Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 483816, 17 pages
doi:10.1155/2011/483816
Research Article
Nonlocal Cauchy Problem for
Nonautonomous Fractional Evolution Equations
Fei Xiao
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Correspondence should be addressed to Fei Xiao, [email protected]
Received 28 November 2010; Accepted 29 January 2011
Academic Editor: Toka Diagana
Copyright q 2011 Fei Xiao. 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 investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional
evolution equations d
q
ut/dt
q
−Atutft, K
1
ut, K
2
ut, ,K
n
ut,t∈ I 0,T,
u0A
−1
0guu
0
K
1
u
t
,
K
2
u
t
, ,
K
n
u
t
,t∈ I
0,T
,
u
0
k
i
t, s
u
s
ds, 1.2
the positive functions k
i
t, s are continuous on D {t, s ∈ R
2
:0≤ s ≤ t ≤ T} and
K
∗
i
sup
t∈0,T
t
0
k
i
t, s
ds < ∞. 1.3
dependent of t,
A2 the operator Atλ
−1
exists in LX for any λ with Re λ ≤ 0and
A
t
λ
−1
≤
C
|
λ 1
|
,t∈
0,T
. 1.5
A3 There exists constant γ ∈ 0, 1 and C such that
1
,t
2
,s∈
0,T
. 1.6
Under condition A2,eachoperator−As, s ∈ 0,T generates an analytic semigroup
exp−tAs, t>0, and there exists a constant C such that
A
n
s
exp
−tA
s
≤
C
t
n
, 1.7
η − s
γ−1
g
s
ds dη B
q, γ
t
0
t − s
qγ−1
g
s
ds, 2.1
where Bq, γ is a Beta function.
Definition 2.2. Let B be a bounded set of seminormed linear space Y. The Kuratowski’s
measure of noncompactness for brevity, α-measure of B is defined as
α
B
ds
t
0
u
s
ds : u ∈ H
, 2.3
for t ∈ I,whereHs{us ∈ X : u ∈ H}.
The following lemma will be needed.
Lemma 2.4 see 27. If H ⊂ CI,X is a bounded, equicontinuous set, then
1 αHsup
t∈I
αHt.
2 α
t
0
Hsds ≤
t
0
αHsds,fort ∈ I.
4AdvancesinDifference Equations
Lemma 2.5 see 28. If {u
n
t
0
u
n
s
ds
∞
n1
≤ 2
t
0
α
{
u
n
s
}
∞
n1
ds. 2.5
We need to use the following Sadovskii’s fixed point theorem.
Definition 2.6 see 29.LetP be an operator in Banach space X.IfP is continuous and takes
bounded, sets into bounded sets, and αPH <αH for every bounded set H of X with
η
A
0
A
−1
0
g
u
u
0
dη
t
0
ψ
t − η, η
f
t
0
η
0
ψ
t − η, η
ϕ
η, s
f
s,
K
1
u
s
,
K
2
u
s
q
θA
s
dθ, 2.7
and ξ
q
is a probability density function defined on 0, ∞ such that its Laplace transform is
given by
∞
0
e
−σx
ξ
q
σ
dσ
∞
j0
−x
j
Γ
A
t
− A
τ
ψ
t − τ, τ
,
ϕ
k1
t, τ
t
τ
ϕ
k
t, s
ϕ
1
−1
0
ds.
2.9
To our purpose, the following conclusions will be needed. For the proofs refer to 4.
Lemma 2.9 see 4. 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
. 2.10
Moreover, we have
ϕ
t, η
η
dη
≤ C
2
t
q
1
q
t
γ
B
q, γ 1
: M
t
. 2.12
3. Existence of Solution
Assume that
t
ω
n
i1
v
i
,
t, v
1
,v
2
, ,v
n
∈ I × X × X ×···×X, 3.1
and set T
p,q
max{T
q−1/p
,T
q
}.
f
s, D
1
,D
2
, ,D
n
≤ β
1
t, s
α
D
1
β
2
t, s
α
D
2
≤ ζ
1
t, s, τ
α
D
1
ζ
2
t, s, τ
α
D
2
··· ζ
n
t, s, τ
α
D
n
t, s, τ
dτ ds : ζ
j
< ∞,j 1, 2, ,n.
3.3
B3 g : CI; X → X is continuous and there exists
0 <α
1
<
C M
T
−1
,α
2
≥ 0 3.4
such that
g
u
≤ α
L
p
lim inf
τ →∞
ω
τ
τ
< 1 − α
1
C
M
T
, 3.6
where Ω
p,q
p − 1/pq − 1
p−1/p
,andT
γ
p,q
max{T
p,q
u
u
0
t
0
ψ
t − η, η
U
η
A
0
A
−1
0
g
u
η
, ,
K
n
u
η
dη
t
0
η
0
ψ
t − η, η
ϕ
η, s
f
be a sequence that u
i
→ u as i →∞.Sincef satisfies B1,wehave
f
t,
K
1
u
i
t
,
K
2
u
i
t
, ,
K
n
u
i
Then
F
u
i
t
− F
u
t
≤
A
−1
0
g
u
i
− g
u
dη
t
0
ψ
t − η, η
f
η,
K
1
u
i
η
,
K
2
u
η
, ,
K
n
u
η
dη
t
0
η
, ,
K
n
u
i
s
−f
s,
K
1
u
s
,
K
2
u
s
, ,
− g
u
−→ 0, as i −→ ∞ ;
t
0
ψ
t − η, η
U
η
g
u
i
− g
K
2
u
i
t
, ,
K
n
u
i
t
− f
t,
K
1
u
t
,
K
2
K
j
u
i
t
⎞
⎠
ω
n
j1
K
j
u
t
⎤
⎦
j
u
⎞
⎠
⎤
⎦
.
3.11
Using 2.10 and by means of the Lebesgue dominated convergence theorem, we obtain
t
0
ψ
t − η, η
f
η,
K
1
u
i
η
,
K
2
u
η
, ,
K
n
u
η
dη
≤ C
t
0
t − η
n
u
i
η
−f
η,
K
1
u
η
,
K
2
u
η
, ,
K
f
s,
K
1
u
i
t
,
K
2
u
i
t
, ,
K
n
u
i
t
−f
2
t
0
η
0
t − η
q−1
η − s
γ−1
×
f
s,
K
1
u
i
t
,
2
u
s
, ,
K
n
u
s
ds dη
−→ 0, as i −→ ∞ .
3.13
Therefore, we deduce that
lim
i →∞
F
u
i
− F
u
K
n
u
t
≤ μ
t
ω
⎛
⎝
n
j1
K
∗
j
r
⎞
⎠
. 3.15
Based on 2.12,wedenotethatSt :
t
0
0
u
0
M
t
A
0
u
0
. 3.16
Then for any u ∈ B
r
,byA2, 2.10, 2.11,andLemma 2.1,wehave
Fu
t
≤
S
t
A
0
u
0
t
0
ψ
t − η, η
f
η,
K
1
u
η
0
ψ
t − η, η
ϕ
η, s
f
s,
K
1
u
s
,
K
2
u
s
Mt
A
0
u
0
C
t
0
t − η
q−1
μ
η
ω
⎛
⎝
n
j1
K
∗
j
K
∗
j
r
⎞
⎠
ds dη
≤ α
1
C
Mt
u
α
2
C
M
t
u
0
t
0
t − η
qγ−1
μ
η
dη
,
3.17
where M
1
ω
n
j1
K
∗
j
r.
By means of the H
¨
older inequality, we have
L
p
,
t
0
t − η
γq−1
μ
η
dη ≤ T
p,qγ
Ω
p,qγ
μ
L
p
.
3.18
10 Advances in Difference Equations
Thus
T
A
0
u
0
M
1
Ω
p,q
T
γ
p,q
C C
2
B
q, γ
μ
L
K
1
u
m
t
,
K
2
u
m
t
, ,
K
n
u
m
t
≤ μ
T
u
m
α
2
C
M
T
u
0
M
T
A
0
0
t
m
− η
qγ−1
μ
η
dη
≤ α
1
C
M
T
u
m
α
2
C
C C
2
B
q, γ
μ
L
p
≤ α
1
C
M
T
m α
2
C
M
T
q, γ
μ
L
p
.
3.21
Dividing both sides by m and taking the lower limit as m →∞,weobtain
C
1 CB
q, γ
T
γ
p,q
Ω
p,q
n
j1
K
∗
t
A
−1
0
g
u
u
0
t
0
ψ
t − η, η
U
η
A
0
t
0
ψ
t − η, η
f
η,
K
1
u
η
,
K
2
u
η
, ,
K
n
,
K
2
u
s
, ,
K
n
u
s
ds dη.
3.24
We show that Gu· is equicontinuous.
Let 0 <t
2
<t
1
<Tand u ∈ B
m
.Then
Gu
ψ
t
1
− η, η
− ψ
t
2
− η, η
f
η,
K
1
u
η
,
K
2
u
f
η,
K
1
u
η
,
K
2
u
η
, ,
K
n
u
η
η, s
f
s,
K
1
u
s
,
K
2
u
s
, ,
K
n
u
s
u
s
,
K
2
u
s
, ,
K
n
u
s
ds dη.
3.26
It follows from Lemma 2.9, B1,and3.20 that I
1
,I
3
→ 0, as t
2
η
,
K
2
u
η
, ,
K
n
u
η
dη
≤ CM
1
t
1
t
ψ
t
1
− η, η
ϕ
η, s
f
s,
K
1
u
s
,
K
2
u
s
, ,
γ−1
μ
s
ds dη −→ 0, as t
2
−→ t
1
.
3.28
12 Advances in Difference Equations
Step 5. We show that αFH <αH for every bounded set H ⊂ B
m
.Foranyε>0, we can
take a sequence {h
v
}
∞
v1
⊂ H such that
α
H
≤ 2α
{
h
v
G
{
h
v
}
ε
≤ Cα
g
H
M
T
α
g
H
2sup
t∈I
α
K
n
h
v
η
dη
t
0
η
0
ψ
t − η, η
ϕ
η, s
×f
s,
H
M
T
βα
H
4sup
t∈I
t
0
α
ψ
t − η, η
f
η,
K
1
h
t
0
η
0
α
ψ
t − η, η
ϕ
η, s
f
s,
K
1
h
v
s
,
K
4sup
t∈I
t
0
n
i1
β
i
t, η
K
∗
i
α
{
h
v
}
dη
8sup
t∈I
M
T
βα
H
4
n
i1
β
i
K
∗
i
8
n
i1
ζ
i
K
∗
i
α
H
ε.
3.30
Since ε is arbitrary, we can obtain
α
F
H
≤
C
M
T
β 4
Σ
n
i1
β
i
2ζ
u
− g
u
∗
≤ μ
u − u
∗
,
f
t, v
1
,v
2
, ,v
n
− f
t, w
∈ X
2
,i 1, 2, ,n.
3.32
G2 There exists a constant 0 <δ<1 such that the function Λ : I → R
defined by
Λ
t
μ
C
M
T
C
n
i1
K
∗
i
Γ
t
≤ δ, t ∈ I.
3.33
Theorem 3.2. Assume that (G1), (G2) are satisfied, then 1.1 has a unique mild solution.
Proof. Let F be defined as in Theorem 3.1.Foranyu, u
∗
∈ CI,X,wehave
f
t,
K
1
u
t
,
K
2
u
t
, ,
u
∗
t
≤ l
t
n
i1
K
i
u
t
−
K
i
u
∗
t
≤ μC
u − u
∗
μ
t
0
ψ
t − η, η
U
η
u − u
∗
dη
t
, ,
K
n
u
η
−f
η,
K
1
u
∗
η
,
K
2
u
∗
η, s
f
s,
Ku
s
,
Hu
s
− f
s,
Ku
∗
s
,
∗
i
t
0
t − η
q−1
l
η
dη
C
2
n
i1
K
∗
i
t
0
η
∗
i
Γ
q
I
q
l
t
C
2
n
i1
K
∗
i
Γ
q
Γ
γ
∗
≤ δ
u − u
∗
. 3.36
By the Banach contraction mapping principle, F has a unique fixed point, which is a mild
solution of 1.1.
4. An Example
To illustrate the usefulness of our main result, we consider the following fractional differential
equation:
∂
q
∂t
q
u
t, ξ
b
t, ξ
∂
2
∂ξ
2
u
ds, ξ ∈
0, 1
,
u
t, 0
u
t, 1
0,
u
0,ξ
−
ξ
0
y
0
b
−1
0,x
|
t
1
− t
2
|
γ
, 0 ≤ t
1
≤ t
2
≤ 1. 4.2
Let X L
2
0, 1,R and define At by
D
A
t
H
2
0, 1
∩ H
1
0
z
.
4.3
Then −As generates an analytic semigroup exp−tAs.
Advances in Difference Equations 15
For t ∈ 0, 1, ξ ∈ 0, 1,weset
u
t
ξ
u
t, ξ
,
g
u
sin
u
λ
dx dy,
f
t,
K
1
u
t
,
K
2
u
t
ξ
t
n
n
t
0
ξ
t
0
t − s
u
s, ξ
ds,
K
2
u
t
ξ
t
0
e
−ts
u
1
4
< ∞.
4.5
Moreover, we can get
g
u
≤
1
λ
u
,
α
g
D
≤
1
λ
n
K
1
u
t, ξ
K
2
u
t, ξ
≤
t
n
n
K
∗
1
u
K
∗
2
,u
2
∈ X,
ψ
t − s, s
f
s,
K
1
u
1
s
,
K
2
u
1
s
ξ
Cs
n
n
t − s
q−1
K
1
u
1
s
ξ
−
K
1
u
2
s
ξ
t − s, s
f
s, D
1
,D
2
≤
Cs
n
n
t − s
q−1
α
D
1
α
D
2
B
q, n 1
: β
1
β
2
. 4.10
Similarly, we obtain
α
ψ
t − s, s
ϕ
s, τ
f
τ, D
1
,D
2
≤
C
t
0
s
0
t − s
q−1
s − τ
γ−1
τ
n
dτ ds ≤
C
2
n
B
q, γ
B
q γ,n 1
: ζ
1
ζ
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