D : P C1 J, R → P C1 J, R is completely
continuous.
To conclude the existence of a fixed point of the operator A D, it remains to show
that the set
x
λDx for some λ ∈ 0, 1
ε x ∈ X : x λA
λ
3.13
is bounded.
Let x ∈ ε, for each t ∈ J,
x
xt λDxt λA
λ
t.
3.14
Abstract and Applied Analysis
13
Hence, from A3, we have
|xt| ≤ λ
t
tk
k
t − sα−1 fs, xsds λ|1 Ω1 t|
Γα
i1
λ|T − tk Ω1 t Ω2 t t − tk |
k ti
i1
λ|1 Ω1 t|
k−1
tk − ti ti
ti−1
i1
ti−1
ti
ti−1
ti − sα−1 fs, xsds
Γα
ti − sα−2 fs, xsds
Γα − 1
ti − sα−2 fs, xsds
Γα − 1
T
T
T − sα−1 T − sα−2 λ|Ω1 t|
fs, xs ds λ|Ω2 t|
fs, xsds
Γα
Γα
−
1
tk
tk
− − k x ti
∗ x tk
λ|1 Ω1 t|
Ii
Ik
3.15
λ
λ
i1
λ|Ω1 t|
k−1
i1
− x ti
T − ti Ii∗
λ
− − k−1 k−1
x ti
∗ x ti
λ|Ω2 t|
t − ti Ii∗
Ii
λ
,
λ
λ
i1
i1
xt ≤
M1 T α−1 ∗ M1 T α Ω2 1 p
1 Ω∗1 1 p 2pα Γα 1
Γα
∗
∗
∗
1 Ω1 pM2 M3 Ω1 T Ω2 T pM3 .
Consequently, we conclude the result of our theorem based on the Burton-Kirk fixed point
theorem.
4. An Example
Consider the following impulsive fractional boundary value problem:
sin 2t|xt|
1
t ∈ 0, 1, t /
,
3
t 5 1 |xt|
1
|x1− /3|
|x 1− /3|
1
Δx
,
Δx
,
−
3
15 |x1 /3|
3
10 |x 1− /3|
C
D
3/2
xt 2
x1 5x0 − 2x 0,
,
x 1 x0 4x 0.
4.1
14
Abstract and Applied Analysis
Here, a 5, b −2, c 1, d 4, α 3/2, T 1, p 1. Obviously, L1 1/25, L2 1/15,
L3 1/10, Ω∗1 4/15, Ω∗2 7/15. Further,
L1 T α−1 ∗ L1 T α Ω2 1 p Ω∗1 T Ω∗2 T pL3
1 Ω∗1 1 p1 L2 2pα L3 Γα 1
Γα
4.2
464
173
< 1.
√ 1125 π 450
Since the assumptions of Theorem 3.2 are satisfied, the closed boundary value problem 4.1
has a unique solution on 0, 1. Moreover, it is easy to check the conclusion of Theorem 3.3.
Acknowledgments
The authors would like to express their sincere thanks and gratitude to the reviewers for
their valuable comments and suggestions for the improvement of this paper. The second
author also gratefully acknowledges that this research was partially supported by the
University Putra Malaysia under the Research University Grant Scheme 05-01-09-0720RU.
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