- Research Article
- Open access
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Existence Results for a Fractional Equation with State-Dependent 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 state-dependent 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 state-dependent delay described in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ3_HTML.gif)
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
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ4_HTML.gif)
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 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ7_HTML.gif)
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)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ9_HTML.gif)
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 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
.
In the sequel, for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ11_HTML.gif)
for ,
, are the paths
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ12_HTML.gif)
and oriented counterclockwise. In addition,
are the sets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ13_HTML.gif)
We now define the operator family by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ19_HTML.gif)
Proof.
The Cauchy problem (2.12)-(2.13) is equivalent to the Volterra equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ20_HTML.gif)
and the -resolvent equation (2.6) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ21_HTML.gif)
To prove (2.16), we notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ22_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ23_HTML.gif)
We obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ26_HTML.gif)
where is given by
. From [7, Lemma 3.12], we obtain that
is a classical solution and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ27_HTML.gif)
Moreover, from (2.24) and taking into account that for all
and
, we deduce the existence of constants
,
(which are independent from
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ28_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ29_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ30_HTML.gif)
Now, by making on
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ32_HTML.gif)
from which we deduce the existence of positive constants (independent from
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ34_HTML.gif)
Noting now that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ35_HTML.gif)
from (2.31), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ36_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ37_HTML.gif)
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.
A function is called a mild solution of the neutral system (1.1)-(1.2) on
if
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ40_HTML.gif)
, 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ42_HTML.gif)
and hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ45_HTML.gif)
where ,
is the convex hull of the set
and
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ49_HTML.gif)
since for every
. If
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ50_HTML.gif)
Denoting by the right-hand side of the last inequality, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ51_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ53_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F642013/MediaObjects/13662_2010_Article_61_Equ54_HTML.gif)
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 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.
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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