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, fangli860@gmail.com
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, Hernández and Henrı́quez
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
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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
xt − ht, xt Axt − ht, xt dtq
t
xt φt ∈ P,
βt, sfs, xs, xs ds,
t ∈ 0, T,
0
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 {St}t≥0 of uniformly bounded linear operators on X.
Then, there exists M ≥ 1 such that St ≤ 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
xt θ xt θ, for θ ∈ −∞, 0, φ belongs to P and φ0 0. The fractional derivative is
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, and by CJ, 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
αB inf d > 0 : B has a finite cover by sets of diameter ≤ d .
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 αA αA, where A denotes the closure of A,
3 αA 0 if and only if A is precompact,
4 αλA |λ|αA, λ ∈ R,
5 αA ∪ B max{αA, αB},
6 αA B ≤ αA αB, where A B {x y : x ∈ A, y ∈ B},
7 αA a αA for any a ∈ X,
8 αconvA αA, where convA is the closed convex hull of A.
2.1
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For H ⊂ CJ, X, we define
t
t
Hsds usds : u ∈ H
for t ∈ J,
0
2.2
0
where Hs {us ∈ X : u ∈ H}.
The following lemmas will be needed.
Lemma 2.3 see 17. If H ⊂ CJ, X is a bounded, equicontinuous set, then
αH sup αHt.
t∈J
2.3
1
1
Lemma 2.4 see 18. If {un }∞
n1 ⊂ L J, X and there exists an m ∈ L J, R such that un t ≤
mt, a.e. t ∈ J, then α{un t}∞
n1 is integrable and
t
α
∞ ≤2
un sds
0
t
n1
0
α{un s}∞
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 x0 ∈ P, then xt ∈ P and xt is continuous
in t ∈ J and
xt ≤ Cxt 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 xs C2 tx0 P ,
s∈0,t
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 φ0 ≤ CφP , 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 convNV or V NV ∪ {0} ⇒ αV 0
holds for every subset V of U, then N has a fixed point.
2.7
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Let Ω be a set defined by
Ω x : −∞, T −→ X such that x|−∞, 0 ∈ P, x|J ∈ CJ, X .
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
xt ⎧
⎪
⎪
⎨φt,
⎪
⎪
⎩−Qth 0, φ ht, xt t s
0
t ∈ −∞, 0,
Rt − sβs, τfτ, xτ, xτ dτ ds, t ∈ J
0
2.9
is called a mild solution of 1.1, where
Qt ∞
ξq σStq σdσ,
0
Rt q
2.10
∞
σt
q−1
ξq σSt σdσ
q
0
and ξq is a probability density function defined on 0, ∞ such that
ξq σ 1 −1−1/q −1/q σ
≥ 0,
q σ
q
2.11
where
∞
1
n−1 −qn−1 Γ nq 1
sin nπq ,
q σ −1 σ
π n1
n!
σ ∈ 0, ∞.
2.12
Remark 2.9. According to 22, direct calculation gives that
Rt ≤ Cq,M tq−1 ,
t > 0,
2.13
where Cq,M qM/Γ1 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 ft, ·, · : 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
ft, v, w ≤ μ1 tv μ2 twP ,
t, v, w ∈ J × X × P.
2.14
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H2 For any bounded sets D1 ⊂ X, D2 ⊂ P, and 0 ≤ s ≤ t ≤ T, there exists an integrable
positive function η such that
α Rt − sfτ, D1 , D2 ≤ ηt s, τ αD1 sup αD2 θ ,
2.15
−∞<θ≤0
where ηt s, τ : ηt, s, τ and supt∈J
t s
0 0
ηt s, τdτds : η∗ < ∞.
H3 There exists a constant L > 0 such that
P ,
h t1 , ϕ − h t2 , ϕ ≤ L |t1 − t2 | ϕ − ϕ
t1 , t2 ∈ J, ϕ, ϕ ∈ P.
2.16
H4 For each t ∈ J, βt, s is measurable on 0, t and βt ess sup{|βt, s|, 0 ≤ s ≤ t}
is bounded on J. The map t → Bt is continuous from J to L∞ J, R, here, Bt s βt, s.
H5 There exists M∗ ∈ 0, 1 such that
LC1∗ T q βCq,M μ1 L1 J, R C1∗ μ2 L1 J, R < M∗ ,
q
2.17
where C1∗ supt∈J C1 t, β supt∈J βt.
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
Φxt ⎧
⎪
⎪
⎨φt,
⎪
⎪
⎩−Qth 0, φ ht, xt t s
0
t ∈ −∞, 0,
Rt − sβs, τfτ, xτ, xτ dτ ds, t ∈ J.
0
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
yt ⎧
⎨φt, t ∈ −∞, 0,
⎩0,
Set xt yt zt, t ∈ −∞, T.
t ∈ J.
3.2
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It is clear to see that x satisfies 2.9 if and only if z satisfies z0 0 and for t ∈ J,
zt −Qth 0, φ h t, yt zt t s
0
0
Rt − sβs, τf τ, yτ zτ, yτ zτ dτ ds.
3.3
Let Z0 {z ∈ Ω : z0 0}. For any z ∈ Z0 ,
zZ0 supzt z0 P supzt.
t∈J
t∈J
3.4
Thus, Z0 , · Z0 is a Banach space. Set
Br z ∈ Z0 : zZ0 ≤ r ,
for some r > 0.
3.5
Then, for z ∈ Br , from2.6, we have
yt zt P ≤ yt P zt P
≤ C1 tsup yτ C2 ty0 P C1 tsup zτ C2 tz0 P
0≤τ≤t
0≤τ≤t
C2 tφP C1 tsup zτ
3.6
0≤τ≤t
≤ C2∗ · φP C1∗ r : r ∗ ,
where C2∗ sup0≤η≤T C2 η.
: Z0 → Z0 be
In order to apply Lemma 2.7 to show that Φ has a fixed point, we let Φ
an operator defined by Φzt 0, t ∈ −∞, 0 and for t ∈ J,
t −Qth 0, φ h t, y zt
Φz
t
t s
0
0
Rt − sβs, τf τ, yτ zτ, yτ zτ dτ ds.
3.7
has one. So, it turns out to
Clearly, the operator Φ has a fixed point is equivalent to Φ
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, yt zt, y t zt ≤ μ1 tyt zt μ2 tyt zt P
≤ μ1 tr μ2 tr ∗ .
3.8
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In view of H3,
h t, yt zt ≤ h t, y t zt − ht, 0 ht, 0
≤ Lyt zt P M1
3.9
≤ Lr ∗ M1 ,
where M1 supt∈J ht, 0.
r ⊂ Br . If this is not true,
Next, we show that there exists some r > 0 such that ΦB
then for each positive number r, there exist a function zr · ∈ Br and some t ∈ J such that
r t > r. However, on the other hand, we have from 3.8, 3.9, and H4
Φz
r t
r < Φz
≤ − Qth 0, φ h t, yt zrt t s
Rt − sβs, τf τ, yτ zr τ, yτ zrτ dτ ds
0
0
≤ LMφP MM1 Lr ∗ M1 βCq,M
t s
≤ LMφP MM1 Lr ∗ M1 βrCq,M
βr ∗ Cq,M
t s
0
0
t − sq−1 μ1 τr μ2 τr ∗ dτ ds
0
t s
0
3.10
t − sq−1 μ1 τdτ ds
0
t − sq−1 μ2 τdτ ds
0
T q βCq,M rμ1 L1 J,R r ∗ μ2 L1 J,R .
≤ L MφP r ∗ M1 M 1 q
Dividing both sides of 3.10 by r, and taking r → ∞, we have
LC1∗ T q βCq,M μ1 L1 J,R C1∗ μ2 L1 J,R ≥ 1.
q
3.11
r ⊂ Br .
This contradicts 2.17. Hence, for some positive number r, ΦB
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, yt zk t, yt zkt −→ f t, yt zt, yt zt ,
as k → ∞.
3.12
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In view of 3.6, we have
y t zkt ≤ r ∗ .
P
3.13
Noting that
f t, yt zk t, yt zkt − f t, yt zt, yt zt ≤ 2μ1 tr 2μ2 tr ∗ ,
3.14
we have by the Lebesgue Dominated Convergence Theorem that
k
t
Φz t − Φz
≤ h t, y t zkt − h t, yt zt t s
Rt − sβs, τ f τ, yτ zk τ, yτ zkτ − f τ, yτ zτ, y τ zτ dτ ds
0
0
≤ Lzkt − zt P
βCq,M
t s
0
−→ 0,
0
t − sq−1 f τ, yτ zk τ, y τ zkτ − f τ, yτ zτ, yτ zτ dτ ds
k −→ ∞.
3.15
Therefore, we obtain
k − Φz
lim Φz
k→∞
Z0
0.
3.16
is continuous.
This shows that Φ
Set
·
G ·, y· z·, y · z· : β·, τf τ, yτ zτ, yτ zτ dτ.
3.17
0
Let 0 < t2 < t1 < T and z ∈ Br , then we can see
t2 Φz t1 − Φz
≤ I1 I2 I3 I4 ,
3.18
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where
I1 Qt1 − Qt2 · h 0, φ ,
I2 h t1 , yt1 zt1 − h t2 , yt2 zt2 ,
t2
I3 Rt1 − s − Rt2 − sG s, ys zs, ys zs ds,
0
I4 t1
t2
3.19
Rt1 − sG s, ys zs, ys zs ds.
It follows the continuity of St 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
t 2 ∞ I3 ≤ q
σ t1 − sq−1 − t2 − sq−1 ξq σS t1 − sq σ G s, ys zs, ys zs dσds
0
0
q
t2 ∞
σt2 − sq−1 ξq σS t1 − sq σ − S t2 − sq σ 0
0
× G s, ys zs, ys zs dσ ds
t2 ≤ Cq,M
t1 − sq−1 − t2 − sq−1 G s, ys zs, ys zs ds
0
q
t2 ∞
σt2 − sq−1 ξq σS t1 − sq σ − S t2 − sq σ 0
0
× G s, ys zs, ys zs dσ ds,
≤ β rμ1 L1 J,R r ∗ μ2 L1 J,R × Cq,M
t2 t1 − sq−1 − t2 − sq−1 ds
0
q
t2 ∞
0
q−1
σt2 − s
q q ξq σ S t1 − s σ − S t2 − s σ dσ ds .
0
3.20
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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
St 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
t2
t1 − sq−1 G s, ys zs, ys zs ds
t1
≤ βCq,M rμ1 L1 J,R r ∗ μ2 L1 J,R t1 − sq−1 ds
3.21
t2
−→ 0,
as t2 −→ t1 .
r is equicontinuous.
Thus, ΦB
∪ {0}.
Now, let V be an arbitrary subset of Br such that V ⊂ convΦV
1 zt ht, y zt ,
Set Φ
t
t s
2 z t −Qth 0, φ Rt − sβs, τf τ, yτ zτ, y τ zτ dτ ds.
Φ
0
3.22
0
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 ≤ LαVt ≤ L sup αV t θ Lsup αV τ ≤ LαV ,
−∞<θ≤0
0≤τ≤t
3.24
1 V sup αΦ
1 V t ≤ LαV .
where Vt {zt : z ∈ V }. Therefore, αΦ
t∈J
Moreover, for any ε > 0 and bounded set D, we can take a sequence {vn }∞
n1 ⊂ D such
that αD ≤ 2α{vn } ε see 23, P125. Thus, for {vn }∞
n1 ⊂ V , noting that the choice of V ,
and from Lemmas 2.2–2.4 and H2, we have
Advances in Difference Equations
2 vn ε 2 sup α Φ
2 vn t ε
2 V ≤ 2α Φ
α Φ
11
t∈J
t
s
2 sup α
ε
Rt − s βs, τf τ, yτ vn τ, yτ vnτ dτds
0
t∈J
≤ 4 sup
t∈J
≤ 8 sup
0
s
t !
α Rt − s βs, τf τ, yτ vn τ, yτ vnτ dτ
ds ε
0
0
t s
t∈J
≤ 8β sup
t∈J
≤ 8β sup
t∈J
≤ 8β sup
t∈J
0
0
dτ ds ε
α Rt − sβs, τf τ, yτ vn τ, yτ vnτ
t s
0
0
t s
0
3.25
ηt s, τ α{vn τ} sup α{vn θ τ} dτ ds ε
−∞<θ≤0
0
t s
0
dτ ds ε
α Rt − sf τ, yτ vn τ, yτ vnτ
dτ ds ε
ηt s, τ α{vn } sup α vn μ
0
0≤μ≤τ
≤ 16βα{vn }sup
t∈J
t s
0
ηt s, τdτ ds ε ≤ 16βη∗ αV ε.
0
It follows from Lemma 2.2 that
1V α Φ
2 V ≤ L 16βη∗ αV ε,
αV ≤ α ΦV
≤α Φ
3.26
since ε is arbitrary, we can obtain
αV ≤ L 16βη∗ αV .
3.27
has a fixed point z∗ in Br .
Hence, αV 0. Applying now Lemma 2.7, we conclude that Φ
∗
Let xt yt z t, t ∈ −∞, T, then xt is a fixed point of the operator Φ which is a mild
solution of 1.1.
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4. Application
In this section, we consider the following integrodifferential model:
0
∂q
|vt θ, ξ|
dθ
γ1 θ
vt, ξ − t
∂tq
1 |vt θ, ξ|
−∞
0
∂2
|vt θ, ξ|
2 vt, ξ − t
dθ
γ1 θ
1 |vt θ, ξ|
∂ξ
−∞
t
sk
t − s sin|vs, ξ| ·
k
0
t
t − s
0
0
vt, 0 − t
vt, 1 − t
0
−∞
0
−∞
−∞
s
cos vτ, ξdτds
0
γ2 θ sin s2 |vs θ, ξ| dθds,
γ1 θ
|vt θ, 0|
dθ 0,
1 |vt θ, 0|
γ1 θ
|vt θ, 1|
dθ 0,
1 |vt θ, 1|
vθ, ξ v0 θ, ξ,
4.1
−∞ < 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
DA H 2 0, 1 ∩ H01 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:
ϕP sup ϕθ,
−∞<θ≤0
then we can see that C1 t 1 in 2.6.
∀ ϕ ∈ P,
4.3
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For t ∈ 0, 1, ξ ∈ 0, 1 and ϕ ∈ BUCR− , X, we set
xtξ vt, ξ,
φθξ v0 θ, ξ, θ ∈ −∞, 0,
0
ϕθξ
dθ,
γ1 θ
h t, ϕ ξ t
1 ϕθξ
−∞
4.4
βt, s t − s,
0
t
tk
f t, xt, ϕ ξ sin|xtξ| · cos xsξds γ2 θ sin t2 ϕθξ dθ.
k
0
−∞
Then 4.1 can be reformulated as the abstract 1.1.
Moreover, for t ∈ 0, 1, we can see
tk1
xt t2 ϕP
f t, xt, ϕ ξ ≤
k
0
−∞
γ2 θdθ
4.5
μ1 txt μ2 tϕP ,
0
where μ1 t : tk1 /k, μ2 t : t2 −∞ |γ2 θ|dθ.
For t1 , t2 ∈ 0, 1, ϕ, ϕ ∈ P, we have
0 ϕθξ dθ
γ1 θ
1 ϕθξ −∞ 0 ϕθξ
ϕθξ
−
t2
dθ
γ1 θ
1 ϕθξ 1 ϕθξ
−∞ h t1 , ϕ − h t2 , ϕ ≤ |t1 − t2 |
≤ |t1 − t2 |
0
−∞
γ1 θdθ L |t1 − t2 | ϕ − ϕ
P ,
0
−∞
γ1 θdθ · 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 μ1 L1 0,1,R μ2 L1 0,1,R < M∗ ,
q
4.7
then 4.1 has a mild solution by Theorem 3.1.
For example, if we put
γ1 θ γ2 θ ekθ ,
q 0.5, k 2,
4.8
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√
then L 1/2, Cq,M 1/Γ0.5 1/ π, μ1 L1 0,1,R 1/8, μ2 L1 0,1,R 1/6. Thus, we
see
L
1
Cq,M 2
μ1 L1 0,1,R μ2 L1 0,1,R √
q
2
π
1 1
8 6
!
< 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.
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