Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 915689, 17 pages
doi:10.1155/2011/915689
Research Article
Existence of Solutions to Anti-Periodic Boundary
Value Problem for Nonlinear Fractional Differential
Equations with Impulses
Anping Chen1, 2 and Yi Chen2
1
2
Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411005, China
Correspondence should be addressed to Anping Chen, chenap@263.net
Received 20 October 2010; Revised 25 December 2010; Accepted 20 January 2011
Academic Editor: Dumitru Baleanu
Copyright q 2011 A. Chen and Y. Chen. 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 discusses the existence of solutions to antiperiodic boundary value problem for
nonlinear impulsive fractional differential equations. By using Banach fixed point theorem,
Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some
existence results of solutions are obtained. An example is given to illustrate the main result.
1. Introduction
In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional
differential equations with impulses
C
Dα ut ft, ut,
Δu|ttk Ik utk ,
t ∈ 0, T , t / tk , k 1, 2, . . . , p,
Δu |ttk Jk utk ,
u0 uT 0,
k 1, 2, . . . , p,
1.1
u 0 u T 0,
where T is a positive constant, 1 < α ≤ 2, C Dα denotes the Caputo fractional derivative of
order α, f ∈ C0, T × R, R, Ik , Jk : R → R and {tk } satisfy that 0 t0 < t1 < t2 < · · · < tp <
tp1 T , Δu|ttk utk − ut−k , Δu |ttk u tk − u t−k , utk and ut−k represent the right
and left limits of ut at t tk .
Fractional differential equations have proved to be an excellent tool in the mathematic
modeling of many systems and processes in various fields of science and engineering.
Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control,
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Advances in Difference Equations
electromagnetic, porous media, and so forth. In consequence, the subject of fractional
differential equations is gaining much importance and attention see 1–6 and the references
therein.
The theory of impulsive differential equations has found its extensive applications
in realistic mathematic modeling of a wide variety of practical situations and has emerged
as an important area of investigation in recent years. For the general theory of impulsive
differential equations, we refer the reader to 7, 8. Recently, many authors are devoted to the
study of boundary value problems for impulsive differential equations of integer order, see
9–12.
Very recently, there are only a few papers about the nonlinear impulsive differential
equations and delayed differential equations of fractional order.
Agarwal et al. in 13 have established some sufficient conditions for the existence of
solutions for a class of initial value problems for impulsive fractional differential equations
involving the Caputo farctional derivative. Ahmad et al. in 14 have discussed some
existence results for the two-point boundary value problem involving nonlinear impulsive
hybrid differential equation of fractional order by means of contraction mapping principle
and Krasnoselskii’s fixed point theorem. By the similar way, they have also obtained the
existence results for integral boundary value problem of nonlinear impulsive fractional
differential equations see 15. Tian et al. in 16 have obtained some existence results for the
three-point impulsive boundary value problem involving fractional differential equations by
the means of fixed points method. Maraaba et al. in 17, 18 have established the existence and
uniqueness theorem for the delay differential equations with Caputo fractional derivatives.
Wang et al. in 19 have studied the existence and uniqueness of the mild solution for a
class of impulsive fractional differential equations with time-varying generating operators
and nonlocal conditions.
To the best of our knowledge, few papers exist in the literature devoted to the
antiperiodic boundary value problem for fractional differential equations with impulses. This
paper studies the existence of solutions of antiperiodic boundary value problem for fractional
differential equations with impulses.
The organization of this paper is as follows. In Section 2, we recall some definitions of
fractional integral and derivative and preliminary results which will be used in this paper. In
Section 3, we will consider the existence results for problem 1.1. We give three results, the
first one is based on Banach fixed theorem, the second one is based on Schaefer fixed point
theorem, and the third one is based on the nonlinear alternative of Leray-Schauder type. In
Section 4, we will give an example to illustrate the main result.
2. Preliminaries
In this section, we present some basic notations, definitions, and preliminary results which
will be used throughout this paper.
Definition 2.1 see 4. The Caputo fractional derivative of order α of a function f : 0, ∞ →
R is defined as
C
Dα ft 1
Γn − α
t
t − sn−α−1 f n sds,
0
where α denotes the integer part of the real number α.
n − 1 < α < n, n α 1,
2.1
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3
Definition 2.2 see 4. The Riemann-Liouville fractional integral of order α > 0 of a function
ft, t > 0, is defined as
I α ft 1
Γα
t
t − sα−1 fsds,
2.2
0
provided that the right side is pointwise defined on 0, ∞.
Definition 2.3 see 4. The Riemann-Liouville fractional derivative of order α > 0 of a
continuous function f : 0, ∞ → R is given by
n t
d
1
D ft t − sn−α−1 fsds,
Γn − α dt
0
α
2.3
where n α 1 and α denotes the integer part of real number α, provided that the right
side is pointwise defined on 0, ∞.
For the sake of convenience, we introduce the following notation.
Let J 0, T , J0 0, t1 , Ji ti , ti1 , i 1, 2, . . . , p−1, Jp tp , T . J J \{t1 , t2 , . . . , tp }.
We define PCJ {u : 0, T → R | u ∈ CJ , utk and ut−k exists, and ut−k utk , 1 ≤
k ≤ p}. Obviously, PCJ is a Banach space with the norm u supt∈J |ut|.
Definition 2.4. A function u ∈ PCJ is said to be a solution of 1.1 if u satisfies the equation
c α
D ut ft, ut for t ∈ J , the equations Δu|ttk Ik utk , Δu |ttk Jk utk , k 1, 2, . . . , p, and the condition u0 uT 0, u 0 u T 0.
Lemma 2.5 see 20. Let α > 0; then
I α C Dα ut ut c0 c1 t c2 t2 · · · cn−1 tn−1 ,
2.4
for some ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1.
Lemma 2.6 nonlinear alternative of Leray-Schauder type 21. Let E be a Banach space with
C ⊆ E closed and convex. Assume that U is a relatively open subset of C with 0 ∈ U and A : U → C
is continuous, compact map. Then either
1 A has a fixed point in U, or
2 there exists u ∈ ∂U and λ ∈ 0, 1 with u λAu.
Lemma 2.7 Schaefer fixed point theorem 22. Let S be a convex subset of a normed linear space
Ω and 0 ∈ S. Let F : S → S be a completely continuous operator, and let
ζF {u ∈ S : u λFu, for some 0 < λ < 1}.
Then either ζF is unbounded or F has a fixed point.
2.5
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Lemma 2.8. Assume that y ∈ C0, T , R, T > 0, 1 < α ≤ 2. A function u ∈ PCJ is a solution
of the antiperiodic boundary value problem
C
Dα ut yt,
Δu|ttk Ik utk ,
t ∈ 0, T , t /
tk , k 1, 2, . . . , p,
Δu |ttk Jk utk ,
k 1, 2, . . . , p,
2.6
u 0 u T 0,
u0 uT 0,
if and only if u is a solution of the integral equation
⎧
t
p1 ⎪
1 ti
1
α−1
⎪
⎪
t − s ysds −
ti − sα−1 ysds
⎪
⎪
Γα 0
2Γα i1 ti−1
⎪
⎪
⎪
⎪
⎪
ti
⎪
p
⎪
⎪
1
⎪
⎪
−
−
t
T
ti − sα−2 ysds
i
⎪
⎪
2Γα
−
1
⎪
ti−1
⎪
i1
⎪
⎪
⎪
⎪
p1
p
⎪
⎪
T − 2t ti
1
⎪
α−2
⎪
−
s
ysds
−
t
T − ti Ji uti ⎪
i
⎪
⎪
4Γα − 1 i1 ti−1
2 i1
⎪
⎪
⎪
⎪
⎪
p
p
⎪
⎪
T − 2t 1
⎪
⎪
J
Ii uti , t ∈ 0, t1 ,
−
ut
⎪
i
i
⎪
⎪
4 i1
2 i1
⎪
⎪
⎪
⎪
⎪
t
⎪
k ti
⎪
1 ⎪
α−1
⎪ 1
⎪
ysds
−
s
t
ti − sα−1 ysds
⎪
⎪
Γα
Γα
⎪
tk
i1 ti−1
⎪
⎪
⎪
⎪
⎨
p1
k
1 ti
1
ut −
ti − sα−1 ysds t − ti ⎪
2Γα i1 ti−1
Γα − 1 i1
⎪
⎪
⎪
⎪
⎪
ti
⎪
⎪
⎪
⎪
⎪
×
ti − sα−2 ysds
⎪
⎪
⎪
ti−1
⎪
⎪
⎪
⎪
ti
p
⎪
⎪
1
⎪
⎪
⎪
−
−
t
T
ti − sα−2 ysds
i
⎪
⎪
2Γα
−
1
⎪
t
i−1
i1
⎪
⎪
⎪
⎪
⎪
p1
t
k
⎪
i
⎪
T − 2t ⎪
α−2
⎪
−
s
ysds
t
t − ti Ji uti ⎪
i
⎪
⎪
4Γα − 1 i1 ti−1
⎪
i1
⎪
⎪
⎪
⎪
p
p
⎪
⎪
1
T − 2t ⎪
⎪
−
Ji uti −
t
T
J
ut
⎪
i i
i
⎪
2 i1
4 i1
⎪
⎪
⎪
⎪
⎪
⎪
p
k
⎪
⎪
1
⎪
⎪
Ii uti , t ∈ tk , tk1 , 1 ≤ k ≤ p.
⎩ Ii uti −
2 i1
i1
2.7
Proof. Assume that y satisfies 2.6. Using Lemma 2.5, for some constants c0 , c1 ∈ R, we have
1
ut I yt − c0 − c1 t Γα
α
t
0
t − sα−1 ysds − c0 − c1 t,
t ∈ 0, t1 .
2.8
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5
Then, we obtain
u t 1
Γα − 1
t
t − sα−2 ysds − c1 ,
t ∈ 0, t1 .
2.9
0
If t ∈ t1 , t2 , then we have
1
ut Γα
t
t − sα−1 ysds − d0 − d1 t − t1 ,
t1
1
u t Γα − 1
t
2.10
t − s
α−2
ysds − d1 ,
t1
where d0 , d1 ∈ R are arbitrary constants. Thus, we find that
u t−1 1
Γα
t1
t1 − sα−1 ysds − c0 − c1 t1 ,
0
u t1 −d0 ,
t1
1
−
u t1 t1 − sα−2 ysds − c1 ,
Γα − 1 0
u t1 −d1 .
2.11
In view of Δu|tt1 ut1 − ut−1 I1 ut1 and Δu |tt1 u t1 − u t−1 J1 ut1 , we have
−d0 1
Γα
t1
0
1
−d1 Γα − 1
t1 − sα−1 ysds − c0 − c1 t1 I1 ut1 ,
t1
2.12
t1 − sα−2 ysds − c1 J1 ut1 .
0
Hence, we obtain
ut 1
Γα
t
t − sα−1 ysds t1
t − t1
Γα − 1
t1
1
Γα
t1
t1 − sα−1 ysds
0
t1 − sα−2 ysds t − t1 J1 ut1 0
I1 ut1 − c0 − c1 t,
t ∈ t1 , t2 .
2.13
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Advances in Difference Equations
Repeating the process in this way, the solution ut for t ∈ tk , tk1 can be written as
ut 1
Γα
t
t − sα−1 ysds tk
k
1 Γα i1
ti
ti − sα−1 ysds
ti−1
ti
k
k
1
t − ti ti − sα−2 ysds t − ti Ji uti Γα − 1 i1
ti−1
i1
k
Ii uti − c0 − c1 t,
2.14
t ∈ tk , tk1 , k 1, 2, . . . , p.
i1
On the other hand, by 2.14, we have
1
uT Γα
T
1 ysds Γα i1
p
T − s
α−1
tp
ti
ti − sα−1 ysds
ti−1
ti
p
p
1
T − ti ti − sα−2 ysds T − ti Ji uti Γα − 1 i1
ti−1
i1
p
Ii uti − c0 − c1 T,
2.15
i1
1
u T Γα − 1
T
T − s
α−2
tp
p ti
1
ysds ti − sα−2 ysds
Γα − 1 i1 ti−1
p
Ji uti − c1 .
i1
By the boundary conditions u0 uT 0, u 0 u T 0, we obtain
p1 ti
1 c0 2Γα i1
ti−1
α−1
ti − s
ti
p
1
ysds T − ti ti − sα−2 ysds
2Γα − 1 i1
ti−1
p1 ti
p
T
T
α−2
Ji uti −
ti − s ysds −
4Γα − 1 i1 ti−1
4 i1
1
1
Ii uti ,
T − ti Ji uti 2 i1
2 i1
p
p
2.16
p1 ti
p
1
1
Ji uti .
c1 ti − sα−2 ysds 2Γα − 1 i1 ti−1
2 i1
Substituting the values of c0 and c1 into 2.8, 2.14, respectively, we obtain 2.7.
Conversely, we assume that u is a solution of the integral equation 2.7. By a
direct computation, it follows that the solution given by 2.7 satisfies 2.6. The proof is
completed.
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3. Main Result
In this section, our aim is to discuss the existence and uniqueness of solutions to the problem
1.1.
Theorem 3.1. Assume that
H1 there exists a constant L1 > 0 such that |ft, u − ft, v| ≤ L1 |u − v|, for each t ∈ J and
all u, v ∈ R;
H2 there exist constants L2 , L3 > 0 such that Ik u − Ik v ≤ L2 |u − v|, Jk u − Jk v ≤
L3 |u − v|, for each t ∈ J and all u, v ∈ R, k 1, 2, . . . , p.
If
3p 5 T α 7 p 1 T α
3
7T
L2 L3 < 1,
p
2Γα 1
4Γα
2
4
L1
3.1
then problem 1.1 has a unique solution on J.
Proof. We transform the problem 1.1 into a fixed point problem. Define an operator T :
PCJ → PCJ by
1
T ut Γα
t
t − s
α−1
tk
p1 ti
1 −
2Γα i1
1 tk
fs, usds tk − sα−1 fs, usds
Γα 0<tk <t tk−1
ti − sα−1 fs, usds
ti−1
tk
1
t − tk tk − sα−2 fs, usds
Γα − 1 0<tk <t
tk−1
−
ti
p
1
T − ti ti − sα−2 fs, usds
2Γα − 1 i1
ti−1
p1 T − 2t ti
ti − sα−2 fs, usds
4Γα − 1 i1 ti−1
t − tk Jk utk −
0<tk <t
1
Ii uti ,
2 i1
p
Ik utk −
0<tk <t
p
p
1
T − 2t Ji uti T − ti Ji uti 2 i1
4 i1
3.2
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where PCJ is with the norm u supt∈J |ut|. Let u, v ∈ PCJ; then for each t ∈ J, we
have
|T ut − T vt|
1
≤
Γα
t
t − sα−1 fs, us − fs, vsds
tk
1 tk
tk − sα−1 fs, us − fs, vsds
Γα 0<tk <t tk−1
p1 ti
1 2Γα i1
ti − sα−1 fs, us − fs, vsds
ti−1
tk
1
t − tk tk − sα−2 fs, us − fs, vsds
Γα − 1 0<tk <t
tk−1
ti
p
1
T − ti ti − sα−2 fs, us − fs, vsds
2Γα − 1 i1
ti−1
p1 |T − 2t| ti
ti − sα−2 fs, us − fs, vsds
4Γα − 1 i1 ti−1
1
T − ti |Ji uti − Ji vti |
2 i1
p
t − tk |Jk utk − Jk vtk | 0<tk <t
p
|T − 2t| |Ji uti − Ji vti | |Ik utk − Ik vtk |
4
0<tk <t
i1
1
|Ii uti − Ii vti |
2 i1
p
L1 u − v
≤
Γα
t
tk
t − s
α−1
p1 ti
3L1 u − v ds 2Γα i1
ti − sα−1 ds
ti−1
p1 3p
7T p
7T L1 u − v ti
L2 u − v L3 u − v
ti − sα−2 ds 4Γα − 1 i1 ti−1
2
4
3 p 1 T α L1
7 p 1 T α L1
T α L1
u − v u − v u − v
≤
Γα 1
2Γα 1
4Γα
3p
7T p
L2 u − v L3 u − v
2
4
3p 5 T α 7 p 1 T α
3
7T
p
L2 L3 u − v.
L1
2Γα 1
4Γα
2
4
3.3
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Therefore,
T u − T v ≤ L1
3p 5 T α 7 p 1 T α
7T
3
L2 L3 u − v.
p
2Γα 1
4Γα
2
4
3.4
Since
L1
3p 5 T α 7 p 1 T α
7T
3
L2 L3 < 1,
p
2Γα 1
4Γα
2
4
3.5
consequently T is a contraction; as a consequence of Banach fixed point theorem, we deduce
that T has a fixed point which is a solution of the problem 1.1.
Theorem 3.2. Assume that
H3 the function f : J × R → R is continuous and there exists a constant N1 > 0 such that
|ft, u| ≤ N1 for each t ∈ J and all u ∈ R;
H4 the functions Ik , Jk : R → R are continuous and there exist constants N2 , N3 > 0 such
that |Ik u| ≤ N2 , |Jk u| ≤ N3 , for all u ∈ R, k 1, 2, . . . , p.
Then the problem 1.1 has at least one solution on J.
Proof. We will use Schaefer fixed-point theorem to prove T has a fixed point. The proof will
be given in several steps.
Step 1. T is continuous.
Let {un } be a sequence such that un → u in PCJ; we have
|T un t − T ut|
t
1
≤
t − sα−1 fs, un s − fs, usds
Γα tk
1 tk
tk − sα−1 fs, un s − fs, usds
Γα 0<tk <t tk−1
p1 ti
1 2Γα i1
ti − sα−1 fs, un s − fs, usds
ti−1
tk
1
t − tk tk − sα−2 fs, un s − fs, usds
Γα − 1 0<tk <t
tk−1
ti
p
1
T − ti ti − sα−2 fs, un s − fs, usds
2Γα − 1 i1
ti−1
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Advances in Difference Equations
p1 |T − 2t| ti
ti − sα−2 fs, un s − fs, usds
4Γα − 1 i1 ti−1
0<tk <t
1
T − ti |Ji un ti − Ji uti |
2 i1
p
t − tk |Jk un tk − Jk utk | p
|T − 2t| |Ji un ti − Ji uti | |Ik un tk − Ik utk |
4
0<tk <t
i1
1
|Ii un ti − Ii uti |
2 i1
t
1
≤
t − sα−1 fs, un s − fs, usds
Γα tk
p1 3 ti
ti − sα−1 fs, un s − fs, usds
2Γα i1 ti−1
p
p1 ti
7T
ti − sα−2 fs, un s − fs, usds
4Γα − 1 i1 ti−1
3
7T |Ii un ti − Ii uti | |Ji un ti − Ji uti |.
2 i1
4 i1
p
p
3.6
Since f, I, J are continuous functions, then we have
T un − T u −→ 0,
n −→ ∞.
3.7
Step 2. T maps bounded sets into bounded sets in PCJ.
Indeed, it is enough to show that for any r > 0, there exists a positive constant L such
that, for each u ∈ Ωr {u ∈ PCJ : u ≤ r}, we have T u ≤ L. By H3 and H4, for each
t ∈ J, we can obtain
1
|T ut| ≤
Γα
t
tk
t − sα−1 fs, usds
1 tk
tk − sα−1 fs, usds
Γα 0<tk <t tk−1
p1 1 ti
ti − sα−1 fs, usds
2Γα i1 ti−1
tk
1
t − tk tk − sα−2 fs, usds
Γα − 1 0<tk <t
tk−1
p
t
i
1
T − ti ti − sα−2 fs, usds
2Γα − 1 i1
ti−1
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p1 |T − 2t| ti
ti − sα−2 fs, usds
4Γα − 1 i1 ti−1
0<tk <t
p
p
1
|T − 2t| |Ji uti | |Ik utk | |Ii uti |
4
2 i1
0<tk <t
i1
N1
≤
Γα
1
T − ti |Ji uti |
2 i1
p
t − tk |Jk utk | t
t − s
α−1
tk
p1 ti
3N1 ds 2Γα i1
ti − sα−1 ds
ti−1
p1 3p
7T p
7T N1 ti
N2 N3
ti − sα−2 ds 4Γα − 1 i1 ti−1
2
4
≤ N1
3p 5 T α 7 p 1 T α
3
7T
N2 N3 .
p
2Γα 1
4Γα
2
4
3.8
Therefore,
T u ≤ N1
3p 5 T α 7 p 1 T α
3
7T
: L.
N2 N3
p
2Γα 1
4Γα
2
4
3.9
Step 3. T maps bounded sets into equicontinuous sets in PCJ.
Let Ωr be a bounded set of PCJ as in Step 2, and let u ∈ Ωr . For each t ∈ J, we can
estimate the derivative T u t:
T u t ≤
1
Γα − 1
t
t − sα−2 fs, usds
tk
tk
1
tk − sα−2 fs, usds
Γα − 1 0<tk <t tk−1
p1 ti
1
ti − sα−2 fs, usds
2Γα − 1 i1 ti−1
1
|Ji uti |
2 i1
p
|Jk utk | 0<tk <t
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N1
Γα − 1
≤
t
t − sα−2 ds tk
p1 3p
3N1 ti
N3
ti − sα−2 ds 2Γα − 1 i1 ti−1
2
N1 T α−1 3 p 1 N1 T α−1 3p
N3
≤
Γα
2Γα
2
3p 5 T α−1
3p
N1 N3 : M.
2Γα
2
3.10
Hence, let t , t ∈ J, t < t ; we have
T u t − T u t t
t
T u sds ≤ M t − t .
3.11
So T Ωr is equicontinuous in PCJ. As a consequence of Steps 1 to 3 together with the
Arzela-Ascoli theorem, we can conclude that T : PCJ → PCJ is completely continuous.
Step 4. A priori bounds.
Now it remains to show that the set
ζT {u ∈ PCJ : u λT u for some 0 < λ < 1}
3.12
is bounded. Let u λT u for some 0 < λ < 1. Thus, for each t ∈ J, we have
λ
ut Γα
t
t − s
α−1
tk
p1 ti
λ −
2Γα i1
λ tk
fs, usds tk − sα−1 fs, usds
Γα 0<tk <t tk−1
ti − sα−1 fs, usds
ti−1
tk
λ
t − tk tk − sα−2 fs, usds
Γα − 1 0<tk <t
tk−1
−
ti
p
λ
T − ti ti − sα−2 fs, usds
2Γα − 1 i1
ti−1
p1 ti
λT − 2t 4Γα − 1 i1
λ
ti − sα−2 fs, usds
ti−1
t − tk Jk utk −
0<tk <t
λ
λ
Ii uti .
2 i1
p
Ik utk −
0<tk <t
p
p
λ
λT − 2t Ji uti T − ti Ji uti 2 i1
4
i1
3.13
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13
For each t ∈ J, by H3 and H4, we have
u ≤ N1
3p 5 T α 7 p 1 T α
3
7T
p
N2 N3 .
2Γα 1
4Γα
2
4
3.14
This shows that the set ζT is bounded. As a consequence of Schaefer fixed-point theorem,
we deduce that T has a fixed point which is a solution of the problem 1.1.
In the following theorem we give an existence result for the problem 1.1 by applying
the nonlinear alternative of Leray-Schauder type and by which the conditions H3 and H4
are weakened.
Theorem 3.3. Assume that H2 and the following conditions hold.
H5 There exists φ ∈ CJ and ψ : 0, ∞ → 0, ∞ continuous and nondecreasing such that
ft, u ≤ φtψ|u|,
t ∈ J, u ∈ R.
3.15
H6 There exist ψ ∗ , ψ ∗ : 0, ∞ → 0, ∞ continuous and nondecreasing such that
|Ik u| ≤ ψ ∗ |u|,
|Jk u| ≤ ψ ∗ |u|, u ∈ R.
3.16
H7 There exists a number M∗ > 0 such that
M∗
φ∗ ψM∗ 3p 5 T α /2Γα 17 p1 T α /4Γα p 3/2ψ ∗ M∗ 7T/4ψ ∗ M∗ > 1,
3.17
where φ∗ sup{φt : t ∈ J}.
Then 1.1 has at least one solution on J.
Proof. Consider the operator T defined in Theorem 3.1. It can be easily shown that T is
continuous and completely continuous. For λ ∈ 0, 1 and each t ∈ J, let u λT u. Then
from H5 and H6, and we have
|ut| ≤
1
Γα
t
t − sα−1 fs, usds
tk
1 tk
tk − sα−1 fs, usds
Γα 0<tk <t tk−1
p1 ti
1 2Γα i1
ti−1
ti − sα−1 fs, usds
14
Advances in Difference Equations
tk
1
t − tk tk − sα−2 fs, usds
Γα − 1 0<tk <t
tk−1
ti
p
1
T − ti ti − sα−2 fs, usds
2Γα − 1 i1
ti−1
p1 |T − 2t| ti
ti − sα−2 fs, usds
4Γα − 1 i1 ti−1
0<tk <t
1
T − ti |Ji uti |
2 i1
p
t − tk |Jk utk | p
p
1
|T − 2t| |Ji uti |
|Ik utk |
|Ii uti |
4
2 i1
0<tk <t
i1
1
≤
Γα
t
t − sα−1 φsψ|us|ds
tk
1 tk
tk − sα−1 φsψ|us|ds
Γα 0<tk <t tk−1
p1 ti
1 2Γα i1
ti − sα−1 φsψ|us|ds
ti−1
tk
1
t − tk tk − sα−2 φsψ|us|ds
Γα − 1 0<tk <t
tk−1
ti
p
1
T − ti ti − sα−2 φsψ|us|ds
2Γα − 1 i1
ti−1
p1 |T − 2t| ti
ti − sα−2 φsψ|us|ds
4Γα − 1 i1 ti−1
1
T − ti ψ ∗ |uti |
2 i1
p
t − tk ψ ∗ |utk | 0<tk <t
p
p
1 ∗
|T − 2t| ∗
ψ |uti | ψ ∗ |utk | ψ |uti |
4
2 i1
0<tk <t
i1
3 p 1 Tα
Tα
φ∗ ψu
≤ φ∗ ψu
Γα 1
2Γα 1
α
7 p1 T
7pT ∗
3p ∗
ψ u
φ∗ ψu
ψ u 4Γα
4
2
3p 5 T α 7 p 1 T α
3 ∗
7T ∗
∗
ψ u p
ψ u .
φ ψu
2Γα 1
4Γα
2
4
3.18
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15
Thus,
φ∗ ψu
u
α
α
3p 5 T /2Γα 17 p1 T /4Γα p 3/2ψ ∗ u7T/4ψ ∗ u
≤ 1.
3.19
Then by H7, there exists M∗ such that u /
M∗ . Let
U {u ∈ PCJ : u < M∗ }.
3.20
The operator T : U → PCJ is a continuous and completely continuous. From the choice of
U, there is no u ∈ ∂U such that u λT u for some 0 < λ < 1. As a consequence of the nonlinear
alternative of Leray-Schauder type, we deduce that T has a fixed point u in U which is a
solution of the problem 1.1. This completes the proof.
4. Example
Let α 3/2, T 2π, p 1. We consider the following boundary value problem:
C
D3/2 ut ft, ut,
1
Δu|t1/2 I u
,
2
u0 u2π 0,
1
0 ≤ t ≤ 2π, t /
,
2
1
Δu |t1/2 J u
,
2
4.1
u 0 u 2π 0,
where
ft, u cos tu
t 202 1 u
u
,
Iu 10 u
,
t, u ∈ J × 0, ∞,
u
Ju .
25 u
4.2
Obviously L1 1/400, L2 1/10, L3 1/25. Further,
3p 5 T α 7 p 1 T α
3
7T
p
L2 L3
L1
2Γα 1
4Γα
2
4
√
√
32 2
7π
1
3
π 14 2π < 1.
400
3
20 50
4.3
16
Advances in Difference Equations
Thus, all the assumptions of Theorem 3.1 are satisfied. Hence, by the conclusion of
Theorem 3.1, the impulsive fractional antiperiodic boundary value problem has a unique
solution on 0, 2π.
Acknowledgments
This work was supported by the Natural Science Foundation of China 10971173, the
Natural Science Foundation of Hunan Province 10JJ3096, the Aid Program for Science and
Technology Innovative Research Team in Higher Educational Institutions of Hunan Province,
and the Construct Program of the Key Discipline in Hunan Province.
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