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Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay
Advances in Difference Equations volume 2011, Article number: 806729 (2011)
Abstract
We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with infinite delay in a Banach space . The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii's fixed point theorem. An application of the abstract results is also given.
1. Introduction
The Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (cf., e.g., [1–15]). This paper is concerned with existence results for nonautonomous fractional integrodifferential equations with infinite delay in a Banach space

where ,
,
is a family of linear operators in
with
and

,
defined by
for
,
belongs to the phase space
, and
. The fractional derivative is understood here in the Riemann-Liouville sense.
In recent years, the fractional differential equations have been proved to be good tools in the investigation of many phenomena in engineering, physics, economy, chemistry, aerodynamics, electrodynamics of complex medium and they have been studied by many researchers (cf., e.g., [13, 14, 16, 17] and references therein). Moreover, many phenomena cannot be described through classical differential equations but the integral and integrodifferential equations in abstract spaces in fields like electronic, fluid dynamics, biological models, and chemical kinetics. So many significant works on this topic have been appeared (cf., e.g., [10, 15, 18–25] and references therein).
In this paper, we study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup
generated by
is compact (see Theorem 3.1). An example is given to show an application of the abstract results.
2. Preliminaries
Throughout this paper, we set , a compact interval in
. We denote by
a Banach space,
the Banach space of all linear and bounded operators on
, and
the space of all
-valued continuous functions on
. We set

Next, we recall the definition of the Riemann-Liouville integral.
Definition 2.1 (see [26]).
The fractional (arbitrary) order integral of the function of order
is defined by

where is the Gamma function. Moreover,
, for all
.
Remark 2.2.
-
(1)
(see [26]),
-
(2)
obviously, for
, it follows from Definition 2.1 that
(2.3)
where is a beta function.
See the following definition about phase space according to Hale and Kato [27].
Definition 2.3.
A linear space consisting of functions from
into
, with seminorm
, is called an admissible phase space if
has the following properties.
-
(1)
If
is continuous on
and
, then
and
is continuous in
, and
(2.4)where
is a constant.
-
(2)
There exist a continuous function
and a locally bounded function
in
, such that
(2.5)for
and
as in (1).
-
(3)
The space
is complete.
Remark 2.4.
Equation (2.4) in (1) is equivalent to , for all
.
Next, we consider the properties of Kuratowski's measure of noncompactness.
Definition 2.5.
Let be a bounded subset of a seminormed linear space
. The Kuratowski's measure of noncompactness(for brevity,
-measure) of
is defined as

From the definition we can get some properties of -measure immediately, see [28].
Lemma 2.6 (see [28]).
Let and
be bounded subsets of
, then
-
(1)
, if
;
-
(2)
, where
denotes the closure of
;
-
(3)
if and only if
is precompact;
-
(4)
,
;
-
(5)
;
-
(6)
, where
;
-
(7)
, for any
.
For , we define

where .
The following lemma will be needed.
Lemma 2.7.
If is a bounded, equicontinuous set, then
(i),
(ii), for
.
For a proof refer to [28].
Lemma 2.8 (see [29]).
If and there exists an
such that
, a.e.
, then
is integrable and

We need to use the following Sadovskii's fixed point theorem here, see [30].
Definition 2.9.
Let be an operator in Banach space
. If
is continuous and takes bounded sets into bounded sets, and
for every bounded set
of
with
, then
is said to be a condensing operator on
.
Lemma 2.10 (Sadovskii's fixed point theorem [30]).
Let be a condensing operator on Banach space
. If
for a convex, closed, and bounded set
of
, then
has a fixed point in
.
In this paper, we denote that is a positive constant, and assume that a family of closed linear operators
satisfying the following.
(A1)The domain of
is dense in the Banach space
and independent of
.
(A2)The operator exists in
for any
with Re 
and

(A3)There exist constants and
such that

Under condition (A2), each operator ,
, generates an analytic semigroup
,
, and there exists a constant
such that

where ,
,
(see [31]).
Let be set defined by

According to [16], a mild solution of (1.1) can be defined as follows.
Definition 2.11.
A function satisfying the equation

is called a mild solution of (1.1), where

and is a probability density function defined on
such that its Laplace transform is given by

where

To our purpose the following conclusions will be needed. For the proofs refer to [16].
Lemma 2.12 (see [16]).
The operator-valued functions and
are continuous in uniform topology in the variables
,
, where
,
, for any
. Clearly,

Moreover, we have

3. Main Results
We need the hypotheses as follows:
(H1) satisfies
is measurable for all
and
is continuous for a.e.
, and there exist a positive function
and a continuous nondecreasing function
, such that

and set ,
(H2)for any bounded sets ,
, and
,

where ,
and
,
(H3)there exists , with
such that

where ,
and
.
Theorem 3.1.
Suppose that (H1)–(H3) are satisfied, and if , then for (1.1) there exists a mild solution on
.
Proof.
Consider the operator defined by

It is easy to see that is well-defined.
Let be the function defined by

Let ,
.
It is easy to see that satisfies
and

if and only if satisfies

and ,
.
Let . For any
,

Thus is a Banach space.
We define the operator by
,
and

Obviously, the operator has a fixed point if and only if
has a fixed point. So it turns out to prove that
has a fixed point.
Let be a sequence such that
in
as
. Since
satisfies (H1), for almost every
, we get

For , we can prove that
is continuous. In fact,

Let , and noting (2.4), (2.5), we have

Moreover,

Noting that in
, we can see that there exists
such that
or
sufficiently large. Therefore, we have

where

In view of (2.17) and the Lebesgue Dominated Convergence Theorem ensure that

Similarly,by (2.17) and (2.18), we have

Therefore, we deduce that

This means that is continuous.
We show that maps bounded sets of
into bounded sets in
. For any
, we set
. Now, for
, by (3.12), (3.13), and (H1), we can see

where .
Then for any , by (2.17), (2.18), (3.19), and Remark 2.2, we have

where .
Noting that the Hölder inequality, we have

Thus

This means .
Next, we show that there exists such that
. Suppose contrary that for every
there exist
and
such that
. However, on the other hand, similar to the deduction of (3.20) and noting

we have

where .
Dividing both sides of (3.24) by , and taking
, we have

This contradicts (3.3). Hence for some positive number ,
.
Let and
, then

where

It follows from Lemma 2.12, (H1) and (3.23) that ,
, as
.
For , from (2.17), (3.23), and (H1), we have

Similarly, by (2.17), (2.18), (H1), and Remark 2.2, we have

So, the set is equicontinuous.
For every bounded set and any
, we can take a sequence
such that
(see [32]), thus from Lemmas 2.6–2.8, and 2.12 and (H2), we have

since is arbitrary, we can obtain

In view of the Sadovskii's fixed point theorem, we conclude that has at least one fixed point
in
. Let
,
, then
is a fixed point of the operator
which is a mild solution of (1.1).
Now we assume that
(H1') there exists a positive function , such that

(H2') there exists a constant , with
, such that the function
defined by

Theorem 3.2.
Assume that (H1') and (H2') are satisfied, then (1.1) has a unique mild solution.
Proof.
Let be defined as in Theorem 3.1. For any
,
, we have

Thus, from (2.17), (2.18), Definition 2.1 and Remark 2.2, we have

So, we get

and the result follows from the contraction mapping principle.
Example 3.3.
We consider the following model:

where ,
,
,
is a continuous function and is uniformly Hölder continuous in
, that is, there exist
and
such that

,
are continuous functions, and
.
Set and define
by

Then generates an analytic semigroup
satisfying assumptions (A1)–(A3) (see [33]).
Let the phase space be
, the space of bounded uniformly continuous functions endowed with the following norm:

then we can see that in (2.5).
For ,
and
, we set

where

now .
Then the above equation (3.37) can be written in the abstract form as (1.1).
Moreover,

where ,
satisfy (H1).
For any ,
,

Therefore, for any bounded sets ,
, we have

 Moreover,

Similarly, we obtain

Suppose further that
-
(1)
there exists
such that
,
-
(2)
,
then (3.37) has a mild solution by Theorem 3.1.
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Acknowledgments
The author is grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (no. 2009ZC054M).
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Li, F. Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay. Adv Differ Equ 2011, 806729 (2011). https://doi.org/10.1155/2011/806729
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DOI: https://doi.org/10.1155/2011/806729
Keywords
- Banach Space
- Probability Density Function
- Mild Solution
- Fractional Differential Equation
- Dominate Convergence Theorem