- Research Article
- Open Access

# Existence Results for a Fractional Equation with State-Dependent Delay

- José Paulo Carvalho dos Santos
^{1}, - Claudio Cuevas
^{2}and - Bruno de Andrade
^{3}Email author

**2011**:642013

https://doi.org/10.1155/2011/642013

© José Paulo Carvalho dos Santos et al. 2011

**Received:**26 August 2010**Accepted:**7 March 2011**Published:**14 March 2011

## Abstract

We provide sufficient conditions for the existence of mild solutions for a class of abstract fractional integrodifferential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.

## Keywords

- Banach Space
- Classical Solution
- Bounded Linear Operator
- Mild Solution
- Fractional Differential Equation

## 1. Introduction

where is the smallest integer greater than or equal to and , , . The history given by belongs to some abstract phase space defined axiomatically, and and are appropriated functions.

Functional differential equations with state-dependent delay appear frequently in applications as model of equations, and for this reason, the study of this type of equations has received great attention in the last years. The literature devoted to this subject is concerned fundamentally with first-order functional differential equations for which the state belong to some finite dimensional space, see among other works, [1–10]. The problem of the existence of solutions for partial functional differential equations with state-dependent delay has been recently treated in the literature in [11–15]. On the other hand, existence and uniqueness of solutions for fractional differential equations with delay was recently studied by Maraaba et al. in [16, 17]. In [18], the authors provide sufficient conditions for the existence of mild solutions for a class of fractional integrodifferential equations with state-dependent delay. However, the existence of mild solutions for the class of fractional integrodifferential equations with state-dependent delay of the form (1.1)-(1.2) seems to be an unread topic.

The plan of this paper is as follows. The second section provides the necessary definitions and preliminary results. In particular, we review some of the standard properties of the -resolvent operators (see Theorem 2.11). We also employ an axiomatic definition for the phase space which is similar to those introduced in [19]. In the third section, we use fixed-point theory to establish the existence of mild solutions for the problem (1.1). To show how easily our existence theory can be used in practice, in the fourth section, we illustrate an example.

## 2. Preliminaries

and we simplify this notation to when no confusion about the space arises.

- (A)If , , is continuous on and , then for every the following conditions hold:
- (i)
is in ,

- (ii)
,

- (iii)
,

where is a constant; , is continuous, is locally bounded, and are independent of .

- (i)
- (A1)
For the function in (A), the function is continuous from into .

- (B)
The space is complete.

Example 2.1 (the phase space ).

The space satisfies axioms (A), (A1), (B). Moreover, when and , we can take , , and , for (see [19, Theorem ] for details).

For additional details concerning phase space we refer the reader to [19].

has an associated -resolvent operator of bounded linear operators on .

Definition 2.2.

- (a)
The function is strongly continuous and for all and .

- (b)

for every .

- (P1)The operator is a closed linear operator with dense in . There is positive constants , such that
where , , for some , and for all .

- (P2)
For all , is a closed linear operator, , and is strongly measurable on for each . There exists (the notation stands for the set of all locally integrable functions from into ) such that exists for and for all and . Moreover, the operator-valued function has an analytical extension (still denoted by ) to such that for all , and , as .

- (P3)
There exists a subspace dense in and positive constant , such that , , for every and all .

The following result has been established in [20, Theorem 2.1].

Theorem 2.3.

Assume that conditions (P1)–(P3) are fulfilled, then there is a unique -resolvent operator for problem (2.3)-(2.4).

In what follows, we always assume that the conditions (P1)–(P3) are verified.

We consider now the nonhomogeneous problem.

where and . In the sequel, is the operator function defined by (2.11). We begin by introducing the following concept of classical solution.

Definition 2.4.

A function , is called a classical solution of (2.12)-(2.13) on if ; the condition (2.13) holds and (2.12) is verified on .

Definition 2.5.

for each .

The proof of the next result is in [20]. For reader's convenience, we will give the proof.

Lemma 2.6.

If the function is exponentially bounded in , then is exponentially bounded in .

Proof.

where .

The next result follows from Lemma 2.6. We will omit the proof.

Lemma 2.7.

If the function is exponentially bounded in , then is exponentially bounded in .

We now establish a variation of constants formula for the solutions of (2.12)-(2.13). The proof of the next result is in [20]. For reader's convenience, we will give the proof.

Theorem 2.8.

Proof.

It is clear from the preceding definition that is a solution of problem (2.3)-(2.4) on for .

Definition 2.9.

The proof of the next result is in [20]. For reader's convenience, we will give the proof.

Theorem 2.10.

Let and . If , then the mild solution of (2.12)-(2.13) is a classical solution.

Proof.

we conclude that is a classical solution of (2.12)-(2.13). The proof is finished.

The proof of the next result is in [20]. For reader's convenience, we will give the proof.

Theorem 2.11.

Let and . If , then the mild solution of (2.12)-(2.13) is a classical solution.

Proof.

With similar arguments as in the proof of Theorem 2.10, we conclude that is a classical solution of (2.12)-(2.13). We omit additional details. The proof is completed.

To establish our existence results, we need the following Lemma.

Lemma 2.12.

Let . If is compact for some , then and are compact for all .

Proof.

From the resolvent identity, it follows that is compact for every . We have from [20, Lemma 2.2] that is a compact operator for ; therefore, is a compact operator for .

From [20, Lemma 2.5], we have, is uniformly continuous for , for any fixed, we can select points , such that if , we obtain , for all and .

where is compact and , then we observe that as . This permits us to conclude that is relatively compact in . This proves that is a compact operator for all .

For completeness, we include the following well-known result.

Theorem 2.13 (Leray-Schauder alternative, [21, Theorem ]).

Let be a closed convex subset of a Banach space with . Let be a completely continuous map. Then, has a fixed point in or the set is unbounded.

## 3. Existence Results

In this section, we study the existence of mild solutions for system (1.1)-(1.2). Along this section, is a positive constant such that and for every . We adopt the notion of mild solutions for (1.1)-(1.2) from the one given in [20].

Definition 3.1.

To prove our results, we always assume that is continuous and that . If , we define as the extension of to such that . We define such that where is the extension of , such that for .

- (H1)The function verifies the following conditions:
- (i)
the function is continuous for every , and for every , the function is strongly measurable,

- (ii)
there exist and a continuous nondecreasing function , such that , for all .

- (i)
- (H2)
For all , and , the set is bounded in .

- (Hφ)
into , and there exists a continuous and bounded function , such that for every .

Remark 3.2.

The condition is frequently verified by continuous and bounded functions.

Remark 3.3.

In the rest of this section, and are the constants and .

Lemma 3.4 (see [13, Lemma 2.1]).

, where .

Theorem 3.5.

Let conditions , , and hold, and assume that . If , then there exists a mild solution of (1.1)-(1.2) on .

Proof.

which contradicts our assumption.

for .

It is easy to see that is continuous and a contraction on . Next, we prove that is completely continuous from into .

Step 1.

which proves that is relatively compact in .

Step 2.

The set is equicontinuous on .

which shows that the set of functions is right equicontinuity at . A similar procedure permits us to prove the right equicontinuity at zero and the left equicontinuity at . Thus, is equicontinuous. By using a similar procedure to proof of the [13, Theorem 2.3], we prove that that is continuous on , which completes the proof is completely continuous.

The existence of a mild solution for (1.1)-(1.2) is now a consequence of [22, Theorem ]. This completes the proof.

Theorem 3.6.

where , then there exists a mild solution of (1.1)-(1.2) on .

Proof.

This inequality and (3.11) permit us to conclude that the set of functions is bounded, which in turn shows that is bounded.

By using a similar procedure allows to proof Theorem 3.5, we obtain that is completely continuous. By Theorem 2.13, the proof is ended.

## 4. Example

In this system, , , are positive numbers and . To represent this system in the abstract form (1.1)-(1.2), we choose the spaces and , see Example 2.1 for details. In the sequel, is the operator given by with domain . It is well known that is the infinitesimal generator of an analytic semigroup on . Hence, is sectorial of type and (P1) is satisfied. We also consider the operator , , for . Moreover, it is easy to see that conditions (P2)-(P3) in Section 2 are satisfied with and , where is the space of infinitely differentiable functions that vanish at and .

Under the above conditions, we can represent the system (4.1) in the abstract form (2.12)-(2.13). The following result is a direct consequence of Theorem 3.5.

Proposition 4.1.

Let be such that condition holds, the functions are bounded, and assume that the above conditions are fulfilled, then there exists a mild solution of (4.1) on .

## Declarations

### Acknowledgments

The final version of this paper was finished while the third author was visiting the Universidade Federal de Pernambuco (Recife, Brasil) during Desember 2010-January 2011. The third author would like to thank the Functional Equations Group for their kind invitation and hospitality. The authors are grateful to the referees for pointing out omissions and misprints, and demanding vigorously details and clarifications. J. P. C. dos Santos is partially supported by FAPEMIG/Brazil under Grant no. CEX-APQ-00476-09. C. Cuevas is partially supported by CNPQ/Brazil under Grant no. 300365/2008-0. B. de Andrade is partially supported by CNPQ/Brazil under Grant no. 100994/2011-3.

## Authors’ Affiliations

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