Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 963463, 15 pages
doi:10.1155/2011/963463
Research Article
Existence of Mild Solutions to
Fractional Integrodifferential Equations of
Neutral Type with Infinite Delay
Fang Li
1
and J un Zh ang
2
1
School of Mathematics, Yunnan Normal University, Kunming 650092, China
2
Department of Mathematics, Central China Normal University, Wuhan 430079, China
Correspondence should be addressed to Fang Li, [email protected]
Received 5 December 2010; Accepted 30 January 2011
Academic Editor: Jin Liang
Copyright q 2011 F. Li and J. Zhang. 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 solvability of the fractional integrodifferential equations of neutral type with infinite
delay in a Banach space X. An existence result of mild solutions to such problems is o btained
under the conditions in respect of Kuratowski’s measure of noncompactness. As an application of
the abstract result, we show the existence of solutions for an integrodifferential equation.
1. Introduction
The fractional differential equations are valuable tools in the modeling of many phenomena
in various fields of science and engineering; so, they attracted many researchers cf., e.g.,
1–6 and references therein. On the other hand, the integrodifferential equations a rise
in various applications such as viscoelasticity, heat equations, and many other physical
t
A
x
t
− h
t, x
t
t
0
β
t, s
f
s, x
s
,x
s
understood here in the Caputo sense.
The aim of our paper is to study the solvability of 1.1 and present the existence of
mild solution of 1.1 based on Kuratowski’s measures of noncompactness. Moreover, an
example is presented to show an application of the abstract results.
2. Preliminaries
Throughout this paper, we set J :0,T and denote by X a real Banach space, by LX the
Banach space of all linear and bounded operators on X,andbyCJ, X the Banach space of
all X-valued continuous functions on J with the uniform norm topology.
Let us recall the definition of Kuratowski’s measure of noncompactness.
Definition 2.1. Let B be a bounded subset of a seminormed linear space Y. Kuratowski’s
measure of noncompactness of B is defined as
α
B
inf
d>0:B has a finite cover by sets of diameter ≤ d
. 2.1
This measure of noncompactness satisfies some important properties.
Lemma 2.2 see 17. Let A and B be bounded subsets of X.Then,
1 αA ≤ αB if A ⊆ B,
2 αAα
A,whereA denotes the closure of A,
3 αA0 if and only if A is precompact,
4 αλA|λ|αA, λ ∈ R,
5 αA ∪ Bmax{αA,αB},
6 αA B ≤ αAαB,whereA B {x y : x ∈ A, y ∈ B},
7 αA aα
sup
t∈J
α
H
t
.
2.3
Lemma 2.4 see 18. If {u
n
}
∞
n1
⊂ L
1
J, X and t here exists an m ∈ L
1
J, R
such that u
n
t≤
mt,a.e.t∈ J,thenα{u
n
t}
∞
n1
ds.
2.4
The following definition about the phase space is due to Hale and Kato 11.
Definition 2.5. A linear space P consisting of functions from R
−
into X with semi-norm ·
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,thenx
t
∈Pand x
t
is continuous
in t ∈ J and
x
t
≤ Cx
t
P
, 2.5
where C ≥ 0 is a constant.
2 There exist a continuous function C
1
t > 0 and a locally bounded function C
3 The space P is complete.
Remark 2.6. 2.5 in 1 is equivalent to φ0≤Cφ
P
,forallφ ∈P.
The following result will be used later.
Lemma 2.7 see 19, 20. Let U be a bounded, closed, and convex subset of a Banach space X such
that 0 ∈ U,andletN be a continuous mapping of U into itself. If the implication
V
convN
V
or V N
V
∪
{
0
}
⇒ α
V
0 2.7
holds for every subset V of U,thenN has a fixed point.
4AdvancesinDifference Equations
Let Ω be a set defined by
Ω
t
,t∈
−∞, 0
,
−Q
t
h
0,φ
h
t, x
t
t
0
s
0
R
t −s
S
t
q
σ
dσ,
R
t
q
∞
0
σt
q−1
ξ
q
σ
S
t
q
σ
dσ
2.10
∞
n1
−1
n−1
σ
−qn−1
Γ
nq 1
n!
sin
nπq
,σ∈
0, ∞
.
2.12
Remark 2.9. According to 22, direct calculation gives that
R
t
≤C
t
w
P
,
t, v, w
∈ J × X ×P. 2.14
Advances in Difference Equations 5
H2 For any bounded sets D
1
⊂ X, D
2
⊂P,and0≤ s ≤ t ≤ T, there exists an integrable
positive function η such that
α
R
t − s
f
τ, D
1
,D
2
s
0
η
t
s, τdτds : η
∗
< ∞.
H3 There exists a constant L>0suchthat
h
t
1
,ϕ
− h
t
2
, ϕ
≤L
|
t
1
− t
2
|
ϕ − ϕ
P
L
1
J, R
C
∗
1
μ
2
L
1
J, R
<M
∗
,
2.17
where C
∗
1
sup
t∈J
C
1
t, β sup
h
0,φ
h
t, x
t
t
0
s
0
R
t −s
β
s, τ
f
τ, x
τ
Set xt
ytzt, t ∈ −∞,T.
6AdvancesinDifference Equations
It is clear to see that x satisfies 2.9 if and only if z satisfies z
0
0andfort ∈ J,
z
t
−Q
t
h
0,φ
h
t,
y
t
z
t
t
0
{z ∈ Ω : z
0
0}.Foranyz ∈ Z
0
,
z
Z
0
sup
t∈J
z
t
z
0
P
sup
t∈J
z
t
.
3.4
Thus, Z
0
, ·
Z
P
≤ C
1
t
sup
0≤τ≤t
y
τ
C
2
t
y
0
P
C
1
t
sup
0≤τ≤t
≤ C
∗
2
·φ
P
C
∗
1
r : r
∗
,
3.6
where C
∗
2
sup
0≤η≤T
C
2
η.
In order to apply Lemma 2.7 to show that Φ has a fixed point, we let
Φ : Z
0
→ Z
0
be
an operator defined by
s
0
R
t −s
β
s, τ
f
τ, y
τ
z
τ
, y
τ
z
τ
dτ ds.
3.7
Clearly, the operator Φ has a fixed point is equivalent to
Φ has one. So, it turns out to
t
y
t
z
t
μ
2
t
y
t
z
t
P
≤ μ
1
t
r μ
2
t, 0
≤ L
y
t
z
t
P
M
1
≤ Lr
∗
M
1
,
3.9
where M
1
sup
t∈J
ht, 0.
Next, we show that there exists some r>0suchthat
ΦB
r
⊂ B
r
t,
y
t
z
r
t
t
0
s
0
R
t − s
β
s, τ
f
τ, y
τ
t − s
q−1
μ
1
τ
r μ
2
τ
r
∗
dτ ds
≤ LMφ
P
MM
1
Lr
∗
M
1
βrC
q,M
t
τ
dτ ds
≤ L
Mφ
P
r
∗
M
1
M 1
T
q
βC
q,M
q
rμ
1
L
1
J,R
J,R
C
∗
1
μ
2
L
1
J,R
≥ 1.
3.11
This contradicts 2.17. Hence, for some positive number r,
ΦB
r
⊂ B
r
.
Let {z
k
}
k∈N
⊂ B
r
t
z
t
, y
t
z
t
, as k →∞. 3.12
8AdvancesinDifference Equations
In view of 3.6,wehave
y
t
z
k
t
P
≤ r
∗
.
3.13
z
t
, y
t
z
t
≤ 2μ
1
t
r 2μ
2
t
r
∗
, 3.14
we have by the Lebesgue Dominated Convergence Theorem that
− h
t,
y
t
z
t
t
0
s
0
R
t −s
β
s, τ
, y
τ
z
τ
dτ ds
≤ L
z
k
t
− z
t
P
βC
q,M
t
0
τ,
y
τ
z
τ
, y
τ
z
τ
dτ ds
−→ 0,k−→ ∞.
3.15
Therefore, we obtain
lim
k →∞
Φz
k
−
·
0
β
·,τ
f
τ, y
τ
z
τ
, y
τ
z
τ
dτ. 3.17
Let 0 <t
2
<t
1
<Tand z ∈ B
r
,thenwecansee
4
, 3.18
Advances in Difference Equations 9
where
I
1
Q
t
1
− Q
t
2
·
h
0,φ
,
I
2
I
3
t
2
0
R
t
1
− s
− R
t
2
− s
G
s, y
s
− s
G
s,
y
s
z
s
, y
s
z
s
ds.
3.19
It follows the continuity of St in the uniform operator topology for t>0thatI
1
tends
to 0, as t
2
→ t
q−1
−
t
2
− s
q−1
ξ
q
σ
S
t
1
− s
q
σ
G
s, y
s
q
σ
S
t
1
− s
q
σ
− S
t
2
− s
q
σ
×
t
1
− s
q−1
−
t
2
− s
q−1
G
s, y
s
z
s
, y
s
z
t
1
− s
q
σ
− S
t
2
− s
q
σ
×
G
s,
y
s
z
×
C
q,M
t
2
0
t
1
− s
q−1
−
t
2
− s
q−1
ds
σ
− S
t
2
− s
q
σ
dσ ds
.
3.20
10 Advances in Difference Equations
Clearly, the first term on the right-hand side of 3.20 tends to 0 as t
2
→ t
1
. The second term
on the right-hand side of 3.20 tends to 0 as t
2
→ t
1
as a consequence of the continuity of
St in the uniform operator topology for t>0.
z
s
ds
≤ βC
q,M
rμ
1
L
1
J,R
r
∗
μ
2
L
1
J,R
t
1
t
z
t
,
Φ
2
z
t
−Q
t
h
0,φ
t
0
s
0
R
t − s
− h
t,
y
t
z
t
≤Lz
t
− z
t
P
. 3.23
Thus,
α
h
t,
y
t
V
t
≤ Lα
V
Φ
1
V sup
t∈J
α
Φ
1
V t ≤ LαV .
Moreover, for any ε>0andboundedsetD,wecantakeasequence{v
n
}
∞
n1
⊂ D such
that αD ≤ 2α{v
n
}ε see 23, P125.Thus,for{v
n
}
∞
n1
⊂ V , noting that the choice of V ,
and from Lemmas 2.2–2.4 and H2,wehave
Advances in Difference Equations 11
α
Φ
2
0
R
t − s
s
0
β
s, τ
f
τ, y
τ
v
n
τ
,
y
τ
v
nτ
dτds
,
y
τ
v
nτ
dτ
ds ε
≤ 8sup
t∈J
t
0
s
0
α
R
t −s
β
s, τ
f
f
τ, y
τ
v
n
τ
,
y
τ
v
nτ
dτ ds ε
≤ 8β sup
t∈J
t
0
s
0
η
t
s, τ
s, τ
α
{
v
n
}
sup
0≤μ≤τ
α
v
n
μ
dτ ds ε
≤ 16βα
{
v
n
}
sup
t∈J
t
0
1
V
α
Φ
2
V
≤
L 16βη
∗
α
V
ε, 3.26
since ε is arbitrary, we can obtain
α
V
≤
L 16βη
∗
0
−∞
γ
1
θ
|
v
t θ, ξ
|
1
|
v
t θ, ξ
|
dθ
∂
2
∂ξ
2
v
t, ξ
s
k
k
sin
|
v
s, ξ
|
·
s
0
cos v
τ, ξ
dτds
t
0
t − s
0
−∞
γ
2
t θ, 0
|
1
|
v
t θ, 0
|
dθ 0,
v
t, 1
− t
0
−∞
γ
1
θ
|
v
t θ, 1
|
1
|
v
2
0, 1, R and define A by
D
A
H
2
0, 1
∩ H
1
0
0, 1
,
Au u
.
4.2
Then, A generates a compact, analytic semigroup S· of uniformly bounded, linear
operators, and St≤1.
Let the phase space P be BUCR
−
,X, the space of bounded uniformly continuous
functions endowed with the following norm:
ϕ
P
φ
θ
ξ
v
0
θ, ξ
,θ∈
−∞, 0
,
h
t, ϕ
ξ
t
0
−∞
γ
1
t, x
t
,ϕ
ξ
t
k
k
sin
|
x
t
ξ
|
·
t
0
cos x
s
f
t, x
t
,ϕ
ξ
≤
t
k1
k
x
t
t
2
ϕ
P
0
−∞
γ
2
t : t
2
0
−∞
|γ
2
θ|dθ.
For t
1
,t
2
∈ 0, 1, ϕ, ϕ ∈P,wehave
h
t
1
,ϕ
− h
t
2
, ϕ
≤
|
t
1
− t
θ
ξ
dθ
t
2
0
−∞
γ
1
θ
1
ϕ
θ
ξ
dθ
≤
|
t
1
− t
2
|
0
−∞
ϕ − ϕ
P
,
4.6
where L
0
−∞
|γ
1
θ|dθ.
Suppose further that there exists a constant M
∗
∈ 0, 1 such that
L
C
q,M
q
μ
1
L
1
0,1,R
μ
2
1/Γ0.51/
√
π, μ
1
L
1
0,1,R
1/8, μ
2
L
1
0,1,R
1/6. Thus, we
see
L
C
q,M
q
μ
1
L
1
the NSF of Yunnan Province 2009ZC054M. J. Zhang is supported by Tianyuan Fund of
Mathematics in China 11026100.
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