Báo cáo hóa học: " Research Article Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay" pptx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (526.91 KB, 15 trang )
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 Li1 and Jun Zhang2
1
2
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Department of Mathematics, Central China Normal University, Wuhan 430079, China
Correspondence should be addressed to Fang Li,
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 obtained
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 arise
in various applications such as viscoelasticity, heat equations, and many other physical
phenomena cf., e.g., 7–10 and references therein . Moreover, the Cauchy problem for
various delay equations in Banach spaces has been receiving more and more attention during
the past decades cf., e.g., 7, 10–15 and references therein .
Neutral functional differential equations arise in many areas of applied mathematics
and for this reason, the study of this type of equations has received great attention in the last
few years cf., e.g., 12, 14–16 and references therein . In 12, 16 , Hern´ ndez and Henr´quez
a
ı
studied neutral functional differential equations with infinite delay. In the following, we
will extend such results to fractional-order functional differential equations of neutral type
with infinite delay. To the authors’ knowledge, few papers can be found in the literature for
2
Advances in Difference Equations
the solvability of the fractional-order functional integrodifferential equations of neutral type
with infinite delay.
In the present paper, we will consider the following fractional integrodifferential
equation of neutral type with infinite delay in Banach space X:
dq
x t − h t, xt
dtq
t
A x t − h t, xt
β t, s f s, x s , xs ds,
t ∈ 0, T ,
0
x t
φ t ∈ P,
1.1
t ∈ −∞, 0 ,
where T > 0, 0 < q < 1, P is a phase space that will be defined later see Definition 2.5 . A is
a generator of an analytic semigroup {S t }t≥0 of uniformly bounded linear operators on X.
Then, there exists M ≥ 1 such that S t ≤ M. h : 0, T × P → X, f : 0, T × X × P → X,
β : D → R D
{ t, s ∈ 0, T × 0, T : t ≥ s} , and xt : −∞, 0 → X defined by
x t θ , for θ ∈ −∞, 0 , φ belongs to P and φ 0
0. The fractional derivative is
xt θ
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 L X the
Banach space of all linear and bounded operators on X, and by C J, 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
inf d > 0 : B has a finite cover by sets of diameter ≤ d .
α B
This measure of noncompactness satisfies some important properties.
Lemma 2.2 see 17 . Let A and B be bounded subsets of X. Then,
1 α A ≤ α B if A ⊆ B,
2 αA
α A , where A denotes the closure of A,
3 αA
0 if and only if A is precompact,
|λ|α A , λ ∈ R,
4 α λA
5 α A∪B
max{α A , α B },
6 αA
B ≤α A
7 αA
a
8 α convA
α B , where A
B
{x
y : x ∈ A, y ∈ B},
α A for any a ∈ X,
α A , where convA is the closed convex hull of A.
2.1
Advances in Difference Equations
3
For H ⊂ C J, X , we define
t
t
H s ds
0
u s ds : u ∈ H
for t ∈ J,
2.2
0
where H s
{u s ∈ X : u ∈ H}.
The following lemmas will be needed.
Lemma 2.3 see 17 . If H ⊂ C J, X is a bounded, equicontinuous set, then
α H
sup α H t .
2.3
t∈J
Lemma 2.4 see 18 . If {un }∞ 1 ⊂ L1 J, X and there exists an m ∈ L1 J, R
n
m t , a.e. t ∈ J, then α {un t }∞ 1 is integrable and
n
∞
t
α
0
t
≤2
un s ds
0
n 1
≤
such that un t
α {un s }∞ 1 ds.
n
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 ·
is called an admissible phase space if P has the following properties.
P
1 If x : −∞, T → X is continuous on J and x0 ∈ P, then xt ∈ P and xt is continuous
in t ∈ J and
x t
≤ C xt
P,
2.5
where C ≥ 0 is a constant.
2 There exist a continuous function C1 t > 0 and a locally bounded function C2 t ≥ 0
in t ≥ 0 such that
xt
P
≤ C1 t sup x s
C2 t x0
s∈ 0,t
P,
2.6
for t ∈ 0, T and x as in 1 .
3 The space P is complete.
Remark 2.6. 2.5 in 1 is equivalent to φ 0
≤C φ
P,
for all φ ∈ 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, and let N be a continuous mapping of U into itself. If the implication
V
convN V or V
N V ∪ {0} ⇒ α V
holds for every subset V of U, then N has a fixed point.
0
2.7
4
Advances in Difference Equations
Let Ω be a set defined by
x : −∞, T −→ X such that x| −∞, 0 ∈ P, x|J ∈ C J, X
Ω
.
2.8
Motivated by 4, 5, 21 , we give the following definition of mild solution of 1.1 .
Definition 2.8. A function x ∈ Ω satisfying the equation
⎧
⎪φ t ,
⎪
⎨
x t
t ∈ −∞, 0 ,
t
0
⎪−Q t h 0, φ
⎪
⎩
s
0
h t, xt
R t − s β s, τ f τ, x τ , xτ dτ ds, t ∈ J
2.9
is called a mild solution of 1.1 , where
∞
Q t
ξq σ S tq σ dσ,
0
2.10
∞
Rt
q
σt
q−1
q
ξq σ S t σ dσ
0
and ξq is a probability density function defined on 0, ∞ such that
1 −1− 1/q
σ
q
ξq σ
σ −1/q ≥ 0,
q
2.11
where
q
σ
1 ∞
−1
πn 1
n−1 −qn−1 Γ
σ
nq 1
sin nπq ,
n!
σ ∈ 0, ∞ .
2.12
Remark 2.9. According to 22 , direct calculation gives that
Rt
where Cq,M
qM/Γ 1
≤ Cq,M tq−1 ,
t > 0,
2.13
q .
We list the following basic assumptions of this paper.
H1 f : J × X × P → X satisfies f ·, v, w : J → X is measurable, for all v, w ∈ X × P
and f t, ·, · : X × P → X is continuous for a.e. t ∈ J, and there exist two positive
functions μi · ∈ L1 J, R
i 1, 2 such that
f t, v, w
≤ μ1 t v
μ2 t w
P,
t, v, w ∈ J × X × P.
2.14
Advances in Difference Equations
5
H2 For any bounded sets D1 ⊂ X, D2 ⊂ P, and 0 ≤ s ≤ t ≤ T, there exists an integrable
positive function η such that
α R t − s f τ, D1 , D2
≤ ηt s, τ
where ηt s, τ : η t, s, τ and supt∈J
α D1
t s
0 0
sup α D2 θ
−∞<θ≤0
,
2.15
ηt s, τ dτds : η∗ < ∞.
H3 There exists a constant L > 0 such that
h t1 , ϕ − h t2 , ϕ
ϕ−ϕ
≤ L |t1 − t2 |
P
,
t1 , t2 ∈ J, ϕ, ϕ ∈ P.
2.16
H4 For each t ∈ J, β t, s is measurable on 0, t and β t
ess sup{|β t, s |, 0 ≤ s ≤ t}
is bounded on J. The map t → Bt is continuous from J to L∞ J, R , here, Bt s
β t, s .
H5 There exists M∗ ∈ 0, 1 such that
∗
LC1
∗
where C1
T q βCq,M
q
supt∈J C1 t , β
μ1
∗
C1 μ2
L1 J, R
L1 J, R
< M∗ ,
2.17
supt∈J β t .
3. Main Result
In this section, we will apply Lemma 2.7 to show the existence of mild solution of 1.1 . To
this end, we consider the operator Φ : Ω → Ω defined by
Φx t
⎧
⎪φ t ,
⎪
⎨
t ∈ −∞, 0 ,
⎪−Q t h 0, φ
⎪
⎩
t
s
0
0
h t, xt
R t − s β s, τ f τ, x τ , xτ dτ ds, t ∈ J.
3.1
It follows from H1 , H3 , and H4 that Φ is well defined.
It will be shown that Φ has a fixed point, and this fixed point is then a mild solution of
1.1 .
Let y · : −∞, T → X be the function defined by
y t
Set x t
y t
⎧
⎨φ t , t ∈ −∞, 0 ,
⎩0,
z t , t ∈ −∞, T .
t ∈ J.
3.2
6
Advances in Difference Equations
0 and for t ∈ J,
It is clear to see that x satisfies 2.9 if and only if z satisfies z0
t
−Q t h 0, φ
zt
s
0
h t, yt
0
zt
R t − s β s, τ f τ, y τ
z τ , yτ
zτ dτ ds.
3.3
{z ∈ Ω : z0
Let Z0
0}. For any z ∈ Z0 ,
z
·
Thus, Z0 ,
Z0
sup z t
Z0
z0
sup z t .
P
t∈J
3.4
t∈J
is a Banach space. Set
z ∈ Z0 : z
Br
Z0
≤r ,
for some r > 0.
3.5
Then, for z ∈ Br , from 2.6 , we have
yt
zt
P
≤ yt
zt
P
P
≤ C1 t sup y τ
C2 t y0
0≤τ≤t
C2 t φ
∗
≤ C2 · φ
∗
where C2
P
C1 t sup z τ
C2 t z0
0≤τ≤t
P
3.6
C1 t sup z τ
P
0≤τ≤t
∗
C1 r : r ∗ ,
P
sup0≤η≤T C2 η .
In order to apply Lemma 2.7 to show that Φ has a fixed point, we let Φ : Z0 → Z0 be
an operator defined by Φz t
0, t ∈ −∞, 0 and for t ∈ J,
Φz t
−Q t h 0, φ
t
0
s
h t, yt
zt
3.7
R t − s β s, τ f τ, y τ
z τ , yτ
0
zτ dτ ds.
Clearly, the operator Φ has a fixed point is equivalent to Φ has one. So, it turns out to
prove that Φ has a fixed point.
Now, we present and prove our main result.
Theorem 3.1. Assume that (H1)–(H5) are satisfied, then there exists a mild solution of 1.1 on
−∞, T provided that L 16βη∗ < 1.
Proof. For z ∈ Br , t ∈ J, from 3.6 , we have
f t, y t
z t , yt
zt
≤ μ1 t y t
≤ μ1 t r
zt
μ2 t r ∗ .
μ2 t yt
zt
P
3.8
Advances in Difference Equations
7
In view of H3 ,
h t, yt
≤ h t, y t
zt
zt − h t, 0
≤ L yt
zt
≤ Lr ∗
h t, 0
M1 ,
M1
P
3.9
supt∈J h t, 0 .
where M1
Next, we show that there exists some r > 0 such that Φ Br ⊂ Br . If this is not true,
then for each positive number r, there exist a function zr · ∈ Br and some t ∈ J such that
Φzr t > r. However, on the other hand, we have from 3.8 , 3.9 , and H4
r<
Φzr
t
≤ − Q t h 0, φ
t
s
0
h t, yt
zr
t
0
≤ LM φ
≤ LM φ
R t − s β s, τ f τ, y τ
P
MM1
M1
zr
τ
s
0
P
t
Lr ∗
MM1
t−s
s
0
P
t
t
Lr ∗
M1
dτ ds
0
βCq,M
s
0
βr ∗ Cq,M
≤L M φ
zr τ , yτ
q−1
μ2 τ r ∗ dτ ds
μ1 τ r
3.10
0
βrCq,M
t−s
q−1
μ1 τ dτ ds
0
r∗
t−s
q−1
μ2 τ dτ ds
M1 M
1
T q βCq,M
r μ1
q
r ∗ μ2
L1 J,R
L1 J,R
.
Dividing both sides of 3.10 by r, and taking r → ∞, we have
∗
LC1
T q βCq,M
q
μ1
L1 J,R
∗
C1 μ2
L1 J,R
≥ 1.
3.11
This contradicts 2.17 . Hence, for some positive number r, Φ Br ⊂ Br .
Let {zk }k∈N ⊂ Br with zk → z in Br as k → ∞. Since f satisfies H1 , for almost every
t ∈ J, we get
f t, y t
zk t , yt
zk −→ f t, y t
t
z t , yt
zt ,
as k → ∞.
3.12
8
Advances in Difference Equations
In view of 3.6 , we have
zk
t
yt
P
≤ r ∗.
3.13
Noting that
zk − f t, y t
t
zk t , yt
f t, y t
z t , yt
zt
≤ 2μ1 t r
2μ2 t r ∗ ,
3.14
we have by the Lebesgue Dominated Convergence Theorem that
Φzk
t − Φz t
≤ h t, y t
t
s
0
zk − h t, yt
t
zt
0
≤ L zk − zt
t
zk − f τ, y τ
τ
z τ , yτ
zτ
dτ ds
zk τ , y τ
zk − f τ, y τ
τ
z τ , yτ
zτ
dτ ds
P
t
s
0
0
βCq,M
−→ 0,
zk τ , yτ
R t − s β s, τ f τ, y τ
t−s
q−1
f τ, y τ
k −→ ∞.
3.15
Therefore, we obtain
lim Φzk − Φz
k→∞
Z0
0.
3.16
This shows that Φ is continuous.
Set
G ·, y ·
z · ,y
·
·
z·
:
β ·, τ f τ, y τ
z τ , yτ
0
zτ dτ.
3.17
Let 0 < t2 < t1 < T and z ∈ Br , then we can see
Φz t1 − Φz t2
≤ I1
I2
I3
I4 ,
3.18
Advances in Difference Equations
9
where
I1
Q t1 − Q t2
I2
h t1 , y t 1
t2
I3
· h 0, φ
,
zt1 − h t2 , yt2
zt2
,
R t1 − s − R t2 − s G s, y s
0
t1
I4
R t1 − s
G s, y s
t2
z s , ys
z s , ys
3.19
zs ds ,
ds.
zs
It follows the continuity of S t in the uniform operator topology for t > 0 that I1 tends
to 0, as t2 → t1 . The continuity of h ensures that I2 tends to 0, as t2 → t1 .
For I3 , we have
t2
∞
0
I3 ≤ q
0
t1 − s
q−1
σ t2 − s
q−1
σ
∞
t2
q
− t2 − s
q−1
ξq σ S t1 − s q σ G s, y s
z s , ys
zs
dσds
S t1 − s q σ − S t2 − s q σ
ξq σ
0
0
× G s, y s
t2
≤ Cq,M
t1 − s
q−1
z s , ys
zs
− t2 − s
q−1
q−1
S t1 − s q σ − S t2 − s q σ
G s, y s
0
t2
∞
0
dσ ds
z s , ys
zs
ds
0
σ t2 − s
q
ξq σ
× G s, y s
≤ β r μ1
r ∗ μ2
L1 J,R
t2
× Cq,M
t1 − s
q−1
z s , ys
zs
dσ ds,
L1 J,R
− t2 − s
q−1
ds
0
t2
∞
q
0
σ t2 − s
q−1
ξq σ
S t1 − s q σ − S t2 − s q σ
dσ ds .
0
3.20
10
Advances in Difference Equations
Clearly, the first term on the right-hand side of 3.20 tends to 0 as t2 → t1 . The second term
on the right-hand side of 3.20 tends to 0 as t2 → t1 as a consequence of the continuity of
S t in the uniform operator topology for t > 0.
In view of the assumption of μi s i 1, 2 and 3.8 , we see that
I4 ≤ Cq,M
t1
t1 − s
q−1
G s, y s
z s , ys
t2
≤ βCq,M r μ1
−→ 0,
L1
J,R
t1
r ∗ μ2
L1
J,R
zs
ds
t1 − s
q−1
ds
3.21
t2
as t2 −→ t1 .
Thus, Φ Br is equicontinuous.
Now, let V be an arbitrary subset of Br such that V ⊂ conv Φ V ∪ {0} .
Set Φ1 z t
h t, yt
zt ,
−Q t h 0, φ
t
s
0
Φ2 z t
0
R t − s β s, τ f τ, y τ
z τ , yτ
zτ dτ ds.
3.22
Noting that for z, z ∈ V , we have
h t, yt
zt − h t, yt
zt
≤ L zt − zt
P.
3.23
Thus,
α h t, yt
Vt
≤ Lα Vt ≤ L sup α V t
−∞<θ≤0
θ
Lsup α V τ
0≤τ≤t
≤ Lα V ,
3.24
supt∈J α Φ1 V t ≤ Lα V .
where Vt {zt : z ∈ V }. Therefore, α Φ1 V
Moreover, for any ε > 0 and bounded set D, we can take a sequence {vn }∞ 1 ⊂ D such
n
that α D ≤ 2α {vn } ε see 23 , P125 . Thus, for {vn }∞ 1 ⊂ V , noting that the choice of V ,
n
and from Lemmas 2.2–2.4 and H2 , we have
Advances in Difference Equations
α Φ2 V
≤ 2α
Φ2 vn
ε
11
2 sup α
Φ2 vn t
ε
t∈J
t
2 sup α
t∈J
≤ 8 sup
t∈J
≤ 8β sup
t∈J
≤ 8β sup
t∈J
≤ 8β sup
t∈J
t
β s, τ f τ, y τ
vn τ , yτ
vnτ dτds
vn τ , yτ
vnτ dτ
0
0
0
t
s
0
0
α
t
s
0
0
t
s
0
vn τ , yτ
ds
dτ ds
vnτ
ε
ε
3.25
s
0
R t − s β s, τ f τ, y τ
0
t
ε
s
R t−s
α
s
β s, τ f τ, y τ
0
t∈J
≤ 4 sup
R t−s
0
α R t − s f τ, y τ
ηt s, τ
α {vn τ }
ηt s, τ
α {vn }
vn τ , yτ
vnτ
sup α {vn θ
ε
τ } dτ ds
−∞<θ≤0
sup α vn μ
dτ ds
dτ ds
ε
ε
0≤μ≤τ
≤ 16βα {vn } sup
t∈J
t
s
ηt s, τ dτ ds
0
ε ≤ 16βη∗ α V
ε.
0
It follows from Lemma 2.2 that
α V ≤ α ΦV
≤ α Φ1 V
α Φ2 V
≤ L
16βη∗ α V
ε,
3.26
since ε is arbitrary, we can obtain
α V ≤ L
16βη∗ α V .
3.27
Hence, α V
0. Applying now Lemma 2.7, we conclude that Φ has a fixed point z∗ in Br .
y t z∗ t , t ∈ −∞, T , then x t is a fixed point of the operator Φ which is a mild
Let x t
solution of 1.1 .
12
Advances in Difference Equations
4. Application
In this section, we consider the following integrodifferential model:
∂q
v t, ξ − t
∂tq
0
−∞
γ1 θ
0
∂2
v t, ξ − t
∂ξ2
t
t−s
0
t
v t, 1 − t
v θ, ξ
−∞
t−s
0
−∞
γ1 θ
0
−∞
1
|v t θ, ξ |
dθ
|v t θ, ξ |
γ1 θ
v0 θ, ξ ,
s
cos v τ, ξ dτds
0
γ2 θ sin s2 |v s
0
−∞
γ1 θ
sk
sin|v s, ξ | ·
k
0
v t, 0 − t
|v t θ, ξ |
dθ
1 |v t θ, ξ |
θ, ξ | dθds,
|v t θ, 0 |
dθ
|v t θ, 0 |
0,
|v t θ, 1 |
dθ
1 |v t θ, 1 |
4.1
0,
1
−∞ < θ ≤ 0,
where 0 ≤ t ≤ 1, ξ ∈ 0, 1 , k ∈ N, γ1 , γ2 : −∞, 0 → R, v0 : −∞, 0 × 0, 1 → R are continuous
0
functions, and −∞ |γi θ |dθ < ∞ i 1, 2 .
Set X L2 0, 1 , R and define A by
D A
1
H 2 0, 1 ∩ H0 0, 1 ,
Au
4.2
u.
Then, A generates a compact, analytic semigroup S · of uniformly bounded, linear
operators, and S t ≤ 1.
Let the phase space P be BUC R− , X , the space of bounded uniformly continuous
functions endowed with the following norm:
ϕ
then we can see that C1 t
P
1 in 2.6 .
sup ϕ θ ,
−∞<θ≤0
∀ ϕ ∈ P,
4.3
Advances in Difference Equations
13
For t ∈ 0, 1 , ξ ∈ 0, 1 and ϕ ∈ BUC R− , X , we set
x t ξ
φ θ ξ
v t, ξ ,
θ ∈ −∞, 0 ,
v0 θ, ξ ,
0
h t, ϕ ξ
t
−∞
γ1 θ
tk
sin|x t ξ | ·
k
dθ,
ϕ θ ξ
1
4.4
t − s,
β t, s
f t, x t , ϕ ξ
ϕ θ ξ
t
0
cos x s ξ ds
−∞
0
γ2 θ sin t2 ϕ θ ξ
dθ.
Then 4.1 can be reformulated as the abstract 1.1 .
Moreover, for t ∈ 0, 1 , we can see
0
tk 1
x t
k
≤
f t, x t , ϕ ξ
t2 ϕ
μ1 t x t
P
γ2 θ dθ
−∞
μ2 t ϕ
4.5
P,
0
where μ1 t : tk 1 /k, μ2 t : t2 −∞ |γ2 θ |dθ.
For t1 , t2 ∈ 0, 1 , ϕ, ϕ ∈ P, we have
h t1 , ϕ − h t2 , ϕ
≤ |t1 − t2 |
0
−∞
γ1 θ
ϕθ ξ
0
t2
−∞
≤ |t1 − t2 |
γ1 θ
ϕ θ ξ
1
0
γ1 θ dθ
L |t1 − t2 |
−
ϕ θ ξ
0
−∞
dθ
ϕ θ ξ
1
ϕ−ϕ
P
−∞
ϕ θ ξ
1
dθ
ϕ θ ξ
γ1 θ dθ · ϕ − ϕ
4.6
P
,
0
where L
−∞ |γ1 θ |dθ.
Suppose further that there exists a constant M∗ ∈ 0, 1 such that
L
Cq,M
q
μ1
L1 0,1 ,R
μ2
L1 0,1 ,R
q
0.5, k
< M∗ ,
4.7
2,
4.8
then 4.1 has a mild solution by Theorem 3.1.
For example, if we put
γ1 θ
γ2 θ
ekθ ,
14
then L
see
Advances in Difference Equations
1/2, Cq,M
L
Cq,M
q
1/Γ 0.5
μ1
L1 0,1 ,R
√
1/ π, μ1
μ2
1/8, μ2
L1 0,1 ,R
L1 0,1 ,R
1
2
2
√
π
1
8
L1 0,1 ,R
1
6
1/6. Thus, we
< 0.9 < 1.
4.9
Acknowledgments
The authors are grateful to the referees for their valuable suggestions. F. Li is supported by
the NSF of Yunnan Province 2009ZC054M . J. Zhang is supported by Tianyuan Fund of
Mathematics in China 11026100 .
References
1 R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of
fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750,
25 pages, 2010.
2 H. M. Ahmed, “Boundary controllability of nonlinear fractional integrodifferential systems,”
Advances in Difference Equations, vol. 2010, Article ID 279493, 9 pages, 2010.
3 A. Alsaedi and B. Ahmad, “Existence of solutions for nonlinear fractional integro-differential
equations with three-point nonlocal fractional boundary conditions,” Advances in Difference Equations,
vol. 2010, Article ID 691721, 10 pages, 2010.
4 M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution
equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002.
5 M. M. El-Borai, “On some stochastic fractional integro-differential equations,” Advances in Dynamical
Systems and Applications, vol. 1, no. 1, pp. 49–57, 2006.
6 G. M. Mophou and G. M. N’Gu´ r´ kata, “Existence of the mild solution for some fractional differential
e e
equations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009.
7 J. Liang, T.-J. Xiao, and J. van Casteren, “A note on semilinear abstract functional differential and
integrodifferential equations with infinite delay,” Applied Mathematics Letters, vol. 17, no. 4, pp. 473–
477, 2004.
8 J. Liang and T.-J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,”
Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
9 J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal problems for integrodifferential equations,” Dynamics of
Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 6, pp. 815–824, 2008.
10 T.-J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite
delay in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442–
e1447, 2009.
11 J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj,
vol. 21, no. 1, pp. 11–41, 1978.
12 E. Hern´ ndez and H. R. Henr´quez, “Existence results for partial neutral functional-differential
a
ı
equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 221, no. 2,
pp. 452–475, 1998.
13 J. Liang and T. J. Xiao, “Functional-differential equations with infinite delay in Banach spaces,”
International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 497–508, 1991.
14 G. M. Mophou and G. M. N’Gu´ r´ kata, “Existence of mild solutions of some semilinear neutral
e e
fractional functional evolution equations with infinite delay,” Applied Mathematics and Computation,
vol. 216, no. 1, pp. 61–69, 2010.
15 G. M. Mophou and G. M. N’Gu´ r´ kata, “A note on a semilinear fractional differential equation of
e e
neutral type with infinite delay,” Advances in Difference Equations, vol. 2010, Article ID 674630, 8 pages,
2010.
16 E. Hern´ ndez and H. R. Henr´quez, “Existence of periodic solutions of partial neutral functionala
ı
differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol.
221, no. 2, pp. 499–522, 1998.
Advances in Difference Equations
15
17 J. Bana´ and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure
s
and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
18 H.-P. Heinz, “On the behaviour of measures of noncompactness with respect to differentiation and
integration of vector-valued functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no.
12, pp. 1351–1371, 1983.
19 R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications, vol. 141 of Cambridge
Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 2001.
20 S. Szufla, “On the application of measure of noncompactness to existence theorems,” Rendiconti del
Seminario Matematico della Universit` di Padova, vol. 75, pp. 1–14, 1986.
a
21 Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis:
Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010.
22 F. Mainardi, P. Paradisi, and R. Gorenflo, “Probability distributions generated by fractional diffusion
equations,” in Econophysics: An Emerging Science, J. Kertesz and I. Kondor, Eds., Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2000.
23 D. Bothe, “Multivalued perturbations of m-accretive differential inclusions,” Israel Journal of
Mathematics, vol. 108, pp. 109–138, 1998.
