Advances in Difference Equations volume 2011, Article number: 642013 (2011)
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.
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 state-dependent 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 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.
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.
If 
, 
, 
is continuous on 
and 
, then for every 
the following conditions hold:
is in 
,
,
,
where 
is a constant; 
, 
is continuous, 
is locally bounded, and 
are independent of 
.
For the function 
in (A), the function 
is continuous from 
into 
.
The space 
is complete.
Example 2.1 (the phase space 
).
Let 
, and let 
be a nonnegative measurable function which satisfies the conditions (g-5), (g-6) 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 
, Lebesgue-measurable, 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.
The function 
is strongly continuous and 
for all 
and 
.
For 
, 
, and
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.
The operator 
is a closed linear operator with 
dense in 
. There is positive constants 
, such that
where 
, 
, 
for some 
, and 
for all 
.
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 
.
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 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.
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.
The function 
verifies the following conditions:
the function 
is continuous for every 
, and for every 
, the function 
is strongly measurable,
there exist 
and a continuous nondecreasing function 
, such that 
, for all 
.
For all 
, 
and 
, the set 
is bounded in 
.
The function 
is well defined and continuous from the set
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 right-hand 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.
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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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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
dos Santos, J.P.C., Cuevas, C. & de Andrade, B. Existence Results for a Fractional Equation with State-Dependent Delay. Adv Differ Equ 2011, 642013 (2011). https://doi.org/10.1155/2011/642013
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DOI: https://doi.org/10.1155/2011/642013