Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 806729, 18 pages
doi:10.1155/2011/806729
Research Article
Solvability of Nonautonomous Fractional
Integrodifferential Equations with Infinite Delay
Fang Li
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Correspondence should be addressed to Fang Li,
Received 4 September 2010; Revised 19 October 2010; Accepted 29 October 2010
Academic Editor: Toka Diagana
Copyright q 2011 Fang Li. 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 nonautonomous
fractional integrodifferential equations with infinite delay in a Banach space X. The existence of
mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii’s
fixed point theorem. An application of the abstract results is also given.
1. Introduction
The Cauchy problem for various delay equations in Banach spaces has been receiving more
and more attention during the past decades cf., e.g., 1–15. This paper is concerned with
existence results for nonautonomous fractional integrodifferential equations with infinite
delay in a Banach space X
d
q
u
t
dt
,
u
t
φ
t
,t∈
−∞, 0
,
1.1
where T>0, 0 <q<1, {At}
t∈0,T
is a family of linear operators in X, K ∈ CD, R
with
D {t, s ∈ R
2
:0≤ s ≤ t ≤ T} and
sup
t∈0,T
t
0
K
Gu
t
:
t
0
K
t, s
u
s
ds, G
∗
: sup
t∈J
t
0
K
t, s
ds < ∞.
2.1
Next, we recall the definition of the Riemann-Liouville integral.
2.2
where Γ is the Gamma function. Moreover, I
ν
1
I
ν
2
I
ν
1
ν
2
, for all ν
1
,ν
2
> 0.
Remark 2.2. 1 I
ν
: L
1
0,T → L
1
0,Tsee 26,
2 obviously, for g ∈ L
1
J, R, it follows from Definition 2.1 that
t
0
ds,
2.3
where Bq, γ is a beta function.
See the following definition about phase space according to Hale and Kato 27.
Advances in Difference Equations 3
Definition 2.3. A linear space P consisting of functions from R
−
into X, with seminorm ·
P
,
is called an admissible phase space if P has the following properties.
1 If x : −∞,T → X is continuous on J and x
0
∈P, then x
t
∈Pand x
t
is continuous
in t ∈ J,and
x
t
≤ L
x
t
C
2
t
x
0
P
,
2.5
for t ∈ 0,T and x as in 1.
3 The space P is complete.
Remark 2.4. Equation 2.4 in 1 is equivalent to φ0≤Lφ
P
, for all φ ∈P.
Next, we consider the properties of Kuratowski’s measure of noncompactness.
Definition 2.5. Let B be a bounded subset of a seminormed linear space Y . The Kuratowski’s
measure of noncompactnessfor brevity, α-measure of B is defined as
α
B
inf
d>0:B has a finite cover by sets of diameter ≤ d
. 2.6
, for t ∈ J, 2.7
where Hs{us ∈ X : u ∈ H}.
4 Advances in Difference Equations
The following lemma will be needed.
Lemma 2.7. If H ⊂ CJ, X is a bounded, equicontinuous set, then
i αHsup
t∈J
αHt,
ii α
t
0
Hsds ≤
t
0
αHsds,fort ∈ J.
For a proof refer to 28.
Lemma 2.8 see 29. If {u
n
}
∞
n1
⊂ L
1
J, X and there exists an m ∈ L
1
J, R
such that u
s
}
∞
n1
ds.
2.8
We need to use the following Sadovskii’s fixed point theorem here, see 30.
Definition 2.9. Let P be an operator in Banach space X.IfP is continuous and takes bounded
sets into bounded sets, and αPB <αB for every bounded set B of X with αB > 0, then
P is said to be a condensing operator on X.
Lemma 2.10 Sadovskii’s fixed point theorem 30. Let P be a condensing operator on Banach
space X.IfP H ⊆ H for a convex, closed, and bounded set H of X,thenP has a fixed point in H.
In this paper, we denote that C is a positive constant, and assume that a family of
closed linear operators {At}
t∈J
satisfying the following.
A1 The domain DA of {At}
t∈J
is dense in the Banach space X and independent
of t.
A2 The operator Atλ
−1
exists in LX for any λ with Re λ ≤ 0and
A
2
A
−1
s
≤ C
|
t
1
− t
2
|
γ
,t
1
,t
2
,s∈ J. 2.10
Under condition A2, each operator −As, s ∈ J, generates an analytic semigroup
exp−tAs, t>0, and there exists a constant C such that
A
n
J
∈ C
J, X
. 2.12
According to 16, a mild solution of 1.1 can be defined as follows.
Definition 2.11. A function u ∈ Ω satisfying the equation
u
t
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
φ
η
0
ψ
t − η, η
ϕ
η, s
f
s, Gu
s
,u
s
ds dη, t ∈ J
2.13
is called a mild solution of 1.1, where
ψ
t, s
q
∞
0
dσ
∞
j0
−x
j
Γ
1 qj
, 0 <q≤ 1,x>0,
ϕ
t, τ
∞
k1
ϕ
k
t, τ
,
2.15
where
k
t, s
ϕ
1
s, τ
ds, k 1, 2,
2.16
To our purpose the following conclusions will be needed. For the proofs refer to 16.
Lemma 2.12 see 16. The operator-valued functions ψt−η, η and Atψt−η,η are continuous
in uniform topology in the variables t, η,where0 ≤ η ≤ t − ε, 0 ≤ t ≤ T, for any ε>0. Clearly,
ψ
t − η, η
≤ C
t − η
q−1
.
2.17
Moreover, we have
t, v, w
≤ μ
t
W
v
w
P
,
t, v, w
∈ J × X ×P,
3.1
and set T
p,q
: T
q−1/p
,
2
t, s
sup
−∞<θ≤0
α
D
2
θ
,
α
ψ
t − s, s
ϕ
s, τ
f
τ,D
1
,D
2
t
0
β
i
t, sds : β
i
< ∞ i 1, 2,sup
t∈J
t
0
s
0
β
i
t, s, τdτ ds : β
i
< ∞ i
3, 4 and D
2
θ{wθ : w ∈ D
2
},
H3 there exists
M,with0< M<1 such that
C
1 CB
τ
<
M,
3.3
where M
p,q
:p − 1/pq − 1
p−1/p
, C
∗
1
sup
0≤η≤T
C
1
η and T
p,q,γ
max{T
p,q
,
T
p,qγ
}.
Theorem 3.1. Suppose that H1–H3 are satisfied, and if 4G
∗
β
1
2β
3
β
,t∈
−∞, 0
,
t
0
ψ
t − η, η
f
η, Gu
η
,u
η
dη
t
0
η
0
φ
t
,t∈
−∞, 0
,
0,t∈ J.
3.5
Advances in Difference Equations 7
Let ut
xtyt, t ∈ −∞,T.
It is easy to see that y satisfies y
0
0and
y
t
t
0
ψ
t − η, η
f
η, s
f
s, G
x
s
y
s
, x
s
y
s
ds dη, t ∈ J
3.6
if and only if u satisfies
u
t
t
f
s, Gu
s
,u
s
ds dη, t ∈ J
3.7
and utφt, t ∈ −∞, 0.
Let Y
0
{y ∈ Ω : y
0
0}. For any y ∈ Y
0
,
y
Y
0
sup
t∈J
0
is a Banach space.
We define the operator
Φ : Y
0
→ Y
0
by
Φyt0, t ∈ −∞, 0 and
Φy
t
t
0
ψ
t − η, η
f
η, G
s, G
x
s
y
s
, x
s
y
s
ds dη, t ∈ J.
3.9
Obviously, the operator Φ has a fixed point if and only if
Φ has a fixed point. So it
turns out to prove that
Φ has a fixed point.
Let {y
k
}
k∈N
be a sequence such that y
x
t
y
t
, x
t
y
t
, as k −→ ∞ . 3.10
8 Advances in Difference Equations
For t ∈ −∞,T, we can prove that
Φ is continuous. In fact,
Φy
k
t
y
k
η
,
x
η
y
k
η
− f
η, G
x
η
y
η
, x
η
y
η
s
y
k
s
,
x
s
y
k
s
− f
s, G
x
s
y
s
, x
t
P
y
t
P
≤ C
1
t
sup
0≤τ≤t
x
τ
C
2
t
P
C
2
t
φ
P
C
1
t
sup
0≤τ≤t
y
τ
≤ C
∗
2
y
t
≤
t
0
K
t, τ
x
τ
y
τ
dτ
t, G
x
t
y
k
t
,
x
t
y
k
t
− f
t, G
x
t
y
t
t
x
t
y
k
t
P
W
G
x
t
ω
2
k
t
,
3.14
Advances in Difference Equations 9
where
ω
1
k
t
W
G
∗
ε G
∗
y
Y
0
W
G
∗
y
Y
0
C
∗
2
φ
P
C
∗
1
y
Y
0
x
η
y
k
η
− f
η, G
x
η
y
η
, x
η
y
η
dη
≤ C
− f
η, G
x
η
y
η
, x
η
y
η
dη
−→ 0, as k −→ ∞ .
3.16
Similarly,by 2.17 and 2.18 , we have
t
0
η
0
s
− f
s, G
x
s
y
s
,
x
s
y
s
ds dη
≤ C
2
t
x
s
y
k
s
− f
s, G
x
s
y
s
, x
s
y
s
ds dη
−→ 0, as k −→ ∞ .
3.17
Therefore, we deduce that
{y ∈ Y
0
: y
Y
0
≤ r}.Now,fory ∈ B
r
,by3.12, 3.13,andH1, we can see
f
t, G
x
t
y
t
,
x
t
y
t
Φy
t
≤
t
0
ψ
t − η, η
f
η, G
x
η
y
s, G
x
s
y
s
, x
s
y
s
ds dη
≤ C
t
0
t − η
q−1
μ
η
ds dη
M
2
C
t
0
t − η
q−1
μ
η
dη C
2
B
q, γ
t
0
t − η
qγ−1
p,q
μ
L
p
J,R
≤ T
p,q
M
p,q
μ
L
p
J,R
,
t
0
t
≤ M
2
M
p,q
T
p,q,γ
C C
2
B
q, γ
μ
L
p
J,R
f
t, G
x
t
y
k
t
,
x
t
y
k
t
≤ μ
t
t
k
≤
M
2
C
t
k
0
t
k
− η
q−1
μ
η
dη C
2
B
B
q, γ
μ
L
p
J,R
,
3.24
where
M
2
WG
∗
k C
∗
2
φ
P
C
∗
lim inf
τ →∞
W
τ
τ
≥ 1.
3.25
This contradicts 3.3. Hence for some positive number k,
ΦB
k
⊂ B
k
.
Let 0 <t
2
<t
1
<Tand y ∈ B
k
, then
Φy
t
2
0
ψ
t
1
− η, η
− ψ
t
2
− η, η
f
η, G
x
η
y
η
η
y
η
, x
η
y
η
dη,
I
3
t
2
0
η
0
ψ
s
ds dη,
I
4
t
1
t
2
η
0
ψ
t
1
− η, η
ϕ
η, s
f
2
,from2.17, 3.23,andH1, we have
I
2
t
1
t
2
ψ
t
1
− η, η
f
η, G
x
η
y
η
1
.
3.28
Similarly, by 2.17, 2.18, H1,andRemark 2.2, we have
I
4
t
1
t
2
η
0
ψ
t
1
− η, η
ϕ
η, s
f
s, G
t
1
− η
q−1
η
0
η − s
γ−1
μ
s
ds dη −→ 0, as t
2
−→ t
1
.
3.29
So, the set {
Φy· : y ∈ B
k
} is equicontinuous.
12 Advances in Difference Equations
For every bounded set H ⊂ B
k
ε
2sup
t∈J
α
t
0
ψ
t − η, η
f
η, G
x
η
h
n
η
,
x
η
n
s
,
x
s
h
ns
ds dη
ε
≤ 4sup
t∈J
t
0
α
ψ
t − η, η
f
η, G
x
ϕ
η, s
f
s, G
x
s
h
n
s
,
x
s
h
ns
ds dη
ε
≤ 4sup
t∈J
θ η
dη
8sup
t∈J
t
0
η
0
β
3
t, η, s
G
∗
α
{
h
n
}
t, η
G
∗
α
{
h
n
}
β
2
t, η
sup
0≤τ≤η
α
{
h
n
τ
}
sup
0≤τ≤s
α
{
h
n
τ
}
ds dη
ε
≤ 4
G
∗
β
1
2β
3
β
H
ε,
3.30
since ε is arbitrary, we can obtain
α
ΦH
≤ 4
G
∗
β
1
2β
3
β
2
2β
4
α
H
t, v
2
,w
2
≤ l
t
v
1
− v
2
w
1
− w
2
P
,
C
∗
1
Γ
q
I
q
l
t
CΓ
γ
I
γq
l
t
≤ , t ∈ J. 3.33
Theorem 3.2. Assume that H1’ and H2’ are satisfied, then 1.1 has a unique mild solution.
Proof. Let
Φ be defined as in Theorem 3.1. For any y, y
t
y
∗
t
,
x
t
y
∗
t
≤ l
t
G
y
t
y − y
∗
Y
0
C
1
t
sup
0≤τ≤t
y
τ
− y
∗
τ
≤
G
−
Φy
∗
t
≤
t
0
ψ
t − η, η
f
η, G
∗
η
dη
t
0
η
0
ψ
t − η, η
ϕ
η, s
f
,
x
s
y
∗
s
ds dη
≤ C
G
∗
C
∗
1
y − y
∗
Y
0
·
t
C
G
∗
C
∗
1
·
Γ
q
I
q
l
t
CΓ
q
Γ
γ
I
γq
Φyt −
Φy
∗
t
Y
0
<
y − y
∗
Y
0
,
3.36
t
0
t − s
v
s, ξ
ds
·
t
0
e
−|
s
0
v
t, 1
0,
v
θ, ξ
v
0
θ, ξ
, −∞ <θ≤ 0,
3.37
where 0 ≤ t ≤ 1, ξ ∈ 0, 1, n ∈ N, at, ξ is a continuous function and is uniformly H
¨
older
continuous in t, that is, there exist C>0and
γ ∈ 0, 1 such that
a
t
1
,ξ
− a
|ζθ|dθ < ∞.
Set X L
2
0, 1, R and define At by
D
A
t
H
2
0, 1
∩ H
1
0
0, 1
,
A
t
u −a
t, ξ
t1in2.5.
Advances in Difference Equations 15
For t ∈ 0, 1, ξ ∈ 0, 1 and ϕ ∈ BUCR
−
,X,weset
u
t
ξ
v
t, ξ
,
φ
θ
ξ
v
0
θ, ξ
,θ∈
−∞, 0
·
t
0
e
−|Gusξ|
ds
t
n
n
2
0
−∞
ζ
θ
sin
t
2
ϕ
θ
ξ
sup
t∈0,1
t
0
t − sds 1/2 < ∞.
Then the above equation 3.37 can be written in the abstract form as 1.1.
Moreover,
f
t, Gu
t
,ϕ
ξ
≤
t
n1
n
2
Gu
t
n1
,t
n2
0
−∞
|
ζ
θ
|
dθ
Gu
t
ϕ
P
|ζθ|dθ}, Wzz/n
2
satisfy H1.
For any u
1
,u
2
∈ X, ϕ, ϕ ∈P,
ψ
t − s, s
f
s, Gu
1
s
,ϕ
ξ
− ψ
t − s, s
− Gu
2
s
s
n
s
0
Gu
1
τ
− Gu
2
τ
dτ
C
n
2
dθ.
3.44
Therefore, for any bounded sets D
1
⊂ X, D
2
⊂P, we have
α
ψ
t − s, s
f
s, D
1
,D
2
≤
2C
n
2
t − s
q−1
θ
dθ.
3.45
16 Advances in Difference Equations
Moreover,
2C
n
2
sup
t∈0,1
t
0
t − s
q−1
s
n1
ds
2C
n
2
sup
t∈
0,1
n2
0
−∞
|
ζ
θ
|
dθds
C
n
2
B
q, n 3
0
−∞
|
ζ
θ
|
dθ : β
2
.
n1
2C
2
n
2
α
D
1
C
2
n
2
τ
0
−∞
|
ζ
θ
|
dθ sup
θ≤0
α
2C
2
n
2
sup
t∈
0,1
t
qγn1
B
q, γ
B
q γ,n 2
2C
2
n
2
B
q, γ
B
0
−∞
|
ζ
θ
|
dθ
C
2
n
2
B
q, γ
B
q γ,n 3
0
−∞
|
ζ
θ
The author is grateful to the referees for their valuable suggestions. This work is supported
by the NSF of Yunnan Province no. 2009ZC054M.
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