Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 158789, 10 pages
doi:10.1155/2010/158789
Research Article
Existence and Uniqueness of Mild Solution for
Fractional Integrodifferential Equations
Fang Li
1
and Gaston M. N’Gu
´
er
´
ekata
2
1
School of Mathematics, Yunnan Normal University, Kunming 650092, China
2
Department of Mathematics, Morgan State University, 1700 East Cold Spring Lane,
Baltimore, MD 21251, USA
Correspondence should be addressed to Fang Li, [email protected]
Received 1 April 2010; Accepted 17 June 2010
Academic Editor: Tocka Diagana
Copyright q 2010 F. Li and G. M. N’Gu
´
er
´
ekata. 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 the existence and uniqueness of mild solution of a class of nonlinear fractional
t
dt
q
Au
t
f
t, u
t
t
0
a
t − s
g
s, u
s
ds, t ∈
t
u
0
t
0
R
t − s
f
s, u
s
K
u
s
ds 2.1
for t ∈ I is called a mild solution of 1.1, where
Q
σ
S
t
q
σ
dσ,
K
u
t
t
0
a
t − s
g
s, u
s
∞
0
σξ
q
σdσ 1 cf., 23, we can see that
R
t
≤qMt
q−1
,t>0.
2.4
In this paper, we use f
p
to denote the L
p
norm of f whenever f ∈ L
p
0,T for some
p with 1 ≤ p<∞. C0,T,X denotes the Banach space of all continuous functions 0,T →
X endowed with the sup-norm given by u
∞
: sup
t∈I
u for u ∈ C0,T,X.Seta
T
:
q
: I → R
,0<q<1, satisfies
L
q
t
Mt
q
·
L La
T
≤ ω<1,t∈
0,T
. 3.2
Theorem 3.1. Let −A be the infinitesimal generator of a strongly continuous semigroup {St}
t≥0
with St≤M, t ≥ 0. If the maps f and g satisfy (H1), L
q
t satisfies (H2), and
L ≤ γ
M · T
q
f
s, u
s
K
u
s
ds.
3.4
Set sup
t∈0,T
ft, 0 M
1
,sup
t∈0,T
gt, 0 M
2
.
Choose r such that
r ≥
M
1 − γ
∞
≤ r
}
. 3.6
4 Advances in Difference Equations
Then for u ∈ B
r
, we have
Fu
t
≤Q
t
u
0
t
0
R
t − s
·f
u
s
ds
≤ Mu
0
qM
t
0
t − s
q−1
f
s, u
s
− f
s, 0
f
s
0
a
s − τ
g
τ,u
τ
dτ
≤
s
0
|
a
s − τ
Fu
t
≤Mu
0
MT
q
Lr M
1
Lr M
2
a
T
≤ r, 3.9
for t ∈ 0,T. Hence F : B
r
→ B
r
.
Let u and v be two elements in C0,T,X. Then
s, v
s
K
u
s
− K
v
s
ds
≤ qM
t
0
t − s
q−1
f
τ
dτ
ds
≤ Mt
q
·
L La
T
u − v
L
q
t
u − v.
3.10
So
Fut − Fvt
∞
≤ L
q
T
t − s
1−q
belongs to L
1
0,t
, R
, 3.12
and set T
p,q
: max{T
q−1/p
,T
q
}.
Let −A be the infinitesimal generator of a compact semigroup S· of uniformly
bounded linear operators. Then there exists a constant M ≥ 1 such that St≤M for
t ≥ 0.
Theorem 3.2. If the maps g and f satisfy (H1), (H3), respectively, and
L ≤ λ
M · T
p,q
· a
T
t
: Q
t
u
0
t
0
R
t − s
K
u
s
ds.
3.14
Choose r such that
r ≥
M
1 − λ
T
.
Let B
r
{u ∈ C0,T,X |u
∞
≤ r} be the closed convex and nonempty subset of
the space C0,T,X.
Letting u, v ∈ B
r
, we have
Φv
t
Ψu
t
≤
t
0
R
t − s
qM
t
0
t − s
q−1
f
s, v
s
ds
qM
t
0
t − s
q−1
K
u
s
ds.
q−1
f
s, v
s
ds
qM
t
0
t − s
q−1
K
u
s
ds
≤ Mu
0
MT
p,q
qM
Ψv
t
≤qM
t
0
t − s
q−1
K
u
s
− K
v
s
ds
≤ qM
t
dτ
ds
≤ MT
q
· a
T
· Lu − v
∞
≤ λu − v
∞
.
3.18
So, we know that Ψ is a contraction mapping.
Set Ut{Φut | u ∈ B
r
}.
Fix t ∈ 0,T. For 0 <ε<t,set
Φ
ε
u
t
s
∞
0
σξ
q
σ
S
t − s
q
σ − ε
q
σ
dσ ds.
3.19
Since St is compact for each t ∈ 0,T,thesetsU
ε
t{Φ
ε
ut | u ∈ B
r
} are relatively
≤ qM · M
p,q
·μ
L
p
loc
I,R
· ε
q−1/p
,
3.20
which implies that Ut is relatively compact in X.
Next, we prove that Φut is equicontinuous.
Advances in Difference Equations 7
For 0 <t
2
<t
1
<T, we have
Φu
t
1
−
t
2
0
R
t
2
− s
f
s, u
s
ds
t
2
− s
f
s, u
s
ds
≤ q
t
2
0
∞
0
σ
s, u
s
dσ ds
t
1
t
2
R
t
1
− s
f
s, u
s
ds
q
− s
q
σ
− S
t
2
− s
q
σ
f
s, u
s
dσ ds
I
1
I
f
s, u
s
ds
≤ qM
t
2
0
t
1
− s
q−1
−
t
2
− s
q−1
f
s, u
s
ds ≤ qM
t
1
t
2
t
1
− s
q−1
μ
s
ds.
3.23
Clearly, the last term tends to 0 as t
2
→ t
1
. Hence I
2
σ
S
t
1
− s
q
σ
− S
t
2
− s
q
σ
f
s, u
s
dσ ds
t
1
− s
q
σ
− S
t
2
− s
q
σ
dσ ds.
3.24
The right-hand side of 3.24 tends to 0 as t
2
→ t
1
as a consequence of the continuity of St
in the uniform operator topology for t>0 by the compactness of St.SoI
3
→ 0ast
2
→ t
−→ f
s, u
s
,n−→ ∞. 3.25
Noting the continuity of f,weget
Φu
n
t
−
Φu
t
t
t − s
q−1
f
s, u
n
s
− f
s, u
s
ds
≤ MT
q
f
·,u
n
·
− f
·,u
t
≤L
u − v,L
> 0, 3.28
holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.
Actually, from what we have just proved, 1.1 has a mild solution ut and
u
t
Q
t
u
0
t
0
R
t − s
f
f
s, u
s
− f
s, v
s
K
u
s
− K
v
s
ds
≤ qM
D
A
H
2
0, 1
∩ H
1
0
0, 1
,
Au −u
.
3.31
Then −A generates a compact, analytic semigroup S· of uniformly bounded linear
operators.
Let t, s ∈ 0,T × 0, 1, ξ ∈ X,andletC, r
0
be positive constants. We set
g
t, ξ
s
s
|
,
a
t
t,
3.32
q 1/2, and p 3.
It is not hard to see that g and f satisfy H1, H3, respectively, and if
C · T
p,q
· T
2
2
≤ λ<1, 3.33
then 1.1 has a unique mild solution by Theorem 3.2 and Remark 3.3.
Acknowledgments
The authors are grateful to the referees for their valuable suggestions. The first author is
supported by the NSF of Yunnan Province 2009ZC054M.
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