Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 901942, 18 pages
doi:10.1155/2012/901942
Research Article
A Nonlocal Cauchy Problem for Fractional
Integrodifferential Equations
Fang Li,1 Jin Liang,2 Tzon-Tzer Lu,3 and Huan Zhu4
1
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
4
Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
2
Correspondence should be addressed to Jin Liang, jliang@ustc.edu.cn
Received 12 February 2012; Accepted 3 March 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Fang Li et al. 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.
This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential
equations in a separable Banach space X. We establish an existence theorem for mild solutions
to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point
theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal
Cauchy problem for a concrete integrodifferential equation is obtained.
1. Introduction
Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations
with nonlocal initial data. Such a Cauchy problem has better effects than the normal Cauchy
problem with the classical initial data when we deal with many concrete problem coming
from engineering and physics cf., e.g., 1–10 and references therein. Therefore, the study
of this type of Cauchy problem is important and significant. Actually, as we have seen from
the just mentioned literature, there have been many significant developments in this field.
On the other hand, fractional differential and integrodifferential equations arise from
various real processes and phenomena appeared in physics, chemical technology, materials,
earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology,
medicine, and economics. They have in recent years been an object of investigations with
much increasing interest. For more information on this subject see for instance 9, 11–18 and
references therein.
2
Journal of Applied Mathematics
Throughout this paper, X is a separable Banach space; LX is the Banach space of
all linear bounded operators on X; A is the generator of an analytic and uniformly bounded
semigroup {T t}t≥0 on X with T tLX ≤ M for a constant M > 0, and Ca, b, X is the
space of all X-valued continuous functions ona, b with the supremum norm as follows:
xa,b : max{xt : t ∈ a, b},
for any x ∈ Ca, b, X.
1.1
Let 0 < q < 1, T > 0, Δ {t, s ∈ 0, T × 0, T : t ≥ s}, f : 0, T × C0, T , X →
X, and h : Δ × C 0, T , X → X. The nonlocal Cauchy problem for abstract fractional
integrodifferential equations, with which we are concerned, is in the following form:
c
Dq xt Axt ft, xt t
kt, sht, s, xsds,
t ∈ 0, T ,
0
1.2
x0 gx x0 ,
where k and g are given functions to be specified later and the fractional derivative is
understood in the Caputo sense, this means that, the fractional derivative is understood in
the following sense:
c
Dq xt : L Dq xt − x0,
t > 0, 0 < q < 1,
1.3
and where
L
d
1
Dq xt : Γ 1 − q dt
t
t − s−q xsds,
t > 0, 0 < q < 1
1.4
0
is the Riemann-Liouville derivative of order q of xt, where Γ· is the Gamma function.
Our main purpose is to establish an existence theorem for the mild solutions to
the nonlocal Cauchy problem based on a special measure of noncompactness under weak
assumptions on the nonlinearity f and the semigroup {T t}t≥0 generated by A.
2. Existence Result and Proof
As usual, we abbreviate uLp 0,T , R with uLp , for any u ∈ Lp 0, T , R .
As in 16, 17, we define the fractional integral of order q with the lower limit zero for
a function f ∈ AC0, ∞ as
1
I q ft Γ q
t
t − sq−1 fsds,
t > 0, 0 < q < 1,
2.1
0
provided the right side is point-wise defined on 0, ∞.
Now we recall some very basic concepts in the theory of measures of noncompactness
and condensing maps see, e.g., 19, 20.
Journal of Applied Mathematics
3
Definition 2.1. Let E be a Banach space, 2E the family of all nonempty subsets of E, A, ≥ a
partially ordered set, and α : 2E → A. If for every Ω ∈ 2E :
αcoΩ αΩ,
2.2
then we say that α is a measure of noncompactness in E.
Definition 2.2. Let E be a Banach space, and F : Y ⊆ E → E is continuous. Let α be a measure
of noncompactness in E such that
i for any Ω0 , Ω1 ∈ 2E with Ω0 ⊂ Ω1 ,
αΩ0 ≤ αΩ1 ;
2.3
α{a0 } ∪ Ω αΩ.
2.4
ii for every a0 ∈ E, Ω ∈ 2E ,
If for every bounded set Ω ⊆ Y which is not relatively compact,
2.5
αFΩ < αΩ,
then we say that F is condensing with respect to the measure of noncompactness α
or α-condensing.
Definition 2.3. Let
∞
1
n−1 −qn−1 Γ nq 1
q σ sin nπq ,
−1 σ
π n1
n!
σ ∈ 0, ∞
2.6
be a one-sided stable probability density, and
ξq σ 1 −1−1/q −1/q ,
σ
q σ
q
σ ∈ 0, ∞.
2.7
For any z ∈ X, we define operators {Y t}t≥0 and {Zt}t≥0 by
Y tz ∞
ξq σT tq σzdσ,
0
Ztz q
2.8
∞
σt
0
q−1
ξq σT t σzdσ.
q
4
Journal of Applied Mathematics
If a continuous function x : 0, T → X satisfies
xt Y t gx x0 t
Zt − s fs, xs axs ds,
t ∈ 0, T ,
2.9
0
then the function x is called a mild solution of 1.2.
Our main result is as follows.
Theorem 2.4. Assume that
1 f·, w and h·, ·, w are measurable for each w ∈ C0, T , X; kt, · is measurable for
each t ∈ 0, T ;
2 ft, · is continuous for a.e. t ∈ 0, T ; g is completely continuous; ht, s, · is continuous
for a.e. t, s ∈ Δ; the map t → kt : kt, · is continuous from 0, T to L∞ 0, T , R;
3 there exist two positive functions μ·, η· ∈ Lp 0, T, R p > 1/q > 1 and two positive
functions m·, · and ζ·, · on Δ with
t
sup
t∈0,T t
∗
mt, sds : m < ∞,
0
sup
t∈0,T ζt, sds : ζ∗ < ∞,
2.10
0
such that
ft, w
≤ μtw a.e. t ∈ 0, T ,
ht, s, w ≤ mt, sw a.e. t, s ∈ Δ,
2.11
for all w ∈ C0, T , X, and
χ ft, D ≤ ηtχD,
χht, s, D ≤ ζt, sχD,
for any bounded set D
noncompactness:
⊂
a.e. t ∈ 0, T,
a.e. t, s ∈ Δ,
2.12
C0, T , X, where χ is the Hausdorff measure of
χΩ inf{ε > 0 : Ω has a finite ε-net}.
2.13
4 g· satisfies
gx
≤ b,
∀ x ∈ C0, T , X,
2.14
for a positive constant b, and
kt : ess sup{|kt, s|, 0 ≤ s ≤ t}
2.15
Journal of Applied Mathematics
5
is bounded on 0, T .
Then the mild solutions set of problem 1.2 is a nonempty compact subset of the space
C0, T , X, in the case of
p − 1 p−1/p q−1/p qM
μ
p < 1.
T
L
Γ 1 q pq − 1
2.16
Proof. First of all, let us prove our definition of the mild solution to problem 1.2 is well
defined and reasonable. Actually, the proof is basic. We present it here for the completeness
of the proof as well as the convenience of reading.
Write
axt t
kt, sht, s, xsds,
0
xλ
∞
e−λt xtdt,
0
fλ
2.17
∞
e
−λt
ft, xtdt,
0
aλ ∞
e−λt axtdt.
0
Clearly, the nonlocal Cauchy problem 1.2 can be written as the following equivalent integral
equation:
1
xt gx x0 Γ q
t
t − sq−1 Axs fs, xs axs ds,
t ∈ 0, T ,
2.18
0
provided that the integral in 2.18 exists. Formally taking the Laplace transform to 2.18, we
have
xλ
1
1
1 q fλ
gx x0 q Axλ
aλ .
λ
λ
λ
Therefore, if the related integrals exist, then we obtain
xλ
λq−1 λq − A−1 gx x0 λq − A−1 fλ
aλ
λq−1
q
∞
∞
0
0
q
e−λ s T s gx x0 ds ∞
0
q
λtq−1 e−λt T tq gx x0 dt
q
e−λ s T s fλ
aλ ds
2.19
6
Journal of Applied Mathematics
∞
∞
q
e−λτ qτ q−1 T τ q e−λt ft, xt axt dt dτ
0
0
∞
1
d −λtq
e
T tq gx x0 dt
λ 0 dt
∞ ∞
∞
−λτσ
q−1
q
−λt
e
qτ q σT τ e
ft, xt axt dt dσ dτ
−
0
0
∞ ∞
0
q
0
e−λtσ σq σT tq gx x0 dσdt
0
∞ ∞
e
0
∞
θq
σT
q
σq
σq
−λθ θ
0
q−1
∞
∞
e
−λt
ft, xt axt dt dσ dθ
0
tq gx x0 dσ dt
e
q σT
σq
0
0
∞ ∞ ∞
q−1
τ − tq −λτ τ − t
q
e
q σT
ft, xt axt dτ dt dσ
σq
σq
0
0
t
−λt
∞
e
−λt
0
q
0
e
∞
e
0
q σT
∞ ∞ τ
0
∞
0
−λt
∞
−λτ
0
tq gx
x
dσ
dt
0
σq
τ − tq−1
τ − tq ft, xt axt dt dτ dσ
q σT
σq
σq
q σT
0
tq gx
x
dσ
dt
0
σq
t ∞
∞
t − sq−1
t − sq −λt
q
e
q σT
fs, xs axs dσ ds dt.
σq
σq
0
0 0
2.20
Now using the uniqueness of the Laplace transform cf., e.g., 21, Theorem 1.1.6, we deduce
that
xt ∞
q σT
0
q
t ∞
0
q
0
∞
0
0
t ∞
0
tq gx x0 dσ
q
σ
t − sq−1
t − sq
fs, xsdσ ds
σT
q
σq
σq
t − sq−1
t − sq
q σT
axsdσ ds
σq
σq
ξq σT tq σ gx x0 dσ
Journal of Applied Mathematics
q
t ∞
0
q
7
σt − sq−1 ξq σT t − sq σ fs, xsdσ ds
0
t ∞
0
σt − sq−1 ξq σT t − sq σ axsdσ ds.
0
2.21
Consequently, we see that the mild solution to problem 1.2 given by Definition 2.3 is well
defined.
Next, we define the operator F : C0, T , X → C0, T , X as follows:
Fxt Y t gx x0 t
Zt − s fs, xs axs ds,
t ∈ 0, T .
2.22
0
It is clear that the operator F is well defined.
The operator F can be written in the form F F1 F2 , where the operators Fi , i 1, 2
are defined as follows:
F1 xt Y t gx x0 ,
F2 xt t
t ∈ 0, T ,
Zt − s fs, xs axs ds,
t ∈ 0, T .
2.23
0
The following facts will be used in the proof.
1
∞
ξq σdσ 1,
2.24
Y t ≤ Const;
2.25
0
which implies that
2
∞
σ ν ξq σdσ 0
∞
0
1
Γ1 ν
q σdσ ,
σ qν
Γ 1 qν
ν ∈ 0, 1,
2.26
which implies that
qM q−1
t ,
Zt ≤ Γ 1q
t > 0.
2.27
8
Journal of Applied Mathematics
Let {xn }n∈N ⊂ C0, T , X such that
lim xn − x0,T 0,
n→∞
2.28
for an x ∈ C0, T , X. Then by the assumptions, we know that for almost every t ∈ 0, T and t, s ∈ Δ:
lim ft, xn t ft, xt,
n→∞
lim ht, s, xn s ht, s, xs.
2.29
n→∞
Therefore, for sufficiently large n, we have
ft, xn t − ft, xt
≤ μt 1 2x
0,T ,
ht, s, xn s − ht, s, xs ≤ mt, s 1 2x0,T ,
t
t
kt, sht, s, xn sds − kt, sht, s, xsds
≤ m∗ k∗ 1 2x0,T ,
0
0
2.30
where
k∗ : sup kt.
t∈0,T 2.31
Hence,
t
t
lim kt, sht, s, xn sds − kt, sht, s, xsds
0.
n → ∞
0
0
2.32
Thus,
F2 xn − F2 x0,T −→ 0,
as n −→ ∞,
since 2.27 implies that
s
t
Zt − s fs, xn s ks, τhs, τ, xn τdτ
0
0
s
− fs, xs ks, τhs, τ, xτdτ ds
0
2.33
Journal of Applied Mathematics
qM
≤ Γ 1q
t
9
t − sq−1 fs, xn s − fs, xs
0
s
ds.
ks,
τhs,
τ,
x
−
hs,
τ,
xτdτ
τ
n
0
2.34
By 2.33 and our assumptions, we see that F is continuous.
Since χ is the Hausdorff measure of noncompactness in X, we know that χ is
monotone, nonsingular, invariant with respect to union with compact sets, algebraically
semiadditive, and regular. This means that
i for any Ω0 , Ω1 ∈ 2E with Ω0 ⊂ Ω1 ,
χΩ0 ≤ χΩ1 ;
2.35
χ{a0 } ∪ Ω χΩ;
2.36
ii for every a0 ∈ E, Ω ∈ 2E ,
iii for every relatively compact set D ⊂ E, Ω ∈ 2E ,
χ{D} ∪ Ω χΩ;
2.37
χΩ0 Ω1 ≤ χΩ0 χΩ1 ;
2.38
iv for each Ω0 , Ω1 ∈ 2E ,
v χΩ 0 is equivalent to the relative compactness of Ω.
Noting that for any ψ ∈ L1 0, T , X, we have
t
lim sup
L → ∞ t∈0,T e−Lt−s ψsds 0.
2.39
0
So, there exists a positive constant L such that
qM
sup
Γ 1 q t∈0,T t
t
0
L1 <
0
qMk∗ ζ∗
sup
Γ 1 q t∈0,T qM
sup
Γ 1 q t∈0,T t − sq−1 ηse−Lt−s ds
t
t − sq−1 e−Lt−s ds
L2 <
0
t − sq−1 μs m∗ k∗ e−Lt−s ds
1
,
3
1
,
3
L3 <
2.40
1
.
3
10
Journal of Applied Mathematics
For every bounded subset Ω ⊂ C0, T , X, we define
mod c Ω : lim sup max vt1 − vt2 ,
δ → 0 v∈Ω |t1 −t2 |≤δ
ΨΩ : sup e−Lt χΩt ,
2.41
t∈0,T αΩ : ΨΩ, mod c Ω.
Then mod c Ω is the module of equicontinuity of Ω, and α is a measure of noncompactness
in the space C0, T , X with values in the cone R2 .
Let Ω ⊂ C0, T , X be a nonempty, bounded set such that
αFΩ ≥ αΩ.
2.42
By the assumptions and the continuity of T t in the uniform operator topology for t > 0, we
get
mod c F1 Ω 0.
2.43
Clearly,
fs, xs
axs ≤ μs m∗ k∗ x
2.44
0,T .
Let δ > 0, t1 , t2 ∈ 0, T such that 0 < t1 − t2 ≤ δ and x ∈ Ω. Then
t2
t1
Zt2 − s fs, xs axs ds
Zt1 − s fs, xs axs ds −
0
0
≤ x0,T t
2
Zt1 − s − Zt2 − s μs m∗ k∗ ds 0
t1
∗ ∗
Zt1 − s μs m k ds
t2
t ∞
2
≤ qx0,T t1 − sq−1 − t2 − sq−1 σξq σ
T t1 − sq σ μs m∗ k∗ dσ ds
0
0
t2 ∞
0
t2 − sq−1 σξq σ
T t1 − sq σ − T t2 − sq σ 0
qMx0,T t1
× μs m k dσ ds t1 − sq−1 μs m∗ k∗ ds
Γ 1q
t2
∗ ∗
Journal of Applied Mathematics
qMx0,T t2 ≤
t1 − sq−1 − t2 − sq−1 μs m∗ k∗ dσ ds
Γ 1q
0
t1
q−1 ∗ ∗
μs m k ds
t1 − s
11
t2
t2 ∞
0
t2 − sq−1 σξq σ
T t1 − sq σ − T t2 − sq σ μs m∗ k∗ dσ ds.
0
2.45
It is not hard to see that the right-hand side of 2.45 tend to 0 as t2 → t1 . Thus, the set
{F2 x· : x ∈ Ω} is equicontinuous, then mod c F2 Ω 0. Combining with 2.43, we have
mod c FΩ 0, which implies mod c Ω 0 from 2.42. Next, we show that ΨΩ 0.
It is easy to see that
ΨF1 Ω 0.
2.46
For any t ∈ 0, T , we define
2 Ωt :
F
t
Zt − sfs, xsds : x ∈ Ω .
2.47
0
We consider the multifunction s ∈ 0, t Gs:
Gs Zt − sfs, xs : x ∈ Ω .
2.48
Obviously, G is integrable, that is, G admits a Bochner integrable selection g : 0, h → E, and
gt ∈ Gt,
for a.e. t ∈ 0, h.
2.49
From 2.27 and our assumptions, it follows that G is integrably bounded, that is, there exists
a function ∈ L1 0, h, E such that
Gt : sup{g : g ∈ Gt} ≤ t,
a.e. t ∈ 0, h.
2.50
12
Journal of Applied Mathematics
Moreover, we have the following estimate for a.e. s ∈ 0, t:
qM
χGs ≤ t − sq−1 χ fs, Ωs
Γ 1q
qM
≤ t − sq−1 ηsχΩs
Γ 1q
qM
t − sq−1 ηseLs e−Ls χΩs
Γ 1q
2.51
qM
≤ t − sq−1 ηseLs ΨΩ.
Γ 1q
Therefore, since X is a separable Banach space, we know by 20, Theorem 4.2.3 that
t
t
qM
2 Ωt χ
χ F
Gsds ≤ t − sq−1 ηseLs ds · ΨΩ.
Γ 1q 0
0
2.52
So
sup e
−Lt
t∈0,T t
qM
χ F2 Ωt ≤ sup
t − sq−1 ηse−Lt−s ds · ΨΩ
Γ 1 q t∈0,T 0
2.53
L1 ΨΩ.
Similarly, if we set
2 Ωt F
t
Zt − saxsds : x ∈ Ω ,
2.54
0
then we see that the multifunction s ∈ 0, t Gs,
Gs
{Zt − saxs : x ∈ Ω}
2.55
is integrable and integrably bounded. Thus, we obtain the following estimate for a.e. s ∈ 0, t:
qMk∗ ζ∗
≤ χ Gs
t − sq−1 eLs ΨΩ,
Γ 1q
sup e
t∈0,T −Lt
t
qMk∗ ζ∗
χ F2 Ωt ≤ sup
t − sq−1 e−Lt−s ds · ΨΩ
Γ 1 q t∈0,T 0
L2 ΨΩ.
2.56
Journal of Applied Mathematics
13
Now, from 2.53 and 2.56, it follows that
ΨFΩ ≤ ΨF1 Ω ΨF2 Ω ≤ L1 L2 ΨΩ LΨΩ,
2.57
< 1. Then by 2.42, we get ΨΩ 0. Hence αΩ 0, 0. Thus, Ω is relatively
where 0 < L
compact due to the regularity property of α. This means that F is α-condensing.
Let us introduce in the space C0, T , X the equivalent norm defined as
x∗ sup e−Lt xt .
2.58
Br {x ∈ C0, T , X : x∗ ≤ r}.
2.59
t∈0,T Consider the set
Next, we show that there exists some r > 0 such that FBr ⊂ Br . Suppose on the contrary that
for each r > 0 there exist xr · ∈ Br , and some t ∈ 0, T such that Fxr t∗ > r.
From the assumptions, we have
F1 xr t∗ ≤ Mb x0 .
2.60
Moreover,
F2 xr t ≤
t
Zt − s fs, xr s axr s ds
0
qM
≤ Γ 1q
qM
Γ 1q
t
0
t − sq−1 μsxr s m∗ k∗ eLs xr ∗ ds
t
t − sq−1 μseLs e−Ls xr sds
∗ ∗
m k
t
q−1 Ls
e ds · xr ∗
t − s
0
qMr
≤ Γ 1q
2.61
0
t
t − sq−1 μs m∗ k∗ eLs ds.
0
Therefore,
r < sup e−Lt Fxr t
t∈0,T qMr
≤ Mb x0 sup
Γ 1 q t∈0,T t
0
t − s
q−1 ∗ ∗
μs m k e
2.62
−Lt−s
ds.
14
Journal of Applied Mathematics
Dividing both sides of 2.62 by r, and taking r → ∞, we have
qM
sup
Γ 1 q t∈0,T t
t − sq−1 μs m∗ k∗ e−Lt−s ds ≥ 1.
2.63
0
This is a contradiction. Hence for some positive number r, FBr ⊂ Br . According to the
following known fact.
Let M be a bounded convex closed subset of E and F : M → M a α-condensing map. Then
Fix F {x : x Fx} is nonempty.
we see that problem 1.2 has at least one mild solution.
Next, for c ∈ 0, 1, we consider the following one-parameter family of maps:
H : 0, 1 × C0, T , X −→ C0, T , X
2.64
c, x −→ Hc, x cFx.
We will demonstrate that the fixed point set of the family H,
2.65
Fix H {x ∈ Hc, x for some c ∈ 0, 1}
is a priori bounded. Indeed, let x ∈ Fix H, for t ∈ 0, T , we have
xt ≤ M
gx x0 t
Zt − s
fs, xs axs
ds
0
qM
≤ Mb x0 Γ 1q
t
t − sq−1 μsxsds
2.66
0
∗ ∗
m k
t
t − s
0
q−1
sup xτds .
τ∈0,s
Noting that the Hölder inequality, we have
t
t − sq−1 μsds ≤
0
≤
p−1
pq − 1
p−1
pq − 1
p−1/p
p−1/p
· tpq−1/p · μ
Lp
· T q−1/p · μ
Lp .
2.67
Therefore, from 2.66, we obtain
p − 1 p−1/p q−1/p qM
μ
p sup xs
T
xt ≤ Mb x0 L
pq
−
1
Γ 1q
s∈0,t
qMm∗ k∗
Γ 1q
t
0
t − sq−1 sup xτds.
τ∈0,s
2.68
Journal of Applied Mathematics
15
We denote
yt : sup xs.
2.69
s∈0,t
Let t ∈ 0, t such that yt xt. Then, by 2.68, we can see
p − 1 p−1/p q−1/p qM
μ
p yt
T
yt ≤ Mb x0 L
Γ 1 q pq − 1
qMm∗ k∗ t
t − sq−1 ysds.
Γ 1q
0
2.70
By a generalization of Gronwall’s lemma for singular kernels 22, Lemma 7.1.1, we deduce
that there exists a constant κ κq such that
yt ≤
≤
Mb x0 p−1/p q−1/p μ
p
1 − qM/Γ 1 q
p − 1 / pq − 1
T
L
t
∗ ∗
κMb x0 qMm k /Γ 1 q
q−1
p−1/p q−1/p 2 0 t − s ds
μ
p
1 − qM/Γ 1 q
p − 1 / pq − 1
T
L
Mb x0 p−1/p q−1/p μ
p
1 − qM/Γ 1 q
p − 1 / pq − 1
T
L
q
∗ ∗
κMb x0 qMm k /Γ 1 q T
p−1/p q−1/p 2
μ
p
q 1 − qM/Γ 1 q
p − 1 / pq − 1
T
L
2.71
: w.
Hence, supt∈0,T xt ≤ w.
Now we consider a closed ball:
BR x ∈ C0, T , X : x0,T ≤ R ⊂ C0, T , X.
2.72
We take the radius R > 0 large enough to contain the set Fix H inside itself. Moreover, from
the proof above, F : BR → C0, T , X is α-condensing. Consequently, the following known
fact implies our conclusion: Let V ⊂ E be a bounded open neighborhood of zero and F : V → E a
α-condensing map satisfying the boundary condition:
x
/ λFx,
for all x ∈ ∂V and 0 < λ ≤ 1. Then, Fix F is nonempty compact.
2.73
16
Journal of Applied Mathematics
3. Example
In this section, let X L2 0, π, we consider the following nonlocal Cauchy problem for an
integrodifferential problem:
q
∂t ut, ξ ∂2
1
ut, ξ
ut, ξ √
·
k
2
∂ξ
k t 1 ut, ξ
t
√
t − s sin
0
ut, 0 ut, π 0,
s · us, ξ
ds,
t
t ∈ 0, 1
t ∈ 0, 1
3.1
j π
u ti , y
u0, ξ ci ξ, y
dy u0 ξ,
1 u ti , y
i0 0
q
where ∂t is the Caputo fractional partial derivative of order 0 < q < 1; ξ ∈ 0, π; k > 0 is a
constant to be specified later;
u0 ξ ∈ X;
j ∈ N ;
0 < t0 < t1 < · · · < tj < 1;
3.2
ci ·, · i 0, 1, . . . , j are continuous functions and there exists a positive constant b such that
j π
ci ξ, y dy ≤ b.
i0
3.3
0
For t ∈ 0, 1, ξ ∈ 0, π, we set
xtξ ut, ξ,
j π
xti y
gxξ ci ξ, y
dy,
1 xti y
0
i0
kt, s t − s,
√
s · xsξ
ht, s, xsξ sin
,
t
3.4
1
xtξ
.
ft, xtξ √
·
k
k t 1 xtξ
On the other hand, it is known that the operator A Au u with DA H 2 0, π ∩
H01 0, π generates an analytic semigroup and uniformly bounded semigroup {T t}t≥0 on
X with T tLX ≤ 1. Therefore, 3.1 is a special case of 1.2.
Moreover, we have
1 for all t ∈ 0, 1,
1
ft, x
≤ √
x : μtx;
kkt
3.5
Journal of Applied Mathematics
17
2 for any w, w
∈ X,
1
ft, w − ft, w
≤ √
w − w,
kkt
3.6
that is, for any bounded set D ⊂ X,
1
χ ft, D ≤ √
χD,
kkt
3.7
√
s · xsξ ≤ mt, sx,
sin
ht, s, x t
3.8
for a.e. t ∈ 0, 1;
3 for almost all t, s ∈ Δ,
where mt, s :
√
s/t, and
m∗ sup
t
t∈0,1
t √
mt, sds sup
0
0
t∈0,1
t
s
ds 2
;
3
3.9
4
√
≤
ht, s, w − ht, s, w
t
s
w − w,
3.10
that is, for any bounded set D ⊂ X,
χht, s, D ≤ ζt, sχD,
where ζt, s :
√
3.11
s/t, and
t
sup
t∈0,1
0
ζt, sds 2
.
3
3.12
Therefore, Theorem 2.4 implies that the problem 3.1 has at least a mild solution
when
q·
p−1/p
p − 1 / pq − 1
μ
p < 1.
L
Γ 1q
3.13
18
Journal of Applied Mathematics
Acknowledgment
The work was supported partly by the Chinese Academy of Sciences, the NSF of Yunnan
Province 2009ZC054M and the NSF of China 11171210.
References
1 S. Aizicovici and M. McKibben, “Existence results for a class of abstract nonlocal Cauchy problems,”
Nonlinear Analysis, vol. 39, pp. 649–668, 2000.
2 L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of
a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11–19,
1991.
3 E. P. Gatsori, “Controllability results for nondensely defined evolution differential inclusions with
nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 194–211,
2004.
4 J. Liang and Z. Fan, “Nonlocal impulsive Cauchy problems for evolution equations,” Advances in
Difference Equations, vol. 2011, Article ID 784161, 17 pages, 2011.
5 J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations
in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.
6 J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,”
Nonlinear Analysis, vol. 57, no. 2, pp. 183–189, 2004.
7 J. Liang, J. van Casteren, and T. J. Xiao, “Nonlocal Cauchy problems for semilinear evolution
equations,” Nonlinear Analysis, vol. 50, pp. 173–189, 2002.
8 J. Liang and T. J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,”
Computers and Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
9 Z. W. Lv, J. Liang, and T. J. Xiao, “Solutions to fractional differential equations with nonlocal initial
condition in Banach spaces,” Advances in Difference Equations, vol. 2010, Article ID 340349, 10 pages,
2010.
10 T. J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolic
problems,” Nonlinear Analysis, vol. 63, pp. e225–e232, 2003.
11 L. Gaul, P. Klein, and S. Kempfle, “Damping description involving fractional operators,” Mechanical
Systems and Signal Processing, vol. 5, pp. 81–88, 1991.
12 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential
Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The
Netherlands, 2006.
13 F. Li, “Solvability of nonautonomous fractional integrodifferential equations with infinite delay,”
Advances in Difference Equations, vol. 2011, Article ID 806729, 18 pages, 2011.
14 F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,”
Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 52, pp. 1–15, 2011.
15 F. Li, T. J. Xiao, and H. K. Xu, “On nonlinear neutral fractional integrodifferential inclusions with
infinite delay,” Journal of Applied Mathematics, vol. 2012, Article ID 916543, 19 pages, 2012.
16 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,
John Wiley & Sons, New York, NY, USA, 1993.
17 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,
Academic Press, San Diego, Calif, USA, 1999.
18 R. N. Wang, D. H. Chen, and T. J. Xiao, “Abstract fractional Cauchy problems with almost sectorial
operators,” Journal of Differential Equations, vol. 252, pp. 202–235, 2012.
19 J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, vol. 60 of Lecture Notes in Pure and
Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
20 M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential
Inclusions in Banach Spaces, vol. 7 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de
Gruyter, Berlin, Germany, 2001.
21 T. J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
22 D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer, Berlin,
Germany, 1989.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014