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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 404917, 16 pages
doi:10.1155/2011/404917
Research Article
Nonlocal Boundary Value Problem for Impulsive
Differential Equations of Fractional Order
Liu Yang
1, 2
and Haibo Chen
1
1
Department of Mathematics, Central South University, Changsha, Hunan 410075, China
2
Department of Mathematics and Computational Science, Hengyang Normal University,
Hengyang, Hunan 421008, China
Correspondence should be addressed to Liu Yang,
Received 18 September 2010; Accepted 4 January 2011
Academic Editor: Mouffak Benchohra
Copyright q 2011 L. Yang and H. 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.
We study a nonlocal boundary value problem of impulsive fractional differential equations. By
means of a fixed point theorem due to O’Regan, we establish sufficient conditions for the existence
of at least one solution of the problem. For the illustration of the main result, an example is given.
1. Introduction
Fractional differential equations a rise in many engineering and scientific disciplines as
the mathematical modeling of systems and processes in various fields, such as physics,
mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves
derivatives of fractional order. Fractional differential equations also provide an excellent tool
for the description of memory and hereditary properties of many materials and processes. In
consequence, fractional differential equations have emerged as a significant development in
recent years, see 1–3.
As one of the important topics in the research differential equations, the boundary
value problem has attained a great deal of attention from many researchers, see 4–11 and the
references therein. As pointed out in 12, the nonlocal boundary condition can be more use-
ful than the standard condition to describe some physical phenomena. There are three note-
worthy papers dealing with the nonlocal boundary value problem of fractional differential
equations. Benchohra et al. 12 investigated the following nonlocal boundary value problem
c
D
α
u
t
f
t, u
t
0, 0 <t<T, 1 <α≤ 2,
u
0
g
u
,u
T
u
T
,
1.1
where
c
D
α
denotes the Caputo’s fractional derivative.
2AdvancesinDifference Equations
Zhong and Lin 13 studied the following nonlocal and multiple-point boundary value
problem
c
D
α
u
t
f
t, u
t
0, 0 <t<1, 1 <α≤ 2,
u
0
u
0
g
u
,u
1
u
1
m−2
i1
b
i
u
ξ
i
.
1.2
Ahmad and Sivasundaram 14 studied a class of four-point nonlocal boundary value
problem of nonlinear integrodifferential equations of fractional order by applying some fixed
point theorems.
On the other hand, impulsive differential equations of fractional order play an
important role in theory and applications, see the references 15–21 and references
therein. However, as pointed out in 15, 16, the theory of boundary value problems for
nonlinear impulsive fractional differential equations is still in the initial stages. Ahmad and
Sivasundaram 15, 16 studied the following impulsive hybrid boundary value problems for
fractional differential equations, respectively,
c
D
q
u
t
f
t, u
t
0, 1 <q≤ 2,t∈ J
1
0, 1
\
t
1
,t
2
, ,t
p
,
Δu
t
k
I
k
u
t
−
k
, Δu
t
k
J
k
u
t
−
k
,t
k
∈
0, 1
,k 1, 2, ,p,
u
0
u
0
0,u
1
u
1
0,
1.3
c
D
q
u
t
f
t, u
t
0, 1 <q≤ 2,t∈ J
1
0, 1
\
t
1
,t
2
, ,t
p
,
Δu
t
k
I
k
u
t
−
k
, Δu
t
k
J
k
u
t
−
k
,t
k
∈
0, 1
,k 1, 2, ,p,
αu
0
βu
0
1
0
q
1
u
s
ds, αu
1
βu
1
1
0
q
2
u
s
ds.
1.4
Motivated by the facts mentioned above, in this paper, we consider the following
problem:
c
D
q
u
t
f
t, u
t
,u
t
, 1 <q≤ 2,t∈J
1
0, 1
\
t
1
,t
2
, ,t
p
,
Δu
t
k
I
k
u
t
−
k
, Δu
t
k
J
k
u
t
−
k
,t
k
∈
0, 1
,k 1, 2, ,p,
αu
0
βu
0
g
1
u
,αu
1
βu
1
g
2
u
,
1.5
where J 0, 1, f : J ×
× → is a continuous function, and I
k
,J
k
: →
are continuous functions, Δut
k
ut
k
− ut
−
k
with ut
k
lim
h →0
ut
k
h,ut
−
k
lim
h →0
−
ut
k
h,k 1, 2, ,p, 0 t
0
<t
1
<t
2
< ··· <t
p
<t
p1
1,α>0,β≥ 0, and
g
1
,g
2
:PCJ, → are two continuous functions. We will define PCJ, in Section 2.
To the best of our knowledge, this is the first time in the literatures that a nonlocal
boundary value problem of impulsive differential equations of fractional order is considered.
Advances in Difference Equations 3
In addition, the nonlinear term ft, ut,u
t involves u
t.Evidently,problem1.5 not
only includes boundary value problems mentioned above but also extends them to a much
wider case. Our main tools are the fixed point theorem of O’Regan. Some recent results in the
literatures are generalized and significantly improved see Remark 3.6
The organization of this paper is as follows. In Section 2, we will give some lemmas
which are essential to prove our main results. In Section 3, main results are given, and an
example is presented to illustrate our main results.
2. Preliminaries
At first, we present here the necessary definitions for fractional calculus theory. These
definitions and properties can be found in recent literature.
Definition 2.1 see 1–3. The Riemann-Liouville fractional integral of or der α>0ofa
function y : 0, ∞ →
is given by
I
α
0
y
t
1
Γ
α
t
0
t − s
α−1
y
s
ds, 2.1
where the right side is pointwise defined on 0, ∞.
Definition 2.2 see 1–3. The Caputo fractional derivative of order α>0ofafunctiony :
0, ∞ →
is given by
c
D
α
u
t
1
Γ
n −α
t
0
t − s
n−α−1
y
n
s
ds, 2.2
where n α1, α denotes the integer part of the number α, and the right side is pointwise
defined on 0, ∞.
Lemma 2.3 see 1–3. Let α>0, then the fractional differential equation
c
D
q
ut0 has
solutions
u
t
c
0
c
1
t c
2
t
2
··· c
n−1
t
n−1
,
2.3
where c
i
∈ ,i 0, 1, ,n− 1,nq1.
Lemma 2.4 see 1–3. Let α>0, then one has
I
α
0
c
D
α
u
t
u
t
c
0
c
1
t c
2
t
2
··· c
n−1
t
n−1
,
2.4
where c
i
∈ ,i 0, 1, ,n− 1,nq1.
Second, we define
PCJ,
{x : J → ; x ∈ Ct
k
,t
k1
, ,k 0, 1, ,p 1andxt
k
,xt
−
k
exist
with xt
−
k
xt
k
,k 1, ,p}.
4AdvancesinDifference Equations
PC
1
J, {x ∈ PCJ, ; x
t ∈ Ct
k
,t
k1
, , k 0, 1, ,p 1, x
t
k
,
x
t
−
k
exist, and x
is left continuous at t
k
,k 1, ,p}.LetC PC
1
J, ;it
is a Banach space with the norm x sup
t∈J
{xt
PC
, x
t
PC
},wherex
PC
sup
t∈J
|xt|.
Like Definition 2.1 in 16, we give the following definition.
Definition 2.5. Afunctionu ∈Cwith its Caputo derivative of order q existing on J
1
is a
solution of 1.5 if it satisfies 1.5.
To deal with problem 1.5, we first consider the associated linear problem and give its
solution.
Lemma 2.6. Assume that
J
i
⎧
⎨
⎩
t
0
,t
1
,i 0,
t
i
,t
i1
,i 1, 2, ,p,
X
t
⎧
⎨
⎩
0,t∈ J
0
,
1,t
∈ J
0
.
2.5
For any σ ∈ C0, 1, the solution of the problem
c
D
q
u
t
σ
t
, 1 <q≤ 2,t∈J
1
0, 1
\
t
1
,t
2
, ,t
p
,
Δu
t
k
I
k
u
t
−
k
, Δu
t
k
J
k
u
t
−
k
,t
k
∈
0, 1
,k 1, 2, ,p,
αu
0
βu
0
g
1
u
,αu
1
βu
1
g
2
u
2.6
is given by
u
t
t
t
i
t −s
q−1
σ
s
Γ
q
ds
β
α
− t
1
t
p
1 −s
q−1
σ
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
σ
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
σ
s
Γ
q
ds I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
σ
s
Γ
q −1
ds J
k
u
t
−
k
Advances in Difference Equations 5
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−1
σ
s
Γ
q
ds I
k
u
t
−
k
X
t
0<t
k
<t
t −t
k
t
k
t
k−1
t
k
− s
q−2
σ
s
Γ
q − 1
ds J
k
u
t
−
k
1
α
2
αg
1
u
X
t
αt − β
g
2
u
− g
1
u
, for t ∈ J
i
,i 0, 1, ,p.
2.7
Proof. By Lemmas 2.3 and 2.4, the solution of 2.6 can be written as
u
t
I
q
0
σ
t
− b
0
− b
1
t
t
0
t −s
q−1
Γ
q
σ
s
ds − b
0
− b
1
t, t ∈
0,t
1
,
2.8
where b
0
,b
1
∈ . Taking into account that
c
D
q
I
q
0
utut,I
q
0
I
p
0
utI
pq
0
ut for p, q > 0,
we obtain
u
t
t
0
t −s
q−2
σ
s
Γ
q −1
ds − b
1
.
2.9
Using αu0βu
0g
1
u,weget
u
t
t
0
t −s
q−1
Γ
q
σ
s
ds b
1
β
α
− t
1
α
g
1
u
,t∈
0,t
1
.
2.10
If t ∈ t
1
,t
2
,thenwehave
u
t
t
t
1
t − s
q−1
σ
s
Γ
q
ds − c
0
− c
1
t − t
1
,
2.11
where c
0
,c
1
∈ . In view of the impulse conditions Δut
1
ut
1
−ut
−
1
I
1
ut
−
1
, Δu
t
1
u
t
1
− u
t
−
1
J
1
ut
−
1
,wehave
u
t
t
t
1
t − s
q−1
σ
s
Γ
q
ds
t
1
0
t
1
− s
q−1
σ
s
Γ
q
ds b
1
β
α
− t
1
α
g
1
u
I
1
u
t
−
1
t −t
1
t
1
0
t
1
− s
q−2
σ
s
Γ
q −1
ds J
1
u
t
−
1
,t∈
t
1
,t
2
.
2.12
6AdvancesinDifference Equations
Repeating the process in this way, the solution ut for t ∈ t
k
,t
k1
can be written as
u
t
t
t
k
t −s
q−1
σ
s
Γ
q
ds b
1
β
α
− t
1
α
g
1
u
0<t
k
<t
t
k
t
k−1
t
k
− s
q−1
σ
s
Γ
q
ds I
k
u
t
−
k
0<t
k
<t
t − t
k
t
k
t
k−1
t
k
− s
q−2
σ
s
Γ
q −1
ds J
k
u
t
−
k
,t∈
t
k
,t
k1
.
2.13
Applying the boundary condition αu1βu
1g
2
u,wefindthat
b
1
1
t
p
1 −s
q−1
σ
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
σ
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
σ
s
Γ
q
ds I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
σ
s
Γ
q −1
ds J
k
u
t
−
k
1
α
g
1
u
− g
2
u
.
2.14
Substituting the value of b
1
into 2.10 and 2.13,weobtain2.7.
Now, we introduce the fixed point theorem which was established by O’Regan in 22.
This theorem will be applied to prove our main results in the next section.
Lemma 2.7 see 13, 22. Denote by U an open set in a closed, convex set Y of a Banach space E.
Assume that 0 ∈U. Also assume that F
U is bounded and that F : U→Y is given by F F
1
F
2
,
in which F
1
: U→E is continuous and com pletely continuous and F
2
: U→E is a nonlinear
contraction (i.e., there exists a nonnegative nondecreasing function φ : 0, ∞ → 0, ∞ satisfying
φz <zfor z>0,suchthatF
2
x − F
2
y≤φx −y for all x, y ∈ U, then either
C
1
F has a fixed point u ∈ U,or
C
2
there exists a point u ∈ ∂U and λ ∈ 0, 1 with u λFu,whereU,∂U represent the
closure and boundary of U, respectively.
Advances in Difference Equations 7
3. Main Results
In order to a pply Lemma 2.7 to prove our main results, we first give F, F
1
, F
2
as follows. Let
Ω
r
{u ∈C: u≤r},r>0,
F
1
u
t
t
t
i
t − s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
− t
1
t
p
1 −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
dsI
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q − 1
dsJ
k
u
t
−
k
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds I
k
u
t
−
k
X
t
0<t
k
<t
t − t
k
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds J
k
u
t
−
k
,
for t ∈ J
i
,i 0, 1, ,p,
F
2
u
t
1
α
2
αg
1
u
X
t
αt − β
g
2
u
− g
1
u
, for t ∈ J
i
,i 0, 1, ,p,
F F
1
F
2
.
3.1
Clearly, for any t ∈ J
i
,i 0, 1, ,p,
F
1
u
t
t
t
i
t −s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
−
1
t
p
1 −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
−s
q−2
f
s, x
s
,x
s
Γ
q −1
ds J
k
u
t
−
k
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds J
k
u
t
−
k
,
F
2
u
t
1
α
X
t
g
2
u
− g
1
u
.
3.2
Now, we make the following hypotheses.
8AdvancesinDifference Equations
A
1
f : 0, 1 × × → is continuous. There exists a nonnegative function pt ∈
C0, 1 with pt > 0 on a subinterval of 0, 1. Also there exists a nondecreasing
function ψ : 0, ∞ → 0, ∞ such that |ft, u, v|≤ptψ|u| for any t, u, v ∈
0, 1 ×
× .
A
2
There exist two positive constants l
1
,l
2
such that α β/α
2
l
1
l
2
L<1.
Moreover, g
1
00,g
2
00, and
g
1
u
− g
1
v
≤ l
1
u −v
,
g
2
u
− g
2
v
≤ l
2
u −v
, ∀u, v ∈C. 3.3
A
3
I
k
,J
k
: → are continuous. There exists a positive constant M such that
|
I
k
u
|
≤ M,
|
J
k
u
|
≤ M, k 1, 2, ,p. 3.4
Let
H
1
β
α
1
Mp
β
α
1
2
Mp 2pM,
H
2
Mp
β
α
1
Mp pM,
K
1
1
0
1 −s
q−1
p
s
Γ
q
ds
β
α
1
1
t
p
1 −s
q−1
p
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
p
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
0<t
k
<1
β
α
1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q − 1
ds,
K
2
P
Γ
q
1
t
p
1 −s
q−1
p
s
Γ
q
ds
β
α
1
t
p
1 − s
q−2
p
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
0<t
k
<1
β
α
1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q −1
ds,
3.5
where P max
s∈0,1
ps.
Now, we state our main results.
Advances in Difference Equations 9
Theorem 3.1. Assume that A
1
, A
2
,andA
3
are satisfied; moreover, sup
r∈0,∞
r/H
Kψr > 1/1 −L,whereH max{H
1
,H
2
},K max{K
1
,K
2
}, then the problem 1.5 has at
least one solution.
Proof. The proof will be given in several steps.
Step 1. The operator F
1
: Ω
r
→Cis completely continuous.
Let M
r
max
s∈0,1
{|fs, xs,x
s|,x∈ Ω
r
}.Infact,byA
1
, M
r
can be replaced by
Pψr.Foranyu ∈
Ω
r
,wehave
|
F
1
u
t
|
≤
t
t
i
t −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
1
t
p
1 −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
×
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
J
k
u
t
−
k
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
I
k
u
t
−
k
X
t
0<t
k
<t
t −t
k
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
J
k
u
t
−
k
≤ M
r
1
0
1 −s
q−1
Γ
q
ds
β
α
1
M
r
1
t
p
1 −s
q−1
Γ
q
ds
β
α
M
r
1
t
p
1 −s
q−2
Γ
q −1
ds
0<t
k
<1
M
r
t
k
t
k−1
t
k
− s
q−1
Γ
q
ds M
0<t
k
<1
β
α
1
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q −1
ds M
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
M
r
Γ
q
ds M
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q −1
ds M
10 Advances in Difference Equations
≤ M
r
1
Γ
q 1
β
α
1
M
r
1
Γ
q 1
β
α
M
r
1
Γ
q
p
M
r
1
Γ
q 1
M
p
β
α
1
M
r
1
Γ
q
M
p
M
r
1
Γ
q 1
M
p
M
r
1
Γ
q
M
, for t ∈ J
i
,i 0, 1, ,p,
F
1
u
t
≤
t
t
i
t −s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
1
t
p
1 −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
J
k
u
t
−
k
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
J
k
u
t
−
k
≤ M
r
1
Γ
q
1
t
p
1 −s
q−1
M
r
Γ
q
ds
β
α
1
t
p
1 −s
q−2
M
r
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
M
r
Γ
q
ds M
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q −1
ds M
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q −1
ds M
≤ M
r
1
Γ
q
M
r
1
Γ
q 1
β
α
M
r
1
Γ
q
p
M
r
1
Γ
q 1
M
p
β
α
1
M
r
1
Γ
q
M
p
M
r
1
Γ
q
M
, for t ∈ J
i
,i 0, 1, ,p.
3.6
Advances in Difference Equations 11
These imply that F
1
ut≤B,whereB is a positive constant, that is, F
1
is uniformly
bounded. In addition, for any u ∈
Ω
r
,forall τ
1
,τ
2
∈ J
i
,τ
1
<τ
2
,wecanobtain
|
F
1
u
τ
1
−
F
1
u
τ
2
|
≤
τ
2
τ
1
τ
2
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
τ
1
0
τ
2
− s
q−1
−
τ
1
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
τ
2
− τ
1
1
t
p
1 −s
q−1
f
s, x
s
,x
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
f
s, x
s
,x
s
Γ
q
ds
I
k
u
t
−
k
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
J
k
u
t
−
k
0<t
k
<τ
1
τ
2
− τ
1
t
k
t
k−1
t
k
− s
q−2
f
s, x
s
,x
s
Γ
q −1
ds
J
k
u
t
−
k
≤ M
r
τ
2
− τ
1
q
Γ
q 1
M
r
−
τ
2
− τ
1
q
τ
2
q
−
τ
1
q
1
Γ
q 1
τ
2
− τ
1
1
t
p
1 −s
q−1
M
r
Γ
q
ds
β
α
1
t
p
1 −s
q−2
M
r
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
M
r
Γ
q
ds M
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q − 1
ds M
0<t
k
<τ
1
τ
2
− τ
1
t
k
t
k−1
t
k
− s
q−2
M
r
Γ
q − 1
ds M
,
F
1
u
τ
1
−
F
1
u
τ
2
≤
τ
2
τ
1
τ
2
− s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
τ
1
t
i
τ
1
− s
q−2
−
τ
2
− s
q−2
f
s, x
s
,x
s
Γ
q − 1
ds
≤
τ
2
− τ
1
q−1
Γ
q
M
r
τ
2
− τ
1
q−1
τ
1
− t
i
q−1
−
τ
2
− t
i
q−1
Γ
q
M
r
.
3.7
12 Advances in Difference Equations
Taking into account the uniform continuity of the function t
q
,t
q−1
on 0, 1,wegetthatF
1
is
equicontinuity on
Ω
r
. By the Lemma 5.4.1 in 23,wehaveF
1
Ω
r
as relatively compact. Due
to the continuity of f, I
k
,J
k
, it is clear that F
1
is continuous. Hence, we complete the proof of
Step 1.
Step 2. F
U is bounded.
From sup
r∈0,∞
r/H Kψr > 1/1 − L, it follows that there exists a positive
constant r
0
,suchthat
r
0
H Kψ
r
0
>
1
1 −L
.
3.8
Now, we verify the validity of all the conditions in Lemma 2.6 with respect to the operator
F
1
,F
2
,andF.LetΩ
r
0
U.FromA
2
,wehave
|
F
2
u
t
|
≤
1
α
2
α
1 −t
β
g
1
u
− g
1
0
αt − β
g
2
u
− g
2
0
≤
1
α
2
α β
l
1
r
0
l
2
r
0
, for t ∈ J
i
,
F
2
u
t
1
α
l
2
r
0
l
1
r
0
, for t ∈ J
i
,i 0, 1, ,p.
3.9
Combining with the property that F
1
U is bounded Step 1,wehaveF bounded on U.
Hence, we can assume that F
U≤G, G>0 is a constant.
Step 3. F
2
is a nonlinear contraction.
Let Y
Ω
r
1
,r
1
max{G, r
0
},E C.ByA
2
,weobtain|F
2
ut − F
2
vt|≤
1/α
2
|α1 − tβg
1
u − g
1
v| |αt −βg
2
u − g
2
v| ≤ α β/α
2
l
1
l
2
u − v,
and |F
2
u
t − F
2
v
t|≤1/αl
1
l
2
u − v≤Lu − v, for t ∈ J
i
.SinceL<1, we have
F
2
u −F
2
v≤φu −v,thatis,F
2
is a nonlinear contraction φzLz.
Step 4. C
2
in Lemma 2.7 does not occur.
To this end, we perform the argument by contradiction. Suppose that C
2
holds, then
there exist λ ∈ 0, 1,u∈ ∂Ω
r
0
,suchthatu λFu.Hence,wecanobtainu r
0
and
|
u
|
≤
t
t
i
t −s
q−1
p
s
ψ
r
0
Γ
q
ds
β
α
1
1
t
p
1 −s
q−1
p
s
ψ
r
0
Γ
q
ds
β
α
1
t
p
1 −s
q−2
p
s
ψ
r
0
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
ψ
r
0
Γ
q
ds M
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
p
s
ψ
r
0
Γ
q − 1
ds M
Advances in Difference Equations 13
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−1
p
s
ψ
r
0
Γ
q
ds M
X
t
0<t
k
<t
t − t
k
t
k
t
k−1
t
k
− s
q−2
p
s
ψ
r
0
Γ
q − 1
ds M
1
α
2
α β
l
1
r
0
l
2
r
0
≤
1
α
2
α β
l
1
l
2
r
0
β
α
1
Mp
β
α
1
2
Mp 2pM
ψ
r
0
1
0
1 −s
q−1
p
s
Γ
q
ds
β
α
1
1
t
p
1 −s
q−1
p
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
p
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
0<t
k
<1
β
α
1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q −1
ds
≤ Lr
0
H
1
K
1
ψ
r
0
,
u
≤
t
t
i
t −s
q−2
p
s
ψ
r
0
Γ
q − 1
ds
1
t
p
1 −s
q−1
p
s
ψ
r
0
Γ
q
ds
β
α
1
t
p
1 −s
q−2
p
s
ψ
r
0
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
ψ
r
0
Γ
q
ds M
0<t
k
<1
β
α
1 − t
k
t
k
t
k−1
t
k
− s
q−2
p
s
ψ
r
0
Γ
q − 1
ds M
X
t
0<t
k
<t
t
k
t
k−1
t
k
− s
q−2
p
s
ψ
r
0
Γ
q −1
ds M
1
α
l
1
r
0
l
2
r
0
≤
1
α
2
α β
l
1
l
2
r
0
Mp
β
α
1
Mp pM
14 Advances in Difference Equations
ψ
r
0
P
Γ
q
1
t
p
1 −s
q−1
p
s
Γ
q
ds
β
α
1
t
p
1 −s
q−2
p
s
Γ
q − 1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−1
p
s
Γ
q
ds
×
0<t
k
<1
β
α
1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q −1
ds
0<t
k
<1
t
k
t
k−1
t
k
− s
q−2
p
s
Γ
q −1
ds
≤ Lr
0
H
2
K
2
ψ
r
0
.
3.10
Therefore, r
0
≤ Lr
0
H Kψr
0
. However, it contradicts with 3.8.
Hence, by using Steps 1–4, Lemmas 2.6 and 2.7, F has at least one fixed point u ∈
Ω
r
0
,
which is the solution of problem 1.5.
Next, we will give some corollaries.
Corollary 3.2. Assume that A
1
, A
2
,andA
3
are satisfied; moreover, lim sup
r∈0,∞
r/H
Kψr ∞,whereH max{H
1
,H
2
},K max{K
1
,K
2
}; then the problem 1.5 has at least
one solution.
Assume that,
A
1
sublinear growth, f : 0 , 1 × × → is continuous. There exists a nonnegative
function pt ∈ C0, 1 with pt > 0 on a subinterval of 0, 1. Also there exists a
constant γ ∈ 0, 1,suchthat|ft, u, v|≤pt|u|
γ
for any t, u, v ∈ 0, 1 × × .
Corollary 3.3. Assume that A
1
, A
2
,andA
3
are satisfied, then the problem 1.5 has at least
one solution.
Assume that
B
1
f : 0, 1 × → is continuous. There exists a nonnegative function pt ∈ C0, 1
with pt > 0 on a subinterval of 0, 1. Also there exists a constant γ ∈ 0, 1 such
that |ft, u|≤pt|u|
γ
for any t, u ∈ 0, 1 × ,
B
2
there exist two positive constants l
1
,l
2
such that α β/α
2
l
1
l
2
L<1.
Moreover, q
1
00,q
2
00, and
q
1
u
− q
1
v
≤ l
1
u −v,
q
2
u
− q
2
v
≤ l
2
u −v
, ∀u, v ∈C, 3.11
B
3
I
k
,J
k
: → are continuous. There exists a positive constant M,suchthat
|
I
k
u
|
≤ M,
|
J
k
u
|
≤ M, k 1, 2, ,p. 3.12
Advances in Difference Equations 15
Corollary 3.4. Assume that B
1
, B
2
,andB
3
are satisfied, then the problem 1.4 has at least one
solution.
Assume that
B
1
f : 0, 1 × → is continuous. There exists a nonnegative function pt ∈ C0, 1
with pt > 0 on a subinterval of 0, 1. |ft, u|≤pt for any t, u ∈ 0, 1 ×
.
Corollary 3.5. Assume that B
1
, B
2
,andB
3
are satisfied, then the problem 1.4 has at least one
solution.
Remark 3.6. Compared with Theorem 3.2 in 16, Corollary 3.5 does not need conditions
ft, u − ft, v≤L
1
u − v, I
k
u −I
k
V ≤L
2
u − v,andJ
k
u −J
k
V ≤L
2
u − v.
Moreover, we only need α β/α
2
l
1
l
2
L<1.
Example 3.7. Consider the following problem:
c
D
3/2
u
t
θu
2
sin
2
u
t
, 0 <t<1,t∈J
1
0, 1
\
1
2
,
Δu
1
2
u
2
1
u
2
, Δu
1
2
u
2
2
u
2
,
u
0
u
0
1
0
|
u
s
|
8
|
u
s
|
ds, u
1
u
1
1
0
|
u
s
|
8
|
u
s
|
ds,
3.13
where θ>0. Here, α β 1,p 1,q 3/2. Let ps ≡ θ P and ψuu
2
,thenwecan
see that A
1
holds. Choosing l
1
l
2
1/8,L 1/2, we can easily obtain that A
2
holds. Let
M 1, then we have that A
3
also holds. Moreover, H 8,K4 26
√
2/3
√
πθ.Hence,
we get sup
r∈0,∞
r/H Kψr 1/2
8θ4 26
√
2/3
√
π>1/1 − L2 for any given
0 <θ<3
√
π/1284 26
√
2. Therefore, By Theorem 3.1,theaboveproblem3.13 has at
least one solution for 0 <θ<3
√
π/1284 26
√
2.
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¨
u, “Positive solutions for boundary value problem of nonlinear fractional differential
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