Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay

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

(1.1)

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

(1.2)

, 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

(2.1)

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

(2.2)

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

(2.6)

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

(2.7)

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

(2.8)

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

(2.9)

(A3)There exist constants and such that

(2.10)

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

(2.11)

where , , (see [31]).

Let be set defined by

(2.12)

According to [16], a mild solution of (1.1) can be defined as follows.

Definition 2.11.

A function satisfying the equation

(2.13)

is called a mild solution of (1.1), where

(2.14)

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

(2.15)

where

(2.16)

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,

(2.17)

Moreover, we have

(2.18)

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

(3.1)

and set ,

(H2)for any bounded sets , , and ,

(3.2)

where , and ,

(H3)there exists , with such that

(3.3)

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

(3.4)

It is easy to see that is well-defined.

Let be the function defined by

(3.5)

Let , .

It is easy to see that satisfies and

(3.6)

if and only if satisfies

(3.7)

and , .

Let . For any ,

(3.8)

Thus is a Banach space.

We define the operator by , and

(3.9)

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

(3.10)

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

(3.11)

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

(3.12)

Moreover,

(3.13)

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

(3.14)

where

(3.15)

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

(3.16)

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

(3.17)

Therefore, we deduce that

(3.18)

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

(3.19)

where .

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

(3.20)

where .

Noting that the Hölder inequality, we have

(3.21)

Thus

(3.22)

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

(3.23)

we have

(3.24)

where .

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

(3.25)

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

Let and , then

(3.26)

where

(3.27)

It follows from Lemma 2.12, (H1) and (3.23) that , , as .

For , from (2.17), (3.23), and (H1), we have

(3.28)

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

(3.29)

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

(3.30)

since is arbitrary, we can obtain

(3.31)

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

(3.32)

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

(3.33)

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

(3.34)

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

(3.35)

So, we get

(3.36)

and the result follows from the contraction mapping principle.

Example 3.3.

We consider the following model:

(3.37)

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

(3.38)

, are continuous functions, and .

Set and define by

(3.39)

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:

(3.40)

then we can see that in (2.5).

For , and , we set

(3.41)

where

(3.42)

now .

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

Moreover,

(3.43)

where , satisfy (H1).

For any , ,

(3.44)

Therefore, for any bounded sets , , we have

(3.45)

 Moreover,

(3.46)

Similarly, we obtain

(3.47)

Suppose further that

• (1) there exists such that ,

• (2) ,

then (3.37) has a mild solution by Theorem 3.1.