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Abstract fractional integro-differential equations
involving nonlocal initial conditions in a-norm
Rong-Nian Wang
*
, Jun Liu and De-Han Chen
* Correspondence: rnwang@mail.
ustc.edu.cn
Department of Mathematics,
NanChang University, NanChang,
JiangXi 330031, People’s Republic
of China
Abstract
In the present paper, we deal with the Cauchy problems of abstract fractional
integro-differential equations involving nonlocal initial conditions in a-norm, where
the operator A in the linear part is the generator of a compact analytic semigroup.
New criterions, ensuring the existence of mild solutions, are established. The results
are obtained by using the theory of operator families associated with the function of
Wright type and the semigroup generated by A, Krasnoselkii’s fixed point theorem
and Schauder’s fixed point theorem. An application to a fractional partial integro-
differential equation with nonlocal initial condition is also considered.
Mathematics subject classification (2000)
26A33, 34G1 0, 34G20
Keywords: Cauchy problem of abstract fractional evolution equation, Nonlocal initial
condition, Fixed point theorem, Mild solution, α-norm
1 Introduction
Let (A, D(A)) be the infinitesimal generator of a co mpact analytic semigroup of
bounded linear operators {T(t)}
t≥0
on a real Banach space (X, ||·||) and 0 Î r(A).
Denote by X

a
, the Banach space D(A
a
) endowed with the graph norm ||u||
a
=||A
a
u||
for u Î X
a
. The present paper concerns the study o f the Cauchy problem for abstract
fractional integro-differential equation involving nonlocal initial condition
⎧
⎪
⎪
⎨
⎪
⎪
⎩
c
D
β
t
u(t )=Au(t)+F(t, u( t), u(κ
1
(t )))
+
t

0

K(t − s)G(s, u(s), u(κ
2
(s)))ds, t ∈ [0, T]
,
u
(
0
)
+ H
(
u
)
= u
0
(1:1)
in X
a
,where
c
D
β
t
,0<b < 1, stands for the Caputo fractional derivative of order b,
and K : [0, T] ® ℝ
+
, 
1
, 
2
:[0,T] ®[0, T], F, G : [0, T]×X

a
× X
a
® X, H : C([0, T];
X
a
) ® X
a
are given functions to be specified later. As can be seen, H constitutes a
nonlocal condition.
The fractional calculus that allows us to consider integration and differentiation of
any order, not necessarily integer, has been the object of extensive study for analyzing
not only an omalous diffusion o n fractals (physical objects of fractional dimension, like
some amorphous semiconductors or strongly porous materials; see [1-3] and references
therein), but also fractional phenomena in optimal control (s ee, e.g., [4-6]) . As indi-
cated in [2,5,7] and the related references given there, the advantages of fractional
Wang et al. Advances in Difference Equations 2011, 2011:25
/>© 2011 Wang et al; licensee Springer. This is an Open Access artic le distributed under the terms of the Creative Commons Attributi on
License ( which permi ts unrestri cted use, distribu tion, and reproduction in any medium,
provided the origin al work is properly cited.
derivatives become apparent in modeling mechanical and e lectrical properties of real
materials, as well as i n the description of rheological properties of rocks, and in many
other fields. One of the emerging branches of the study is the Cauchy problems of
abstract differential equations involving fractional derivatives in time. In recent d ec-
ades, there has been a lot of interest in this type of problems, its applications and var-
ious ge neralizations (cf. e .g., [8-11] and references therein). It is significant to s tudy
this class of problems, because, in this way, one is more realistic to describe the mem-
ory and hereditary properties of various materials and processes (cf. [4,5,12,13]).
In particular, much in terest has developed regarding the abstract fractional Cau-
chy problems involving nonlocal initial conditions. For example, by using the frac-

tional power of operators and some fixed point theorems, the a uthors studied the
existence of mild solutions in [14] for fractional differential equations with nonlocal
initial conditions and in [15] for fractional neutral differential equations with nonlo-
cal initial conditions and time delays. The existence of mild solutions for fractional
differential equations with nonlocal initial conditions in a-norm using the contrac-
tion mapping principle and the Schauder’s fixed point theorem have been investi-
gated in [16].
We here mention that the abstract problem with nonlocal initial conditio n was first
considered by Byszewski [17], and the importance of nonlocal initial conditions in dif-
ferent fields has bee n discussed i n [18,19] and the references therein. Deng [19], espe-
cially, gave the following nonlocal initial values:
H(u)=

p
i
=1
C
i
u(t
i
)
,whereC
i
(i =1,
, p)aregivenconstantsand0<t
1
< ··· <t
p-1
<t
p

<+∞ (p Î N), which is used to
describe the diffu sion phenomenon of a small amount of gas in a transp arent tube. In
the past several years theorems about existence, uniqueness and stability of Cauchy
problem for abstract evolution equations with nonlocal initial conditions have been
studied by many authors, see for instance [19-28] and references therein.
In this paper, we will study the existence of mild solutions for the f ractional Cauchy
problem (1.1). New criterions are established. Both Krasnoselkii’s fixed point theorem
and Schauder’ s fixed point theorem, and the theory of operator families a ssociated
with the function of Wright type and the semigroup generated b y A, are employed in
our appro ach. The results obtained are generalizations and continuat ion of the recent
results on this issue.
The paper is organized as follows. In Section 2, some required notations, definitions
and lemmas are given. In Section 3, we present our main results and their proofs.
2 Preliminaries
In this section, we introduce some notations, definitions and preliminary facts which
are used throughout this work.
We first recall some definitions of fractional calculus (see e.g., [ 6,13] for more
details).
Definition 2.1 The Riemann-Liouville fractional integral operator of order b >0of
function f is defined as
I
β
f (t)=
1
(β)
t

0
(t − s)
β−1

f (s)ds
,
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 2 of 16
provided the right-hand side is pointwise defined on [0, ∞), where Γ(·) is the gamma
function.
Definition 2.2 The Caputo fractional derivative of order b >0,m -1<b <m, m Î N,
is defined as
c
D
β
f (t)=I
m−β
D
m
t
f (t)=
1
(m − β)
t

0
(t − s)
m−β−1
D
m
s
f (s)ds
,
where

D
m
t
:=
d
m
dt
m
and f is an abstract function with value in X. If 0<b <1,then
c
D
β
f (t)=
1
(1 − β)
t

0
f

(s)
(t − s)
β
ds
.
Throughout this paper, we let A : D(A) ® X be the infinitesimal generator of a com-
pact analytic semigroup of bounded linear operators {T(t)}
t≥0
on X and 0 Î r(A),
which allows us to define the fractional power A

a
for 0 ≤ a < 1, as a closed linear
operator on its domain D(A
a
) with inverse A
-a
.
Let X
a
denote the Banach space D(A
a
) endowed with the graph norm ||u||
a
=||
A
a
u|| for u Î X
a
and let C([0, T];X
a
) be the Banach space of all continuo us functions
from [0, T] into X
a
with the uniform norm topology
|
u|
α
=sup{ u
(
t

)

α
, t ∈
[
0, T
]
}
.
L (X) stands for the Banach space of all linear and bounded operators on X.LetM
be a constant such that
M =sup{ T(t)
L
(
X
)
, t ∈ [0, ∞)}
.
For k > 0, write

k
= {u ∈ C
(
[0, T]; X
α
)
; |u|
α
≤ k}
.

The following are basic properties of A
a
.
Theorem 2.1 ([29], pp. 69-75)).
(a) T(t):X ® X
a
for each t >0, and A
a
T(t)x = T(t)A
a
x for each x Î X
a
and t ≥ 0.
(b) A
a
T(t) isboundedonXforeveryt>0and there exist M
a
>0and δ >0such
that
|
|A
α
T(t)||
L(X)
≤
M
α
t
α
e

−δt
.
(c) A
-a
is a bounded linear operator in X with D(A
a
)=Im(A
-a
).
(d) If 0<a
1
≤ a
2
, then
X
α
2
.↪
X
α
1
Lemma 2.1. [27]The restriction of T(t) to X
a
is exact ly the part of T(t) in X
a
and is
an immediately compact semigroup in X
a
, and hence it is immediately norm-
continuous.

Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 3 of 16
Define two families
{S
β
(t ) }
t≥0
and
{P
β
(t ) }
t≥
0
of linear operators by
S
β
(t ) x =
∞

0

β
(s)T(t
β
s)xds, P
β
(t ) x =
∞

0

βs
β
(s)T(t
β
s)x d
s
for x Î X, t ≥ 0, where

β
(s)=
1
π
β
∞

n
=1
(−s)
n−1
(1 + βn)
n!
sin(nπβ), s ∈ (0, ∞
)
is the function of Wright type defined on (0, ∞) which satisfies

β
(s) ≥ 0, s ∈ (0, ∞),
∞

0


β
(s)ds =1, an
d
∞

0
s
ζ

β
(s)ds =
(1 + ζ )

(
1+βζ
)
, ζ ∈ (−1, ∞).
(2:1)
The following lemma follows from the results in [15].
Lemma 2.2. The following properties hold:
(1) For every t ≥ 0,
S
β
(t )
and
P
β
(t
)

are linear and bounded operators on X, i.e.,
 S
β
(t ) x ≤ M  x ,  P
β
(t ) x ≤
β
M

(
1+β
)
 x

for all x Î X and 0 ≤ t < ∞.
(2) For every x Î X,
t →
S
β
(t )
x
,
t → P
β
(t ) x
are continuous functions from [0, ∞)
into X.
(3)
S
β

(t )
and
P
β
(t
)
are compact operators on X for t >0.
(4) For all x Î XandtÎ (0, ∞),
 A
α
P
β
(t ) x ≤C
α
t
−αβ
 x

,where
C
α
=
M
α
β(2−α)

(
1+β
(
1−α

))
.
We can also prove the following criterion.
Lemma 2.3. The functions
t → A
α
P
β
(t
)
and
t → A
α
S
β
(t
)
are continuous in the uni-
form operator topology on (0, +∞).
Proof. Let ε > 0 be given. For every r >0,from(2.1),wemaychooseδ
1
, δ
2
>0such
that
M
α
r
αβ
δ

1

0

β
(s)s
−α
ds ≤
ε
6
,
M
α
r
αβ
∞

δ
2

β
(s)s
−α
ds ≤
ε
6
.
(2:2)
Then, we deduce, in view of the fact t ® A
a

T(t)thatiscontinuousintheuniform
operator topology on (0, ∞) ( see [[30], Lemma 2.1]), that there exists a constant δ >
such that
δ
2

δ
1

β
(s)



A
α
T

t
β
1
s

− A
α
T

t
β
2

s




L(X)
ds ≤
ε
3
,
(2:3)
for t
1
, t
2
≥ r and |t
1
- t
2
|<δ.
Wang et al. Advances in Difference Equations 2011, 2011:25
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On the other hand, for any x Î X, we write
S
β
(t
1
)x − S
β
(t

2
)x =
δ
1

0

β
(s)

T

t
β
1
s

x − T

t
β
2
s

x

ds
+
δ
2


δ
1

β
(s)

T

t
β
1
s

x − T

t
β
2
s

x

ds
+
∞

δ
2


β
(s)(T(t
β
1
s)x − T(t
β
2
s)x)ds.
Therefore, using (2.2, 2.3) and Lemma 2.2, we get


A
α
S
β
(t
1
)x − A
α
S
β
(t
2
)x


≤
δ
1


0

β
(s)




A
α
T

t
β
1
s




L(X)
+



A
α
T

t

β
2
s




L(X)

 x  ds
+
δ
2

δ
1

β
(s)  A
α
T

t
β
1
s

− A
α
T


t
β
2
s


L(X)
 x  ds
+
∞

δ
2

β
(s)




A
α
T

t
β
1
s





L(X)
+



A
α
T

t
β
2
s




L(X)

 x  d
s
≤
2M
α
r
αβ
δ

1

0

β
(s)s
−α
 x  ds
+
δ
2

δ
1

β
(s)



T

t
β
1
s

− T

t

β
2
s




L(X)
 x  ds
+
2M
α
r
αβ
∞

δ
2

β
(s)s
−α
 x  ds
≤ ε

x

,
that is,
 A

α
S
β
(t
1
) − A
α
S
β
(t
2
) ≤ε, for t
1
, t
2
≥ rand|t
1
− t
2
| <
δ
which together with the arbitrariness of r > 0 implies that
A
α
P
β
(t
)
is continuous in
the uniform operator topology for t > 0. A similar argument enable us to give the

characterization of continuity on
A
α
P
β
(t
)
. This completes the proof. ■
Lemma 2.4. For every t >0, the restriction of
S
β
(t
)
to X
a
and the restriction of
P
β
(t
)
to
X
a
are compact operators in X
a
.
Proof. First consider the restriction of
S
β
(t

)
to X
a
. For any r > 0 and t > 0, it is suffi-
cient to show that the set
{S
β
(t ) u; u ∈ B
r
}
is relatively compact in X
a
,whereB
r
:= {u Î
X
a
;||u||
a
≤ r}.
Since by Lemma 2.1, the restriction of T(t)toX
a
is compact for t >0inX
a
, for each
t > 0 and ε Î (0, t ),

∞

ε


β
(s)T

t
β
s

uds; u ∈ B
r

=

T

t
β
ε

∞

ε

β
(s)T

t
β
s − t
β

ε

uds; u ∈ B
r

Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 5 of 16
is relatively compact in X
a
. Also, for every u Î B
r
,as
∞

ε

β
(s)T

t
β
s

uds → S
β
(t ) u, ε →
0
in X
a
, we conclude, using the total boundedness, that the set

{S
β
(t ) u; u ∈ B
r
}
is
relatively compact, which implies that the restriction of
S
β
(t )
to X
a
is compact.
Thesameideacanbeusedtoprovethattherestrictionof
P
β
(t
)
to X
a
is also com-
pact. ■
The fo llowing fixed point theorems play a key role in the proofs of our main results,
which can be found in many books.
Lemma 2.5 (Krasnoselskii’s Fixed Point Theorem).LetEbeaBanachspaceandB
be a bounded closed and convex subset of E, and let F
1
, F
2
be maps of B into E such

that F
1
x + F
2
y Î B for every pair x, y Î B. If F
1
is a contraction and F
2
is completely
continuous, then the equation F
1
x + F
2
x = x has a solution on B.
Lemma 2.6 (Schauder Fixed Point T heorem). If B is a closed bounded and convex
subset of a Banach space E and F : B ® B is completely continuous, then F has a fixed
point in B.
3 Main results
Based on the work in [[15], Lemma 3.1 and Definition 3.1], in this paper, we adopt the
following definition of mild solution of Cauchy problem (1.1).
Definition 3.1. By a mild solution of Cauchy problem (1.1), we mean a function u Î
C([0, T]; X
a
) satisfying
u
(t)=S
β
(t)(u
0
− H(u)) +

t

0
(t − s)
β−1
P
β
(t − s)(F(s, u(s), u(κ
1
(s))
)
+
s

0
K(s − τ )G(τ ,u(τ),u(κ
2
(τ )))dτ )ds
for t Î [0, T].
Let us first introduce our basic assumptions.
(H
0
) 
1
, 
2
Î C([0, T]; [0, T]) and K Î C([0, T]; ℝ
+
).
(H

1
) F, G : [0, T]×X
a
× X
a
® X are continuous and for each positive number k Î
N, there exist a co nstant g Î [0, b(1 - a)) and functions 
k
(·) Î L
1/g
(0, T; ℝ
+
), j
k
(·)
Î L
∞
(0, T; ℝ
+
) such that
sup
u
α
,
v
α
≤k
 F(t, u, v) ≤ϕ
k
(t ) and lim inf

k→+∞

ϕ
k

L
1/γ
(0,T)
k
= σ
1
< ∞
,
sup
u
α
,
v
α
≤k
 G(t, u, v) ≤φ
k
(t ) and lim inf
k→+∞
 φ
k

L
∞
(0,T)

k
= σ
2
< ∞.
(H
2
) F, G :[0,T]×X
a
× X
a
® X are continuous and there exist constants L
F
, L
K
such that
 F(t, u
1
, v
1
) − F(t , u
2
, v
2
) ≤L
F
( u
1
− u
2


α
+  v
1
− v
2

α
),
 G
(
t, u
1
, v
1
)
− G
(
t, u
2
, v
2
)
≤L
G
(
 u
1
− u
2


α
+  v
1
− v
2

α
)
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 6 of 16
for all (t, u
1
, v
1
), (t, u
2
, v
2
) Î [0, T]×X
a
× X
a
.
(H
3
) H : C([0, T]; X
a
) ® X
a
is Lipschitz continuous with Lipschitz constant L

H
.
(H
4
) H : C([0, T]; X
a
) ® X
a
is continuous and there i s a h Î (0, T)suchthatfor
any u, w Î C([0, T]; X
a
) satisfying u(t)=w(t)(t Î[h, T]), H(u)=H(w).
(H
5
) There exists a nondecreasing continuous function F : ℝ
+
® ℝ
+
such that for
all u Î Θ
k
,
 H(u)
α
≤ (k), and lim inf
k→+∞
(k)
k
= μ<∞
.

Remark 3.1. Let us note that (H
4
) is the case when the values of the solution u(t) for t
near zero do not affect H( u). We refer to [19]for a case in point.
In the sequel, we set

k :=

T
0
K(t)d
t
. We are now ready to state our main results in
this section.
Theorem 3.1. Let the assumptions (H
0
), (H
1
) and (H
3
) be satisfied. Then, for u
0
Î
X
a
, the fractional Cauchy problem (1.1) has at least one mild solution provided that
ML
H
+ C
α

σ
1
T
(1−α)β−γ

1 − γ
(
1 − α
)
β − γ

1−γ
+
C
α
σ
2

kT
(1−α)β
(
1 − α
)
β
< 1
.
(3:1)
Proof.Letv Î C([0, T]; X
a
) be fixed wit h |v|

a
≡ 0. From (3.1) and (H
1
), it is easy to
see that there exists a k
0
> 0 such that
M( u
0

α
+ L
H
k
0
+  H(ν) 
α
)+C
α

1 − γ
(1 − α)β − γ

1−γ
T
(1−α)β−γ
 ϕ
k
0


L
1/γ
(0,T
)
+
C
α

kT
(1−α)β
(
1 − α
)
β
 φ
k
0

L
∞
(0,T)
≤ k
0
.
Consider a mapping Γ defined on

k
0
by
(u)(t)=S

β
(t )(u
0
− H(u)) +
t

0
(t − s)
β−1
P
β
(t − s)

F( s , u(s), u(κ
1
(s))
)
+
s

0
K(s − τ )G(τ,u(τ),u(κ
2
(τ )))dτ

ds
:=
(

1

u
)(
t
)
+
(

2
u
)(
t
)
, t ∈ [0, T].
It is easy to verify that (Γu)(·) Î C([0, T]; X
a
) for every
u
∈ 
k
0
.Moreover,forevery
pair
v, u ∈ 
k
0
and t Î [0, T], by (H
1
) a direct calculation yields
 (
1

v)(t)+(
2
u)(t)
α
≤S
β
(t )(u
0
− H(v))
α
+
t

0
(t − s)
β−1
 A
α
P
β
(t − s)
L(X)


F( s , u(s), u(κ
1
(s))
)
+
s


0
K(s − τ ) G(τ , u(τ ), u(κ
2
(τ )))dτ


ds
≤ M( u
0

α
+ L
H
k
0
+  H(ν)
α
)
+C
α
t

0
(t − s)
β(1−α)−1
(ϕ
k
0
(s)+


k  φ
k
0

L
∞
(0,T)
)ds
≤ M( u
0

α
+ L
H
k
0
+  H(ν)
α
)
+C
α

1 − γ
(1 − α)β − γ

1−γ
T
(1−α)β−γ
 ϕ

k
0

L
1/γ
(0,T)
C
α

kT
(1−α)β
(1 − α)β
 φ
k
0

L
∞
(0,T)
≤ k
0
.
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 7 of 16
That is,

1
v + 
2
u ∈ 

k
0
for every pair
v, u ∈ 
k
0
. Therefore, the fractional Cauchy
problem (1.1) has a mild solution if and only if the operator equation Γ
1
u + Γ
2
u = u
has a solution in

k
0
.
In what follows, we will show that Γ
1
and Γ
2
satisfy the conditions of Lemma 2.5.
From ( H
3
)and(3.1),weinferthatΓ
1
is a contraction. Next, we show that Γ
2
is com-
pletely continuous on


k
0
.
We first pro ve that Γ
2
is continuous on

k
0
.Let
{u
n
}
∞
n
=1
⊂ 
k
0
be a sequence such
that u
n
® u as n ® ∞ in C([0, T]; X
a
). Theref ore, it fo llows from the continuity of F,
G, 
1
and 
2

that for each t Î [0, T],
F(t, u
n
(t), u
n
(κ
1
(t))) → F(t, u(t), u(κ
1
(t))) as n →∞,
G
(
t, u
n
(
t
)
, u
n
(
κ
1
(
t
)))
→ G
(
t, u
(
t

)
, u
(
κ
2
(
t
)))
as n →∞
.
Also, by (H
1
), we see
t

0
(t − s)
β−1−αβ
 F(s, u
n
(s), u
n
(κ
1
(s))) − F(s, u(s), u(κ
1
(s)))  d
s
≤ 2
t


0
(t − s)
β−1−αβ
ϕ
k
0
(s)ds
≤ 2

1 − γ
(
1 − α
)
β − γ

1−γ
T
(1−α)β−γ
 ϕ
k
0

L
1/γ
(0,T)
,
and
t


0
(t − s)
β−1−αβ
s

0
K(s − τ )  G(τ , u
n
(τ ), u
n
(κ
2
(τ ))
)
−G(τ ,u(τ ),u(κ
2
(τ )))  dτ ds
≤ 2

k  φ
k
0

L
∞
(0,T)
t

0
(t − s)

β−1−αβ
ds
≤
2

kT
(1−α)β
(
1 − α
)
β
 φ
k
0

L
∞
(0,T)
.
Hence, as

(

2
u
n
)(
t
)
−

(

2
u
)(
t
)

α
≤ C
α
t

0
(t − s)
β−1−αβ
 F(s, u
n
(s), u
n
(κ
1
(s))) − F(s, u(s), u(κ
1
(s)))  d
s
+C
α
t


0
(t − s)
β−1−αβ
s

0
K(s − τ )  G(τ , u
n
(τ ), u
n
(κ
2
(τ )))
−G
(
τ ,u
(
τ
)
, u
(
κ
2
(
τ
)))
 dτ ds,
we conclude, using the Lebesgue dominated convergence theorem, that for all t Î [0,
T],


(

2
u
n
)(
t
)
−
(

2
u
)(
t
)

α
→ 0, as n →∞
,
which implies that
|

2
u
n
− 
2
u
|

α
→ 0, as n →∞
.
This proves that Γ
2
is continuous on

k
0
.
It suffice to prove that Γ
2
is compact on

k
0
. For the sake of brevity, we write
N (t, u(t)) = F(t, u(t), u(κ
1
(t ))) +
t

0
K(t − τ)G(τ , u(τ ), u(κ
2
(τ )))dτ
.
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 8 of 16
Let t Î [0, T] be fixed and ε, ε

1
> 0 be small enough. For
u
∈ 
k
0
, we define the map

ε,ε
1
by
(
ε,ε
1
u)(t)=
t
−ε

0
∞

ε
1
βτ
β
(τ )T((t − s)
β
τ )N (s, u(s))dτ ds
= T(ε
β

ε
1
)
t−ε

0
∞

ε
1
βτ
β
(τ )T((t − s)
β
τ − ε
β
ε
1
)N (s, u(s))dτ ds
.
Therefore, from Lemma 2.1 we see that for each t Î (0, T], the set
{
ε,ε
1
u)(t); u ∈ 
k
0
}
is relatively compact in X
a

. Then, as
 (
2
u)(t) − (
ε,ε
1
u)(t)
α
≤




t

0
ε
1

0
βτ(t − s)
β−1

β
(τ )T((t − s)
β
τ )N (s, u(s))dτ ds





α
+






t

t−ε
∞

ε
1
βτ(t − s)
β−1

β
(τ )T((t − s)
β
τ )N (s, u(s))dτ ds






α

≤ βM
α

t

0
(t − s)
β(1−α)−1
(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)ds
ε
1

0
τ
1−α

β
(τ )dτ

+
t

t−ε
(t − s)
β(1−α)−1
(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)ds
∞

ε
1
τ
1−α

β
(τ )dτ

≤ βM

α


1 − γ
(1 − α)β − γ

1−γ
T
(1−α)β−γ
 ϕ
k
0

L
1/γ
(0,T)
+

kT
(1−α)β
(1 − α)β
 φ
k
0

L
∞
(0,T)

ε

1

0
τ
1−α

β
(τ )dτ
+
βM
α
(2 − α)
(1 + β(1 − α))


1 − γ
(1 − α)β − γ

1−γ
 ϕ
k
0

L
1/γ
(0,T)
ε
(1−α)β−
γ
+


k
(1 − α)β
 φ
k
0

L
∞
(0,T)
ε
(1−α)β

→ 0 as ε, ε
1
→ 0
+
in view of (2.1), w e conclude, u sing the total boundedness, that for each t Î [0, T],
the set
{
2
u)(t); u ∈ 
k
0
}
is relatively compact in X
a
.
On the other hand, for 0 <t
1

<t
2
≤ T and ε’ > 0 small enough, we have

(

2
u
)(
t
1
)
−
(

2
u
)(
t
2
)

α
≤ A
1
+ A
2
+ A
3
+ A

4
,
where
A
1
=
t
2

t
1
(t
2
− s)
β−1−αβ
 N (s , u(s))  ds,
A
2
=
t
1
−ε


0
(t
1
− s)
β−1
 A

α
P
β
(t
2
− s) − A
α
P
β
(t
1
− s)
L(X)
 N (s , u(s))  ds
,
A
3
=
t
1

t
1
−ε

(t
1
− s)
β−1


(t
2
− s)
−αβ
+(t
1
− s)
−αβ


N (s, u(s))  ds,
A
4
=
t
1

0



(t
2
− s)
β−1
− (t
1
− s)
β−1




· (t
2
− s)
−αβ
 N (s, u(s))  ds.
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 9 of 16
Therefore, it follows from (H
1
), Lemma 2.2, and Lemma 2.3 that
A
1
≤ C
α
t
2

t
1
(t
2
− s)
β−1−αβ
(ϕ
k
0
(s)+


k  φ
k
0

L
∞
(0,T)
)ds
≤ C
α

1 − γ
(1 − α)β − γ

1−γ
 ϕ
k
0

L
1/γ
(0,T)
(t
2
− t
1
)
(1−α)β−γ
+
C

α

k  φ
k
0

L
∞
(0,T)
(1 − α)β
(t
2
− t
1
)
(1−α)β
,
A
2
≤ sup
s∈[0,t
1
−ε

]
 A
α
P
α
(t

2
− s) − A
α
P
α
(t
1
− s)
L(X)
×
t
1
−ε


0
(t
1
− s)
β−1
(ϕ
k
0
(s)+

k  φ
k
0

L

∞
(0,T)
)ds
≤
⎡
⎣

1 − γ
β − γ

1−γ
 ϕ
k
0

L
1/γ
(0,T)

t
β−γ
1−γ
1
− ε

β−γ
1−γ

1−γ
+


k  φ
k
0

L
∞
(0,T)
β

t
1
β
− ε

β


× sup
s∈[0,t
1
−ε

]
 A
α
P
α
(t
2

− s) − A
α
P
α
(t
1
− s)
L(X)
,
A
3
≤ C
α
t
1

t
1
−ε

(t
1
− s)
β−1

(t
2
− s)
−αβ
+(t

1
− s)
−αβ

×(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)ds
≤ 2C
α
t
1

t
1
−ε

(t
1
− s)
β−1−αβ

(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)ds
≤ C
α

1 − γ
(1 − α)β − γ

1−γ
 ϕ
k
0

L
1/γ
(0,T)
ε

(1−α)β−γ

+
C
α

k  φ
k
0

L
∞
(0,T)
(1 − α)β
ε

(1−α)β
,
A
4
≤ C
α
t
1

0
((t
1
− s)
β−1
− (t
2

− s)
β−1
)(t
2
− s)
−αβ
×(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)ds
≤ C
α
t
1

0
((t
1
− s)
(1−α)β−1
− (t

2
− s)
(1−α)β−1
)(ϕ
k
0
(s)+

k  φ
k
0

L
∞
(0,T)
)d
s
≤ C
α

1 − γ
(1 − α)β − γ

1−γ
 ϕ
k
0

L
1/γ

(0,T)
×
⎡
⎣
t
1
(1−α)β−γ
−

t
2
(1−α)β−γ
1−γ
− (t
2
− t
1
)
(1−α)β−γ
1−γ

1−γ
⎤
⎦
+
2

k
(
1 − α

)
β
 φ
k
0

L
∞
(0,T)

t
1
(1−α)β
− t
2
(1−α)β
+(t
2
− t
1
)
(1−α)β

,
from which it is easy to see that A
i
( i =1,2,3,4)tendstozeroindependentlyof
u
∈ 
k

0
as t
2
- t
1
® 0 and ε’ ® 0. Hence, we can conclude that

(

2
u
)(
t
1
)
−
(

2
u
)(
t
2
)

α
→ 0, as t
2
− t
1

→ 0
,
and the limit is independently of
u
∈ 
k
0
.
For the case when 0 = t
1
<t
2
≤ T, since
 (
2
u)(t
1
) − (
2
u)(t
2
)
α
=




t
2


0
(t
2
− s)
β−1
P
β
(t
2
− s)N (s, u(s))ds




α
≤ C
α
t
2

0
(t
2
− s)
β−1−αβ
(ϕ
k
0
(s)+


k  φ
k
0

L
∞
(0,T)
)ds
≤ C
α

1 − γ
(
1 − α
)
β − γ

1−γ
 ϕ
k

L
1/γ
(0,T)
t
2
(1−α)β−γ
+
C

α

k  φ
k
0

L
∞
(0,T)
(
1 − α
)
β
t
2
(1−α)β
.
||(Γ
2
u)(t
1
)-(Γ
2
u)(t
2
)||
a
can be made small when t
2
is small independently of

u
∈ 
k
0
.
Consequently, the set
{(
2
)(·); ·∈[0, T], u ∈ 
k
0
}
is equicontinuous. Now applying the
Arzela-Ascoli theorem, it follows that Γ
2
is compact on

k
0
.
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 10 of 16
Therefore, a pplying Lemma 2.5, we conclude that Γ has a fixed point, which gives
rise to a mild solution of Cauchy problem (1.1). This completes the proof. ■
The second result of this paper is the following theorem.
Theorem 3.2. Let the assumptions (H
0
), (H
2
), (H

4
) and (H
5
) be satisfied. Then, for u
0
Î X
a
, the fractional Cauchy problem (1.1) has at least one mild solution provided that
Mμ +
2C
α
T
(1−α)β
(L
F
+

kL
G
)
(
1 − α
)
β
< 1
.
(3:2)
Proof. The proof is divided into the following two steps.
Step 1. Assume that w Î C([h, T]; X
a

) is fixed and set

w(t )=

w(t ), t ∈ [η, T],
w(η), t ∈ [0, η]
.
It is clear that w Î C([0, T]; X
a
). We define a mapping Γ
w
on C([0, T]; X
a
)by
(
w
u)(t)=S
β
(t )(u
0
− H(

w)) +
t

0
(t − s)
β−1
P
β

(t − s)(F(s, u(s), u(κ
1
(s))
)
+
s

0
K(s − τ )G(τ,u(τ),u(κ
2
(τ )))dτ )ds, t ∈ [0, T].
Clearly, ( Γ
w
u)(·) Î C([0, T]; X
a
) for every u Î C([0, T]; X
a
). Moreover, for u Î Θ
k
,
from (H
2
), it follows that
 (
w
u)(t)
α
≤S
β
(t)(u

0
− H(

w))
α
+
t

0
(t − s)
β−1
 P
β
(t − s)

F(s, u(s), u( κ
1
(s)))
+
s

0
K(s − τ )G(τ ,u(τ),u(κ
2
(τ )))dτ


α
ds
≤ M( u

0

α
+  H(

w)
α
)
+C
α
t

0
(t − s)
β(1−α)−1

L
F
( u(s)
α
+  u(κ
1
(s))
α
)+  F(s, ν, ν)

+L
G
s


0
K(s − τ )( u(τ )
α
+  u(κ
2
(τ ))
α
+  G(s, ν, ν) )dτ]ds
≤ M( u
0

α
+  H(

w)
α
)+
2kC
α
(L
F
+

kL
G
)t
(1−α)β
(1 − α)β
+
C

α

max
0≤s≤T
 F(s, ν, ν)  +

k max
0≤s≤T
 G(s, ν, ν) 

T
(1−α)β
(
1 − α
)
β
,
where v Î C([0, T]; X
a
) is fixed with |v|
a
≡ 0, which i mplies that th ere exists a inte-
ger k
0
> 0 such that Γ
w
maps

k
0

into itself. In fact, if this is not the case, then for
each k > 0, there would exist u
k
Î Θ
k
and t
k
Î [0, T] such that ||(Γ
w
u
k
)(t
k
)||
a
>k.
Thus, we have
k <  (
w
u
k
)(t
k
)
α
≤ M( u
0

α
+  H(


u)
α
)+
2kC
α
(L
F
+

kL
G
)T
(1−α)β
(1 − α)β
+
C
α

max
0≤s≤T
 F(s, ν, ν)  +

k max
0≤s≤T
 G(s, ν, ν) 

T
(1−α)β
(

1 − α
)
β
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 11 of 16
Dividing on both sides by k and taking the lower limit as k ® +∞, we get
1 ≤
2C
α
(L
F
+

kL
G
)T
(1−α)β
(
1 − α
)
β
,
this contradicts (3.2). Also, for
u
, v ∈ 
k
0
, a direct calculation yields

(


w
u
)(
t
)
−
(

w
v
)(
t
)

α
=




t

0
(t − s)
β−1
P
β
(t − s)


F( s , u(s), u(κ
1
(s))) − F(s, v(s), v(κ
1
(s)))
+
s

0
K (s − τ )(G(τ , u(τ),u(κ
2
(τ ))) − G(τ, v(τ ), v(κ
2
(τ ))))dτ

ds




α
≤ C
α
t

0
(t − s)
β−1−αβ

 F(s, u(s), u(κ

1
(s))) − F(s, v(s), v(κ
1
(s)))  ds
+
s

0
K(s − τ )  G(τ , u(τ ), u(κ
2
(τ ))) − G(τ , v(τ ), v(κ
2
(τ )))  dτ

d
s
≤ C
α
t

0
(t − s)
β−1−αβ

L
F
( u(s) − v(s)
α
+  u(κ
1

(s)) − v(κ
1
(s))
α
)
+L
G
s

0
K(s − τ )( u(τ ) − v(τ)
α
+  u(κ
2
(τ )) − v(κ
2
(τ ))
α
)dτ

ds
≤
2C
α
T
(1−α)β
(L
F
+


kL
G
)
(
1 − α
)
β
|u − v|
α
,
which together with (3.2) implies that Γ
w
is a contraction mapping on

k
0
.Thus,by
the Banach contraction mapping principle, Γ
w
has a unique fixed point
u
w
∈ 
k
0
, i.e.,
u
w
= S
β

(t )(u
0
− H(

w)) +
t

0
(t − s)
β−1
P
β
(t − s)(F(s, u
w
(s), u
w
(κ
1
(s))
)
+
s

0
K(s − τ )G(τ,u
w
(τ ), u
w
(κ
2

(τ )))dτ )ds
for t Î [0, T].
Step 2. Write

η
k
0
= {u ∈ C([η, T]; X
α
);  u(t)
α
≤ k
0
for all t ∈ [η, T]}
.
It is clear that

η
k
0
is a bounded closed convex subset of C([h, T]; X
a
).
Based on the argument in Step 1, we consider a mapping
F
on

η
k
0

defined by
(
Fw
)(
t
)
= u
w
, t ∈ [η, T]
.
It follows from (H
5
) and (3.2) that
F
maps

η
k
0
into itself. Moreover, for
w
1
, w
2
∈ 
η
k
0
,
from Step 1, we have


1 −
2C
α
T
(1−α)β
(L
F
+

kL
G
)
(1 − α)β

|u
w
1
− u
w
2
|
α
≤ M  H ( w
1
) − H(w
2
)
α
,

that is,
sup
t∈
[
η,T
]
 F w
1
(t ) − Fw
2
(t ) 
α
→ 0 as w
1
→ w
2
in 
η
k
0
,
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 12 of 16
which yields that
F
is continuous. Next, we prove that
F
has a fixed point in

η

k
0
.It
will suffice to prov e that
F
is a compact operator. Then, the result follows from
Lemma 2.6.
Let’s decompose the mapping
F = F
1
+ F
2
as
(F
1
w)(t)=S
β
(t )(u
0
− H(

w)),
(
F
2
w)(t)=
t

0
(t − s)

β−1
P
β
(t − s)(F(s, u
w
(s), u
w
(κ
1
(s)))
+
s

0
K(s − τ )G(τ,u
w
(τ ), u
w
(κ
2
(τ )))dτ )ds
.
Since assumption ( H
5
) implies that the set

H(

w); w ∈ 
η

k
0

is bounded in X
a
,itfol-
lows from Lemma 2.4 that for each t Î [h, T],

(F
1
w)(t); w ∈ 
η
k
0

is relat ively com-
pact in X
a
. Also, for h ≤ t
1
≤ t
2
≤ T,


(
S
β
(t
2

) − S
β
(t
1
))(u
0
− H(

w))


α
→ 0 as t
2
− t
1
→
0
independently of
w ∈ 
η
k
0
. This proves that the set

(F
1
w)(·); w ∈ 
η
k

0

is equicontin-
uous. Thus, an application of Arzela-Ascoli’s theorem yields that
F
1
is compact.
Observe that the set
⎧
⎨
⎩
F(t, u(t), u(κ
1
(t))) +
t

0
K(t − τ )G(τ , u(τ),u(κ
2
(τ )))dτ ; t ∈ [0, T], w ∈ 
η
k
0
⎫
⎬
⎭
is bo unded in X.Therefore,usingLemma2.1,Lemma2.2andLemma2.3,itisnot
difficult to prove, similar to the argument with Γ
2
in Theorem 3.1, that

F
2
is compact.
Hence,makinguseofLemma2.6weconcludethat
F
has a fixed point
w
∗
∈ 
η
k
0
.Put
q
= u
w
∗
. Then,
q(t)=S
β
(t )

u
0
− H


w
∗


+
t

0
(t − s)
β−1
P
β
(t − s)

F( s , q(s), q ( κ
1
(s))
)
+
s

0
K (s − τ )G(τ , q(τ), q(κ
2
(τ )))dτ )

ds, t ∈ [0, T].
Since
u
w
∗
= F w
∗
= w

∗
(
t ∈ [η, T]
)
, H
(
w
∗
)
= H
(
q
)
and hence q is a mild solution of
the fractional Cauchy problem (1.1). This completes the proof. ■
4 Example
In this section, we present an example to our abstract results, which do not aim at
generality but indicate how our theorem can be applied to concrete problem.
We consider the partial differential equation with Dirichlet boundary condition and
nonlocal initial condition in the form
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 13 of 16
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪

⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
c
∂
1
2
t
u(t , x)=
∂
2
u(t , x)
∂x
2
+
q
1
(t ) |u(sin(t), x)|
1+|u(sin(t), x)|
+

t

0
q
2
(s)
1+(t − s)
2
|u(s, x)|
1+|u(s, x)|
ds,0≤ t ≤ 1, 0 ≤ x ≤ π
,
u(t ,0) =u(t, π)=0, 0≤ t ≤ 1,
u(0, x)=
1

η
ln[e
u(s,x)
(|u(s, x)| +1)]ds + u
0
(x), 0 ≤ x ≤ π,
(4:1)
where the functions q
1
, q
2
are continuous on [0, 1] and 0 <h <1.
Let X = L
2

[0, π] and the operators
A =
∂
2
∂x
2
: D(A) ⊂ X →
X
be defined by
D(A)={u ∈ X; u, u

are absolutely continuous, u

∈ X,an
d
u
(
0
)
= u
(
π
)
=0}.
Then, A has a discrete spectrum and the eigenvalues are - n
2
,nÎ N, with the corre-
sponding normalized eigenvectors
y
n

(x)=

2
π
sin(nx
)
.Moreover,A generates a com-
pact, a nalytic semigroup {T (t)}
t≥0
. The following results are well also known (see [29]
for more details):
(1)
T(t)u =

∞
n=1
e
−n
2
t
(u, y
n
)y
n
,  T(t)
L
(
X
)
≤ e

−t
for all t ≥
0
.
(2)
A
−
1
2
u =

∞
n=1
1
n
(u, y
n
)y
n
for each u Î X. In particular,




A
−
1
2





L
(
X
)
=
1
.
(3)
A
1
2
u =

∞
n
=1
n(u, y
n
)y
n
with the domain
D

A
1
2

=


u ∈ X;

∞
n=1
n(u, y
n
)y
n
∈ X

.
Denote by E
ζ, b
, the generalized Mittag-Leffler special function (cf., e.g., [4]) defined
by
E
ζ ,β
(t )=
∞

k
=
0
t
k

(
ζ k + β
)

ζ , β>0, t ∈ R
.
Therefore, we have
S
β
(t ) u =
∞

n=1
E
β
(−n
2
t
β
)(u, y
n
)y
n
, u ∈ X;  S
β
(t ) 
L(X)
≤ 1,
P
β
(t ) u =
∞

n=1

e
β
(−n
2
t
β
)(u, y
n
)y
n
, u ∈ X;  P
β
(t ) 
L(X)
≤
β

(
1+β
)
for all t ≥ 0, where E
b
(t):=E
b,1
(t) and e
b
(t):=E
b, b
(t).
The consideration of this section also needs the following result.

Lemma 4.1.[31]If
w ∈ D
(
A
1
2
)
, then w is absolutely continuous, w’ Î X, and
 w

 =




A
1
2
w




.
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 14 of 16
Define
F( t, u(t ), u(κ
1
(t )))(x)=

q
1
(t ) |u(sin(t), x)|
1+|u(sin(t), x)|
,
K(t)=
1
t
2
+1
, κ
1
(t )=sin(t), κ
2
(t )=t
,
G(t , u(t), u(κ
2
(t )))(x)=
q
2
(t ) |u(t, x)|
1+|u(t, x)|
,
H(u)(x)=
1

η
ln[e
u(s,x)

(|u(s, x)| +1)]ds.
Therefore, it is not difficult to verify that
F, G : [0, 1] × X
1
2
× X
1
2
→
X
and
H : C([0, 1]; X
1
2
) → X
1
2
are continuous,
 F(t, u(t), u(κ
1
(t ))) − F(t, v(t), v(κ
1
(t ))) 
≤ μ
1




A

−
1
2




L(X)


u(κ
1
(t )) − v(κ
1
(t ))


1
2
,
 G(t, u(t), u(κ
1
(t ))) − G(t, v(t), v(κ
1
(t )))

≤ μ
2





A
−
1
2




L
(
X
)
 u(t) − v(t)
1
2
,
where μ
i
:= sup
tÎ [0, 1]
|q
i
(t)|, and for any u satisfying
|
u|
1
2
≤

k
,
 H(u)
1
2
=




A
1
2
H(u)(·)




=  H(u)

(·) ≤2(1 − η)
k
in view of Lemma 4.1.
Now, we note that the problem (4.1) can be reformulated as the abstract problem
(1.1) and the assumptions (H
0
), (H
2
), (H
4

) and (H
5
) hold with
α
=
1
2
, T =1, L
F
= μ
1
, L
G
= μ
2
, (k)=2(1− η)k, μ =2(1− η)
.
Thus, when
1 − η +4M
1
2
(μ
1
+ μ
2
) <
1
2
such that condition (3.2) is verified, (4.1)
has at least one mild solution due to Theorem 3.2.

Acknowledgements
We would like to thank the referees for their valuable comments and suggestions. This research was supported in
part by NNSF of China (11101202), NSF of JiangXi Province of China (2009GQS0018), and Youth Foundation of JiangXi
Provincial Education Department of China (GJJ10051).
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 17 December 2010 Accepted: 16 August 2011 Published: 16 August 2011
References
1. Anh Vo, V, Leonenko, NN: Spectral analysis of fractional kinetic equations with randomdata. J Stat Phys. 104, 1349–1387
(2001). doi:10.1023/A:1010474332598
2. Eidelman, SD, Kochubei, AN: Cauchy problem for fractional diffusion equations. J Differ Equ. 199(2), 211–255 (2004).
doi:10.1016/j.jde.2003.12.002
3. Metzler, R, Klafter, J: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 339,
1–77 (2000). doi:10.1016/S0370-1573(00)00070-3
4. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Wang et al. Advances in Difference Equations 2011, 2011:25
/>Page 15 of 16
5. Podlubny, I, Petraš, I, Vinagre, BM, O’Leary, P, Dorčak, L: Analogue realizations of fractional-order controllers: fractional
order calculus and its applications. Nonlinear Dyn. 29, 281–296 (2002). doi:10.1023/A:1016556604320
6. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach,
Yverdon (1993)
7. Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi
F (eds.) Fractals and Fractional Calculus in Continuum Mechanics. pp. 291–348. Springer, New York (1997)
8. Agarwal, RP, Lakshmikantham, V, Nieto, JJ: On the concept of solution for fractional differential equations with
uncertainty. Nonlinear Anal. 72, 2859–2862 (2010). doi:10.1016/j.na.2009.11.029
9. Agarwal, RP, Zhou, Y, He, YY: Existence of fractional neutral functional differential equations. Comput Math Appl. 59,
1095–1100 (2010)
10. Cuevas, C, de Souza, JC: S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl

Math Lett. 22, 865–870 (2009). doi:10.1016/j.aml.2008.07.013
11. Hernández, E, O’Regan, D, Balachandran, K: On recent developments in the theory of abstract differential equations
with fractional derivatives. Nonlinear Anal. 73, 3462–3471 (2010). doi:10.1016/j.na.2010.07.035
12. Hilfer, H: Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapure (2000)
13. Kilbas, AA, Srivastava, HM, Juan Trujillo, J: Theory and applications of fractional differential equations. In North-Holland
Mathematics Studies, vol. 204,Elsevier, Amsterdam (2006)
14. Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl. 11,
4465–4475 (2010). doi:10.1016/j.nonrwa.2010.05.029
15. Zhou, Y, Jiao, F: Existence of mild solutions for fractional neutral evolution equations. Comput Math Appl. 59,
1063–1077 (2010)
16. Diagana, T, Mophou, GM, N’Guérékata, GM: On the existence of mild solutions to some semilinear fractional integro-
differential equations. E J Qual Theory Diff Equ. 58,1–17 (2010)
17. Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy
problem. J Math Anal Appl. 162, 494–505 (1991). doi:10.1016/0022-247X(91)90164-U
18. Byszewski, L, Lakshmikantham, V: Theorem about the existence and uniqueness of a solution of a nonlocal abstract
Cauchy problem in a Banach space. Appl Anal. 40,11–19 (1991). doi:10.1080/00036819008839989
19. Deng, K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J Math Anal
Appl. 179, 630–637 (1993). doi:10.1006/jmaa.1993.1373
20. Aizicovici, S, Staicu, V: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA
Nonlinear Differ Equ Appl. 14, 361–376 (2007). doi:10.1007/s00030-007-5049-5
21. Ezzinbi, K, Fu, X, Hilal, K: Existence and regularity in the -norm for some neutral partial differential equations with
nonlocal conditions. Nonlinear Anal. 67, 1613–1622 (2007). doi:10.1016/j.na.2006.08.003
22. Fan, Z: Impulsive problems for semilinear differential equations with nonlocal conditions. Nonlinear Anal.
72, 1104–1109
(2010). doi:10.1016/j.na.2009.07.049
23. Liang, J, van Casteren, J, Xiao, TJ: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Anal. 50,
173–189 (2002). doi:10.1016/S0362-546X(01)00743-X
24. Liang, J, Liu, JH, Xiao, TJ: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57,
183–189 (2004). doi:10.1016/j.na.2004.02.007
25. Liang, J, Liu, JH, Xiao, TJ: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math

Comput Model. 49, 798–804 (2009). doi:10.1016/j.mcm.2008.05.046
26. Liang, J, Xiao, TJ: Semilinear integrodifferential equations with nonlocal initial conditions. Comput Math Appl. 47,
863–875 (2004). doi:10.1016/S0898-1221(04)90071-5
27. Liu, H, Chang, JC: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. 70,
3076–3083 (2009). doi:10.1016/j.na.2008.04.009
28. N’Guérékata, GM: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions.
Nonlinear Anal. 70, 1873–1876 (2009). doi:10.1016/j.na.2008.02.087
29. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
30. Hernández, E, Rabelloa, M, Henriquez, HR: Existence of solutions for impulsive partial neutral functional differential
equations. J Math Anal Appl. 331, 1135–1158 (2007). doi:10.1016/j.jmaa.2006.09.043
31. Travis, CC, Webb, GF: Partial functional differential equations with deviating arguments in the time variable. J Math Anal
Appl. 56, 397–409 (1976). doi:10.1016/0022-247X(76)90052-4
doi:10.1186/1687-1847-2011-25
Cite this article as: Wang et al.: Abstract fractional integro-differential equations involving nonlocal initial
conditions in a-norm. Advances in Difference Equations 2011 2011:25.
Wang et al. Advances in Difference Equations 2011, 2011:25
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