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Existence Results for a Fractional Equation with StateDependent Delay
Advances in Difference Equations volume 2011, Article number: 642013 (2011)
Abstract
We provide sufficient conditions for the existence of mild solutions for a class of abstract fractional integrodifferential equations with statedependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.
1. Introduction
In the last two decades, the theory of fractional calculus has gained importance and popularity, due to its wide range of applications in varied fields of sciences and engineering as viscoelasticity, electrochemistry of corrosion, chemical physics, optics and signal processing, and so on. The main object of this paper is to provide sufficient conditions for the existence of mild solutions for a class of abstract partial neutral integrodifferential equations with statedependent delay described in the form
where , are closed linear operators defined on a common domain which is dense in a Banach space , and represent the Caputo derivative of defined by
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 statedependent 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 firstorder 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 statedependent 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 statedependent delay. However, the existence of mild solutions for the class of fractional integrodifferential equations with statedependent 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 fixedpoint 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
In what follows, we recall some definitions, notations, and results that we need in the sequel. Throughout this paper, is a Banach space, and , , are closed linear operators defined on a common domain which is dense in ; the notations and represent the resolvent set of the operator and the domain of endowed with the graph norm, respectively. For , we fix , and we represent by the norm of in . Let and be Banach spaces. In this paper, the notation stands for the Banach space of bounded linear operators from into endowed with the uniform operator topology, and we abbreviate this notation to when . Furthermore, for appropriate functions , the notation denotes the Laplace transform of . The notation stands for the closed ball with center at and radius in . On the other hand, for a bounded function and , the notation is given by
and we simplify this notation to when no confusion about the space arises.
We will define the phase space axiomatically, using ideas and notations developed in [19]. More precisely, will denote the vector space of functions defined from into endowed with a seminorm denoted , and such that the following axioms hold.

(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 ).
Let , and let be a nonnegative measurable function which satisfies the conditions (g5), (g6) in the terminology of [19]. Briefly, this means that is locally integrable and there exists a nonnegative, locally bounded function on , such that , for all and , where is a set with Lebesgue measure zero. The space consists of all classes of functions , such that is continuous on , Lebesguemeasurable, and is Lebesgue integrable on . The seminorm in is defined by
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].
To obtain our results, we assume that the integrodifferential abstract Cauchy problem
has an associated resolvent operator of bounded linear operators on .
Definition 2.2.
A one parameter family of bounded linear operators on is called a resolvent operator of (2.3)(2.4) if the following conditions are verified.

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

(b)
For , , and
(2.5)
for every .
The existence of a resolvent operator for problem (2.3)(2.4) was studied in [20]. In this paper, we have considered the following conditions.

(P1)
The operator is a closed linear operator with dense in . There is positive constants , such that
(2.7)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 operatorvalued 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 .
In the sequel, for and ,
for , , are the paths
and oriented counterclockwise. In addition, are the sets
We now define the operator family by
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.
In the rest of this section, we discuss existence and regularity of solutions of
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.
Let , we define the family by
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.
If there are constants, such that , we obtain
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.
Let . Assume that and is a classical solution of (2.12)(2.13) on , then
Proof.
The Cauchy problem (2.12)(2.13) is equivalent to the Volterra equation
and the resolvent equation (2.6) is equivalent to
To prove (2.16), we notice that
Therefore,
We obtain
It is clear from the preceding definition that is a solution of problem (2.3)(2.4) on for .
Definition 2.9.
Let . A function is called a mild solution of (2.12)(2.13) if
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.
To begin with, we study the case in which . Let be the mild solution of (2.12)(2.13) and assume that . It is easy to see that and
where is given by . From [7, Lemma 3.12], we obtain that is a classical solution and satisfies
Moreover, from (2.24) and taking into account that for all and , we deduce the existence of constants , (which are independent from ) such that
Now, we assume that . Let be a sequence in such that in . From [7, Lemma 3.12], we know that , , is a classical solution of (2.12)(2.13) with in the place of . By using the estimate (2.25), we deduce the existence of functions , such that in and in . These facts, jointly with our assumptions on , permit to conclude that in . On the other hand,
we obtain
Now, by making on
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.
Let , there is on such that on and on . Put proceeding as in the proof of Theorem 2.10. It follows that is a classical solution of (2.12)(2.13). Moreover, from [7, Lemma 3.13], we obtain
from which we deduce the existence of positive constants (independent from ) such that
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 .
Therefore, for all , we have that
Noting now that
from (2.31), we find that
Thus,
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 wellknown result.
Theorem 2.13 (LeraySchauder 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.
A function is called a mild solution of the neutral system (1.1)(1.2) on if , , and
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 .
In the sequel, we introduce the following conditions.

(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)
The function is well defined and continuous from the set
(3.2)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]).
Let be continuous on and . If holds, then
, 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.
Let be the extension of to such that on . Consider the space endowed with the uniform convergence topology and define the operator by
for . It is easy to see that . We prove that there exists such that . If this property is false, then for every there exist and such that . Then, from Lemma 3.4, we find that
and hence,
which contradicts our assumption.
Let be such that , in the sequel; and are the numbers defined by and . To prove that is a condensing operator, we introduce the decomposition , where
for .
It is easy to see that is continuous and a contraction on . Next, we prove that is completely continuous from into .
Step 1.
Let , and let be a positive real number such that . We can infer that
where , is the convex hull of the set and , since
which proves that is relatively compact in .
Step 2.
The set is equicontinuous on .
Let and such that for every with . Under these conditions, for and with , we get
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.
Let conditions , , hold, for every , and assume that . If
where , then there exists a mild solution of (1.1)(1.2) on .
Proof.
Let be the operator defined by (3.4). In the sequel we use Theorem 2.13. If , , then from Lemma 3.4, we have that
since for every . If , we obtain that
Denoting by the righthand side of the last inequality, we obtain that
and hence
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
To finish this section, we apply our results to study an integrodifferential equation which arises in the theory of heat equation. Consider the system
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 .
We next consider the problem of the existence of mild solutions for the system (4.1). To this end, we introduce the following functions:
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 .
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Acknowledgments
The final version of this paper was finished while the third author was visiting the Universidade Federal de Pernambuco (Recife, Brasil) during Desember 2010January 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. CEXAPQ0047609. C. Cuevas is partially supported by CNPQ/Brazil under Grant no. 300365/20080. B. de Andrade is partially supported by CNPQ/Brazil under Grant no. 100994/20113.
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Keywords
 Banach Space
 Classical Solution
 Bounded Linear Operator
 Mild Solution
 Fractional Differential Equation