Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay

Abstract

We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space . 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 Henríquez 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 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 :

(1.1)

where , , is a phase space that will be defined later (see Definition 2.5). is a generator of an analytic semigroup of uniformly bounded linear operators on . Then, there exists such that . , , (), and defined by , for , belongs to and . 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 and denote by a real Banach space, by the Banach space of all linear and bounded operators on , and by the Banach space of all -valued continuous functions on with the uniform norm topology.

Let us recall the definition of Kuratowski's measure of noncompactness.

Definition 2.1.

Let be a bounded subset of a seminormed linear space . Kuratowski's measure of noncompactness of is defined as

(2.1)

This measure of noncompactness satisfies some important properties.

Lemma 2.2 (see [17]).

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 ,

• (8) , where is the closed convex hull of .

For , we define

(2.2)

where .

The following lemmas will be needed.

Lemma 2.3 (see [17]).

If is a bounded, equicontinuous set, then

(2.3)

Lemma 2.4 (see [18]).

If and there exists an such that , a.e. , then is integrable and

(2.4)

The following definition about the phase space is due to Hale and Kato [11].

Definition 2.5.

A linear space consisting of functions from into with semi-norm is called an admissible phase space if has the following properties.

• (1) If is continuous on and , then and is continuous in and

  

  (2.5)

  where is a constant.

• (2) There exist a continuous function and a locally bounded function in such that

  

  (2.6)

  for and as in (1).

• (3) The space is complete.

Remark 2.6.

(2.5) in (1) is equivalent to , for all .

The following result will be used later.

Lemma 2.7 (see [19, 20]).

Let be a bounded, closed, and convex subset of a Banach space such that , and let be a continuous mapping of into itself. If the implication

(2.7)

holds for every subset of , then has a fixed point.

Let be a set defined by

(2.8)

Motivated by [4, 5, 21], we give the following definition of mild solution of (1.1).

Definition 2.8.

A function satisfying the equation

(2.9)

is called a mild solution of (1.1), where

(2.10)

and is a probability density function defined on such that

(2.11)

where

(2.12)

Remark 2.9.

According to [22], direct calculation gives that

(2.13)

where .

We list the following basic assumptions of this paper.

(H1) satisfies is measurable, for all and is continuous for a.e. , and there exist two positive functions such that

(2.14)

(H2) For any bounded sets , , and , there exists an integrable positive function such that

(2.15)

where and .

(H3) There exists a constant such that

(2.16)

(H4) For each , is measurable on and is bounded on . The map is continuous from to , here, .

(H5) There exists such that

(2.17)

where , .

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 defined by

(3.1)

It follows from (H1), (H3), and (H4) that is well defined.

It will be shown that has a fixed point, and this fixed point is then a mild solution of (1.1).

Let be the function defined by

(3.2)

Set , .

It is clear to see that satisfies (2.9) if and only if satisfies and for ,

(3.3)

Let . For any ,

(3.4)

Thus, is a Banach space. Set

(3.5)

Then, for , from(2.6), we have

(3.6)

where .

In order to apply Lemma 2.7 to show that has a fixed point, we let be an operator defined by and for ,

(3.7)

Clearly, the operator has a fixed point is equivalent to has one. So, it turns out to prove that 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 provided that .

Proof.

For , , from (3.6), we have

(3.8)

In view of (H3),

(3.9)

where .

Next, we show that there exists some such that . If this is not true, then for each positive number , there exist a function and some such that . However, on the other hand, we have from (3.8), (3.9), and (H4)

(3.10)

Dividing both sides of (3.10) by , and taking , we have

(3.11)

This contradicts (2.17). Hence, for some positive number , .

Let with in as . Since satisfies (H1), for almost every , we get

(3.12)

In view of (3.6), we have

(3.13)

Noting that

(3.14)

we have by the Lebesgue Dominated Convergence Theorem that

(3.15)

Therefore, we obtain

(3.16)

This shows that is continuous.

Set

(3.17)

Let and , then we can see

(3.18)

where

(3.19)

It follows the continuity of in the uniform operator topology for that tends to 0, as . The continuity of ensures that tends to 0, as .

For , we have

(3.20)

Clearly, the first term on the right-hand side of (3.20) tends to 0 as . The second term on the right-hand side of (3.20) tends to 0 as as a consequence of the continuity of in the uniform operator topology for .

In view of the assumption of and (3.8), we see that

(3.21)

Thus, is equicontinuous.

Now, let be an arbitrary subset of such that .

Set ,

(3.22)

Noting that for , we have

(3.23)

Thus,

(3.24)

where . Therefore, .

Moreover, for any and bounded set , we can take a sequence such that (see [23], P125). Thus, for , noting that the choice of , and from Lemmas 2.2–2.4 and (H2), we have

(3.25)

It follows from Lemma 2.2 that

(3.26)

since is arbitrary, we can obtain

(3.27)

Hence, . Applying now Lemma 2.7, we conclude that has a fixed point in . Let , then is a fixed point of the operator which is a mild solution of (1.1).

4. Application

In this section, we consider the following integrodifferential model:

(4.1)

where , , , , are continuous functions, and .

Set and define by

(4.2)

Then, generates a compact, analytic semigroup of uniformly bounded, linear operators, and .

Let the phase space be , the space of bounded uniformly continuous functions endowed with the following norm:

(4.3)

then we can see that in (2.6).

For , and , we set

(4.4)

Then (4.1) can be reformulated as the abstract (1.1).

Moreover, for , we can see

(4.5)

where , .

For , , we have

(4.6)

where .

Suppose further that there exists a constant such that

(4.7)

then (4.1) has a mild solution by Theorem 3.1.

For example, if we put

(4.8)

then , , . Thus, we see

(4.9)