Advances in Difference Equations volume 2009, Article number: 981728 (2009)
We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the 
-semigroups theory combined with some suitable fixed point theorems.
Differential equations and inclusions of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, including some applications and recent results, see the monographs of Kilbas et al. [1], Kiryakova [2], Miller and Ross [3], Podlubny [4] and Samko et al. [5], and the papers of Agarwal et al. [6], Diethelm et al. [7, 8], El-Sayed [9–11], Gaul et al. [12], Glockle and Nonnenmacher [13], Lakshmikantham and Devi [14], Mainardi [15], Metzler et al. [16], Momani et al. [17, 18], Podlubny et al. [19], Yu and Gao [20] and the references therein. Some classes of evolution equations have been considered by El-Borai [21, 22], Jaradat et al. [23] studied the existence and uniqueness of mild solutions for a class of initial value problem for a semilinear integrodifferential equation involving the Caputo's fractional derivative.
In this survey paper, we give existence results for various classes of initial value problems for fractional semilinear functional differential equations and inclusions, both cases of finite and infinite delay are considered. More precisely the paper is organized as follows. In the second section we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. In the third section we will be concerned with semilinear functional differential equations with finite as well infinite delay. In the forth section, we consider semilinear functional differential equation of neutral type for the both cases of finite and infinite delay. Section 5 is devoted to the study of functional differential inclusions, we examine the case when the right-hand side is convex valued as well as nonconvex valued. In Section 6, we will be concerned with perturbed functional differential equations and inclusions. In the last section, we give some existence results of extremal solutions in ordered Banach spaces.
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let 
be a Banach space and 
a compact real interval. 
is the Banach space of all continuous functions from 
into 
with the norm
For 
the norm of 
is defined by
For 
the norm of 
is defined by
denotes the Banach space of bounded linear operators from 
into 
with norm
denotes the Banach space of measurable functions 
which are Bochner integrable normed by
Definition 2.1.
A semigroup of class 
is a one parameter family 
satisfying the conditions
for all 
the map 
is strongly continuous, for each 
, that is,
It is well known that the operator 
generates a 
semigroup if 
satisfies
the Hille-Yosida condition, that is, there exists 
and 
such that 
, 
where 
is the resolvent set of 
and 
is the identity operator in 
.
For more details on strongly continuous operators, we refer the reader to the books of Goldstein [24], Fattorini [25], and the papers of Travis and Webb [26, 27], and for properties on semigroup theory we refer the interested reader to the books of Ahmed [28], Goldstein [24], and Pazy [29].
In all our paper we adopt the following definitions of fractional primitive and fractional derivative.
The Riemann-Liouville fractional primitive of order 
of a function 
of order 
is defined by
provided the right side is pointwise defined on 
, and where 
is the gamma function.
For instance, 
exists for all 
, when 
; note also that when 
, then 
and moreover 
The Riemann-Liouville fractional derivative of order 
of a continuous function 
is defined by
Let 
be a metric space. We use the notations
Consider 
given by
where 
Then 
is a metric space and 
is a generalized metric space (see [30]).
A multivalued map 
is said to be measurable if, for each 
, the function 
defined by
is measurable.
Definition 2.4.
A measurable multivalued function 
is said to be integrably bounded if there exists a function 
such that 
a.e. 
for all 
A multivalued map 
is convex (closed) valued if 
is convex (closed) for all 
. 
is bounded on bounded sets if 
is bounded in 
for all 
, that is, 
.
is called upper semicontinuous (u.s.c. for short) on 
if for each 
the set 
is nonempty, closed subset of 
, and for each open set 
of 
containing 
, there exists an open neighborhood 
of 
such that 
is said to be completely continuous if 
is relatively compact for every 
If the multivalued map 
is completely continuous with nonempty compact valued, then 
is u.s.c. if and only if 
has closed graph, that is, 
imply 
Definition 2.5.
A multivalued map 
is said to be Carathéodory if
is measurable for each 
is u.s.c. for almost all 
Furthermore, a Carathéodory map 
is said to be 
-Carathéodory if
for each real number 
, there exists a function 
such that
for a.e. 
and for all 
Definition 2.6.
A multivalued operator 
is called
-Lipschitz if and only if there exists 
such that
contraction if and only if it is 
-Lipschitz with 
has a fixed point if there exists 
such that 
The fixed point set of the multivalued operator 
will be denoted by 
For more details on multivalued maps and the proof of the known results cited in this section we refer interested reader to the books of Deimling [31], Gorniewicz [32], and Hu and Papageorgiou [33].
Essential for the main results of this paper, we state a generalization of Gronwall's lemma for singular kernels [34, Lemma 7.1.1].
Lemma 2.7.
Let 
be continuous functions. If 
is nondecreasing and there are constants 
and 
such that
then there exists a constant 
such that
for every 
In the sequel, the following fixed point theorems will be used. The following fixed point theorem for contraction multivalued maps is due to Covitz and Nadler [35].
Theorem 2.8.
Let 
be a complete metric space. If 
is a contraction, then 
The nonlinear alternative of Leray-Schauder applied to completely continuous operators [36].
Theorem 2.9.
Let 
be a Banach space, and 
convex with 
. Let 
be a completely continuous operator. Then either
has a fixed point, or
the set 
is unbounded.
The following is the multivalued version of the previous theorem due to Martelli [37].
Theorem 2.10.
Let 
be an upper semicontinuous and completely continuous multivalued map. If the set
is bounded, then 
has a fixed point.
To state existence results for perturbed differential equations and inclusions we will use the following fixed point theorem of Krasnoselskii-Scheafer type of the sum of a completely continuous operator and a contraction one due to Burton and Kirk [38].
Theorem 2.11.
Let 
be a Banach space, and 
two operators satisfying
is a contraction;
is completely continuous.
Then either
the operator equation 
has a solution, or
the set 
is unbounded for 
.
Recently Dhage states the multivalued version of the previous theorem.
Let 
be a Banach space, 
and 
two multivalued operators satisfying
is a contraction;
is completely continuous.
Then either
The operator inclusion 
has a solution for 
, or
the set 
is unbounded.
In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on 
, 
, endowed with the uniform norm topology. When the delay is infinite, the notion of the phase space plays an important role in the study of both qualitative and quantitative theory, a usual choice is a seminormed space 
introduced by Hale and Kato [41] and satisfying the following axioms.
(A1) There exist a positive constant 
and functions 
, 
with 
continuous and 
locally bounded, such that for any 
, if 
, 
, and 
is continuous on 
, then for every 
the following conditions hold:
is in 
and 
, and 
are independent of 
.
(A2) For the function 
in 
, 
is a 
-valued continuous function on 
(A3) The space 
is complete.
Denote by
Hereafter are some examples of phase spaces. For other details we refer, for instance, to the book by Hino et al. [42].
Example 2.13.
The spaces 
, and 
.
BC is the space of bounded continuous functions defined from 
to 
BUC is the space of bounded uniformly continuous functions defined from 
to 
We have that the spaces 
, and 
satisfy conditions 
. 
satisfies 
but 
is not satisfied.
Example 2.14.
The spaces 
, and 
.
Let 
be a positive continuous function on 
. We define
We consider the following condition on the function 
.
For all 
Then we have that the spaces 
and 
satisfy conditions 
. They satisfy conditions 
and 
if 
holds.
Example 2.15.
The space 
.
For any real constant 
, we define the functional space 
bys
endowed with the following norm
Then in the space 
the axioms 
are satisfied.
Functional differential and partial differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for functional differential equations is the books by Hale [43] and Hale and Verduyn Lunel [44], Kolmanovskii and Myshkis [45], and Wu [46] and the references therein.
In a series of papers (see [47–50]), the authors considered some classes of initial value problems for functional differential equations involving the Riemann-Liouville and Caputo fractional derivatives of order 
In [51, 52] some classes of semilinear functional differential equations involving the Riemann-Liouville have been considered. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types see [53, 54].
In the following, we consider the semilinear functional differential equation of fractional order of the form
where 
is the standard Riemann-Liouville fractional derivative, 
is a continuous function, 
is a closed linear operator (possibly unbounded), 
a given continuous function with 
, and 
a real Banach space. For any function 
defined on 
and any 
we denote by 
the element of 
defined by
Here 
represents the history of the state from time 
, up to the present time 
.
The reason for studying (3.1) is that it appears in mathematical models of viscoelasticity [55], and in other fields of science [54, 56]. Equation (3.1) is equivalent to solve an integral equation of convolution type. It is also of interest to explore the neighborhood of the diffusion (
). In this survey paper, we use the fractional derivative in the Riemann-Liouville sense. The problems considered in the survey are subject to zero data, which in this case, the Riemann-Liouville and Caputo fractional derivatives coincide. From a practical point of view, in some mathematical models it is more appropriate to consider traditional initial or boundary data. This is what we are considering in this survey.
In all our paper we suppose that the operator 
is the infinitesimal generator of a 
-semigroup 
. Denote by
Before stating our main results in this section for problem (3.1) and (3.2) we give the definition of the mild solution.
Definition 3.1 (see [23]).
One says that a continuous function 
is a mild solution of problem (3.1) and (3.2) if 
and
By using the Banach's contraction principle, we get the following existence result for problem (3.1) and (3.2).
Theorem 3.2.
Let 
continuous. Assume the following.
(H1) There exists a nonnegative constant 
such that
Then there exists a unique mild solution of problem (3.1) and (3.2) on 
Proof.
Transform the IVP (3.1) and (3.2) into a fixed point problem. Consider the operator 
defined by
Let us define the iterates of operator 
by
It will be sufficient to prove that 
is a contraction operator for 
sufficiently large. For every 
we have
Indeed,
Therefore (3.9) is proved for 
. Assuming by induction that (3.9) is valid for 
, then
and then (3.9) follows for 
.
Now, taking 
sufficiently large in (3.9) yield the contraction of operator 
.
Consequently 
has a unique fixed point by the Banach's contraction principle, which gives rise to a unique mild solution to the problem (3.1) and (3.2).
The following existence result is based upon Theorem 2.9.
Theorem 3.3.
Assume that the following hypotheses hold.
(H2) The semigroup 
is compact for 
.
(H3) There exist functions 
such that
Then the problem (3.1) and (3.2) has at least one mild solution on 
Proof.
Transform the IVP (3.1) and (3.2) into a fixed point problem. Consider the operator 
as defined in Theorem 3.2. To show that 
is continuous, let us consider a sequence 
such that 
in 
. Then
Since 
is a continuous function, then we have
Thus 
is continuous. Now for any 
, and each 
we have for each 
Thus 
maps bounded sets into bounded sets in 
.
Now, let 
, 
Thus if 
and 
we have for any 
As 
and 
sufficiently small, the right-hand side of the above inequality tends to zero, since 
is a strongly continuous operator and the compactness of 
for 
implies the continuity in the uniform operator topology [29]. By the Arzelá-Ascoli theorem it suffices to show that 
maps 
into a precompact set in 
.
Let 
be fixed and let 
be a real number satisfying 
. For 
we define
Since 
is a compact operator for 
, the set
is precompact in 
for every 
Moreover
Therefore, the set 
is precompact in 
. Hence the operator 
is completely continuous. Now, it remains to show that the set
is bounded.
Let 
be any element. Then, for each 
,
Then
We consider the function defined by
Let 
such that 
, if 
then by (3.22) we have, for 
(note 
)
If 
then 
and the previous inequality holds.
By Lemma 2.7 we have
Hence
This shows that the set 
is bounded. As a consequence of Theorem 2.9, we deduce that the operator 
has a fixed point which is a mild solution of the problem (3.1) and (3.2).
As an application of our results we consider the following partial functional differential equation of the form
where 
is continuous and 
is a given function.
Let
Take 
and define 
by 
with domain
Then
where 
is the inner product in 
and 
is the orthogonal set of eigenvectors in 
It is well known (see [29]) that 
is the infinitesimal generator of an analytic semigroup 
in 
and is given by
Since the analytic semigroup 
is compact, there exists a constant 
such that
Also assume that there exist continuous functions 
such that
We can show that problem (3.1) and (3.2) is an abstract formulation of problem (3.27). Since all the conditions of Theorem 3.3 are satisfied, the problem (3.27) has a solution 
on 
In the following we will extend the previous results to the case when the delay is infinite. More precisely we consider the following problem
where 
is the standard Riemann-Liouville fractional derivative, 
is a continuous function, 
the phase space [41], 
is the infinitesimal generator of a strongly continuous semigroup 
, 
a continuous function with 
and 
a real Banach space. For any 
the function 
is defined by
Consider the following space:
where 
is the restriction of 
to 
Let 
be the seminorm in 
defined by
Definition 3.4.
One says that a function 
is a mild solution of problem (3.34) if 
and
The first existence result is based on Banach's contraction principle.
Theorem 3.5.
Assume the following.
(H4) There exists a nonnegative constant 
such that
Then there exists a unique mild solution of problem (3.34) on 
Proof.
Transform the IVP (3.34) into a fixed point problem. Consider the operator 
defined by
For 
, we define the function
Then 
. Set
It is obvious that 
satisfies (3.38) if and only if 
satisfies 
and
Let
For any 
, we have
Thus 
is a Banach space. Let the operator 
defined by
It is obvious that 
has a fixed point is equivalent to 
has a fixed point, and so we turn to proving that 
has a fixed point. As in Theorem 3.2, we show by induction that 
satisfy for any 
, the following inequality:
which yields the contraction of 
for sufficiently large values of 
. Therefore, by the Banach's contraction principle 
has a unique fixed point 
. Then 
is a fixed point of the operator 
, which gives rise to a unique mild solution of the problem (3.34).
Next we give an existence result based upon the nonlinear alternative of Leray-Schauder type.
Theorem 3.6.
Assume that the following hypotheses hold.
(H5) The semigroup 
is compact for 
.
(H6) There exist functions 
such that
Then, the problem (3.34) has at least one mild solution on 
Proof.
Transform the IVP (3.34) into a fixed point problem. Consider the operator 
defined as in Theorem 3.5. We will show that the operator 
is continuous and completely continuous. Let 
be a sequence such that 
in 
. Then
Since 
is a continuous function, then we have
Thus 
is continuous. To show that 
maps bounded sets into bounded sets in 
it is enough to show that for any 
there exists a positive constant 
such that for each 
we have 
Let 
, then
Then we have for each 
Taking the supremum over 
we have
Now let 
, 
thus if 
and 
we have for each 
As 
and 
sufficiently small, the right-hand side of the above inequality tends to zero, since 
is a strongly continuous operator and the compactness of 
for 
implies the continuity in the uniform operator topology (see [29]). By the Arzelá-Ascoli theorem it suffices to show that 
maps 
into a precompact set in 
. Let 
be fixed and let 
be a real number satisfying 
. For 
we define
Since 
is a compact operator for 
, the set
is precompact in 
for every 
Moreover
Therefore, the set 
is precompact in 
. Hence the operator 
is completely continuous. Now, it remains to show that the set
is bounded. Let 
be any element. Then, for each 
,
Then
but
Take the right-hand side of the above inequality as 
, then by (3.60) we have
Using the above inequality and the definition of 
we have
By Lemma 2.7, there exists a constant 
such that we have
Then there exists a constant 
such that 
This shows that the set 
is bounded. As a consequence of the Leray-Schauder Theorem, we deduce that the operator 
has a fixed point, then 
has one which gives rise to a mild solution of the problem (3.34).
To illustrate the previous results, we consider in this section the following model:
where 
are continuous functions.
Consider 
and define 
by 
with domain
Then 
generates a 
semigroup 
(see [29]).
For the phase space 
, we choose the well-known space 
: the space of uniformly bounded continuous functions endowed with the following norm:
If we put for 
and 
Then, problem (3.65) takes the abstract neutral evolution form (3.34).
Neutral differential equations arise in many areas of applied mathematics, an extensive theory is developed, we refer the reader to the book by Hale and Verduyn Lunel [44] and Kolmanovskii and Myshkis [45]. The work for neutral functional differential equations with infinite delay was initiated by Hernández and Henríquez in [57, 58]. In the following, we will extend such results to arbitrary order functional differential equations of neutral type with finite as well as infinite delay. We based our main results upon the Banach's principle and the Leray-Schauder theorem.
First we will be concerned by the case when the delay is finite, more precisely we consider the following class of neutral functional differential equations
Definition 4.1.
One says that a function 
is a mild solution of problem (4.1) if 
and
Our first existence result is based on the Banach's contraction principle.
Theorem 4.2.
Assume the following.
(H7) There exists a nonnegative constant 
such that
(H8) There exists a nonnegative constant 
such that
Then there exists a unique mild solution of problem (4.1) on 
Proof.
Transform the IVP (4.1) into a fixed point problem. Consider the operator 
defined by
As in Theorem 3.2, we show by induction that 
satisfy for any 
, the following inequality:
which yields the contraction of 
for sufficiently large values of 
. Therefore, by the Banach's contraction principle 
has a unique fixed point which gives rise to unique mild solution of problem (4.1).
Next we give an existence result using the nonlinear alternative of Leray-Schauder.
Theorem 4.3.
Assume that the following hypotheses hold.
(H9) The semigroup 
is compact for 
.
(H10) There exist functions 
such that
(H11) The function 
is continuous and completely continuous, and for every bounded set 
, the set 
is equicontinuous in 
.
(H12) There exists constants: 
such that
Then the problem (4.1) has at least one mild solution on 
Proof.
Consider the operator 
as in Theorem 4.2.
To show that the operator 
is continuous and completely continuous it suffices to show, using 
, that the operator 
defined by
is continuous and completely continuous. This can be done following the proof of Theorem 3.3.
Now, it remains to show that the set
is bounded. Let 
be any element. Then, for each 
,
We consider the function defined by
Let 
such that 
, If 
then we have, for 
(note 
)
If 
then 
and the previous inequality holds.
By Lemma 2.7 there exists 
such that
This shows that the set 
is bounded. As a consequence of the Leray-Schauder Theorem, we deduce that the operator 
has a fixed point which gives rise to a mild solution of the problem (4.1).
In the following we will extend our previous results to the case of infinite delay, more precisely we consider the following problem:
Our first existence result is based on the Banach's contraction principle.
Theorem 4.4.
Assume that the following hypotheses hold.
(H13) There exists a nonnegative constant 
such that
(H14) There exists a nonnegative constant 
such that
Then there exists a unique mild solution of problem (4.15) on 
Proof.
Consider the operator 
defined by
In analogy to Theorem 3.2, we consider the operator 
defined by
As in Theorem 3.2, we show by induction that 
satisfy for any 
, the following inequality:
which yields the contraction of 
for sufficiently large values of 
. Therefore, by the Banach's contraction principle 
has a unique fixed point 
. Then 
is a fixed point of the operator 
, which gives rise to a unique mild solution of the problem (4.15).
Next we give an existence result based upon the the nonlinear alternative of Leray-Schauder.
Theorem 4.5.
Assume that the following hypotheses hold.
(H15) The semigroup 
is compact for 
.
(H16) There exist functions 
such that
(H17) The function 
is continuous and completely continuous, and for every bounded set 
, the set 
is equicontinuous in 
.
(H18) There exists constants: 
such that
Then the problem (4.15) has at least one mild solution on 
Proof.
Let 
defined as in Theorem 4.4. We can easily show that the operator 
is continuous and completely continuous. Using 
it suffices to show that the operator 
defined by
is continuous and completely continuous. Now, it remains to show that the set
is bounded.
Let 
be any element. Then, for each 
,
Denote 
as in Theorem 3.6. Then
Then
By Lemma 2.7 there exists a constant 
such that
where
Then there exists a constant 
such that 
This shows that the set 
is bounded. As a consequence of the Leray-Schauder Theorem, we deduce that the operator 
has a fixed point which gives rise to a mild solution of the problem (4.15).
To illustrate the previous results, we consider the following model arising in population dynamics:
where 
and 
and 
are continuous functions. Let 
and consider the operator
defined by
It is well known that 
generates a 
-semigroup (see [29]). For the phase space 
, we choose the well-known space 
: the space of bounded uniformly continuous functions endowed with the following norm:
If we put for 
and 
then (4.30) take the abstract form (4.15). Under appropriate conditions on 
, the problem (4.30) has by Theorem 4.5 a solution.
Differential inclusions are generalization of differential equations, therefore all problems considered for differential equations, that is, existence of solutions, continuation of solutions, dependence on initial conditions and parameters, are present in the theory of differential inclusions. Since a differential inclusion usually has many solutions starting at a given point, new issues appear, such as investigation of topological properties of the set of solutions, and selection of solutions with given properties.
Functional differential inclusions with fractional order are first considered by El Sayed and Ibrahim [59]. Very recently Benchohra et al. [49], and Ouahab [60] have considered some classes of ordinary functional differential inclusions with delay, and in [6, 61] Agarwal et al. considered a class of boundary value problems for differential inclusion involving Caputo fractional derivative of order 
. Chang and Nieto [62] considered a class of fractional differential inclusions of order 
. Here we continue this study by considering partial functional differential inclusions involving the Riemann-Liouville derivative of order 
. The both cases of convex valued and nonconvex valued of the right-hand side are considered, and where the delay is finite as well as infinite. Our approach is based on the 
-semigroups theory combined with some suitable fixed point theorems.
In the following, we will be concerned with fractional semilinear functional differential inclusions with finite delay of the form
where 
is the standard Riemann-Liouville fractional derivative. 
is a multivalued function. 
is the family of all nonempty subsets of 
. 
is a densely defined (possibly unbounded) operator generating a strongly continuous semigroup 
of bounded linear operators from 
into 
is a given continuous function such that 
and 
is a real separable Banach spaces. For 
the norm of 
is defined by
For 
the norm of 
is defined by
Recall that for each 
the set
is known as the set of selections of the multivalued 
.
Definition 5.1.
One says that a continuous function 
is a mild solution of problem (5.1) if there exists 
such that 
and
In the following, we give our first existence result for problem (5.1) with a convex valued right-hand side. Our approach is based upon Theorem 2.10.
Theorem 5.2.
Assume the following.
(H19) 
is Carathéodory.
(H20) The semigroup 
is compact for 
.
(H21) There exist functions 
such that
Then the problem (5.1) has at least one mild solution.
Proof.
Consider the multivalued operator
defined by 
such that
where 
It is obvious that fixed points of 
are mild solutions of problem (5.1). We will show that 
is a completely continuous multivalued operator, u.s.c. with convex values.
It is obvious that 
is convex valued for each 
since 
has convex values.
To show that 
maps bounded sets into bounded sets in 
it is enough to show that there exists a positive constant 
such that for each 
, 
one has 
Indeed, if 
, then there exists 
such that for each 
we have
Using 
we have for each 
,
Then for each 
we have 
.
Now let 
for 
, and let 
, 
If 
and 
we have
where 
. Using the following semigroup identities
we get
As 
and 
sufficiently small, the right-hand side of the above inequality tends to zero, since 
is a strongly continuous operator and the compactness of 
for 
implies the continuity in the uniform operator topology [29]. Let 
be fixed and let 
be a real number satisfying 
. For 
we define
where 
. Since 
is a compact operator, the set
is precompact in 
for every 
Moreover, for every 
we have
Therefore, the set 
is totally bounded. Hence 
is precompact in 
.
As a consequence of the Arzelá-Ascoli theorem we can conclude that the multivalued operator 
is completely continuous.
Now we show that the operator 
has closed graph. Let 
, 
, and 
. We will show that 
.
means that there exists 
such that
We must show that there exists 
such that, for each 
Since 
has compact values, there exists a subsequence 
such that
Since 
is u.s.c., then for every 
, there exist 
such that for every 
, we have
and hence,
Then for each 
Hence,
Now it remains to show that the set
is bounded. Let 
be any element, then there exists 
such that
Then by (H20) and (H21) for each 
we have
Consider the function defined by
Let 
such that 
, If 
then we have, for 
(note 
)
If 
then 
and the previous inequality holds.
By Lemma 2.7 we have
Taking the supremum over 
we get
Hence
and so, the set 
is bounded. Consequently the multivalued operator 
has a fixed point which gives rise to a mild solution of problem (5.1) on 
Now we will be concerned with existence results for problem (5.1) with nonconvex valued right-hand side. Our approach is based on the fixed point theorem for contraction multivalued maps due to Covitz and Nadler Jr. [35].
Theorem 5.3.
Assume that (H19) holds.
There exists 
such that
with
If
then the problem (5.1) has at least one mild solution on 
Proof.
First we will prove that 
for each 
. 
such that 
in 
. Then 
and there exists 
such that for each 
Using the compactness property of the values of 
and the second part of 
we may pass to a subsequence if necessary to get that 
converges weakly to 
(the space endowed with the weak topology). From Mazur's lemma (see [63]) there exists
then there exists a subsequence 
in 
such that 
converges strongly to 
in 
Then for each 
,
So, 
Now Let 
and 
. Then there exists 
such that
Then from 
there is 
such that
Consider the multivalued operator 
defined by
Since the multivalued operator 
is measurable (see [64, proposition III4]) there exists 
a measurable selection for 
. So, 
and
Let us define for each 
Then we have
For 
, the previous inequality is satisfied. Taking the supremum over 
we get
By analogous relation, obtained by interchanging the roles of 
and 
, it follows that
By (5.34) 
is a contraction, and hence Theorem 2.8 implies that 
has a fixed point which gives rise to a mild solution of problem (5.1).
In the following, we will extend the previous results to the case when the delay is infinite. More precisely we consider the following problem:
where 
is the standard Riemann-Liouville fractional derivative. 
is a multivalued function. 
is the phase space [41], 
is the infinitesimal generator of a strongly continuous semigroup 
, 
a continuous function with 
and 
a real Banach space. Consider the following space:
where 
is the restriction of 
to 
Let 
be the seminorm in 
defined by
Definition 5.4.
One says that a function 
is a mild solution of problem (5.46) if 
and there exists 
such that
In the following, we give an existence result for problem (5.46) with convex valued right-hand side. Our approach is based upon Theorem 2.10.
Theorem 5.5.
Assume the following.
(H23) 
is Carathéodory.
(H24) The semigroup 
is compact for 
.
(H25) There exist functions 
such that
Then the problem (5.46) has at least one mild solution.
Proof.
Consider the operator
defined by
where 
.
For 
, we define the function
Then 
. Set
It is obvious that 
satisfies (5.49) if and only if 
satisfies 
and
Let
For any 
, we have
Thus 
is a Banach space. Let the operator 
defined by
where 
.
As in Theorem 5.2, we can show that the multivalued operator 
is completely continuous, u.s.c. with convex values. It remains to show that the set
is bounded.
Let 
be any element, then there exists a selection 
such that
Then for each 
we have
Following the proof of Theorem 3.6, we can show that the set 
is bounded. Consequently, the multivalued operator 
has a fixed point. Then 
has one, witch gives rise to a mild solution of problem (5.46).
Now we give an existence result for problem (5.46) with nonconvex valued right-hand side by using the fixed point Theorem 2.8.
Theorem 5.6.
Assume that (H23) holds. Then
(H26) There exists 
such that
with
If
then the problem (5.46) has at least one mild solution on 
Proof.
As the previous theorem and following steps of the proof of Theorem 5.3.
In this section, we will be concerned with semilinear functional differential equations and inclusion of fractional order and where a perturbed term is considered. Our approach is based upon Burton-Kirk fixed point theorem (Theorem 2.11).
First, consider equations of the form
Definition 6.1.
One says that a continuous function 
is a mild solution of problem (6.1) if 
and
Our first main result in this section reads as follows.
Theorem 6.2.
Assume that the following hypotheses hold.
(H27) The semigroup 
is compact for 
.
(H28) There exist functions 
such that
(H29) There exists a nonnegative constant 
such that
then the problem (6.1) has at least one mild solution on 
Proof.
Transform the problem (6.1) into a fixed point problem. Consider the two operators
defined by
Then the problem of finding the solution of IVP (6.1) is reduced to finding the solution of the operator equation 
We will show that the operators 
and 
satisfies all conditions of Theorem 2.11.
From Theorem 3.6, the operator 
is completely continuous. We will show that the operator 
is a contraction. Let 
, then for each 
Taking the supremum over 
,
which implies by (6.5) that 
is a contraction. Now, it remains to show that the set
is bounded.
Let 
be any element. Then, for each 
,
Then
where
We consider the function defined by
Let 
such that 
. If 
then by the previous inequality we have, for 
(note 
)
If 
then 
and the previous inequality holds.
By Lemma 2.7, there exists a constant 
such that we have
Hence,
This shows that the set 
is bounded. as a consequence of the Theorem 2.11, we deduce that the operator 
has a fixed point which gives rise to a mild solution of the problem (6.1).
Now we consider multivalued functional differential equations of the form
Definition 6.3.
One says that a continuous function 
is a mild solution of problem (6.18) if 
and there exist 
and 
such that
Theorem 6.4.
Assume that the following hypotheses hold.
(H30) The semigroup 
is compact for 
.
(H31) The multifunction 
is measurable, convex valued and integrably bounded for each 
.
(H32) There exists a function 
such that
with
(H33) 
is Carathéodory.
(H34) There exist functions 
such that
Then IVP (6.18) has at least one mild solution on 
.
Proof.
Consider the two multivalued operators
defined by 
such that
defined by 
such that
where 
and 
. We will show that the operator 
is closed, convex, and bounded valued and it is a contraction. Let 
such that 
in 
. Using (H31), we can show that the values of Niemysky operator 
are closed in 
, and hence 
is closed for each 
Now let 
, then there exists 
such that, for each 
we have
Let 
Then, for each 
, we have
Since 
has convex values, one has
and hence 
is convex for each 
Let 
be any element. Then, there exists 
such that
By (H31), we have for all 
where 
is from Definition 2.4. Then 
for all 
. Hence 
is a bounded subset of 
.
As in Theorem 5.3, we can easily show that the multivalued operator 
is a contraction. Now, as in Theorem 5.2 we can show that the operator 
satisfies all the conditions of Theorem 2.12.
It remains to show that the set
is bounded.
Let 
be any element. Then there exists 
and 
such that for each 
,
Then
where
We consider the function defined by
Let 
such that 
. If 
then by the previous inequality we have, for 
(note 
)
If 
then 
and the previous inequality holds.
By Lemma 2.7, there exists a constant 
such that we have
Hence
This shows that the set 
is bounded. As a result, the conclusion (ii) of Theorem 2.12 does not hold. Hence, the conclusion (i) holds and consequently 
has a fixed point which is a mild solution of problem (6.18).
In this section, we present some existence results in ordered Banach spaces using the method of upper and lower mild solutions. Before stating our main results let us introduce some preliminaries.
Definition 7.1.
A nonempty closed subset 
of a Banach space 
is said to be a cone if
,
for 
,
.
A cone 
is called normal if the norm 
is semimonotone on 
, that is, there exists a constant 
such that 
whenever 
. We equip the space 
with the order relation 
induced by a regular cone 
in 
, that is for all 
if and only if 
In what follows will assume that the cone 
is normal. Cones and their properties are detailed in [65, 66]. Let 
be such that 
. Then, by an order interval 
we mean a set of points in 
given by
Definition 7.2.
Let 
be an ordered Banach space. A mapping 
is called increasing if 
for any 
with 
. Similarly, 
is called decreasing if 
whenever 
.
Definition 7.3.
A function 
is called increasing in 
for 
, if 
for each 
for all 
with 
. Similarly 
is called decreasing in 
for 
, if 
for each 
for all 
with 
.
Now suppose that 
is an ordered Banach space and reconsider the initial value problem (3.1) and (3.2) with the same data.
Definition 7.4.
One says that a continuous function 
is a lower mild solution of problem (3.1) and (3.2) if 
and
Similarly an upper mild solution 
of IVP (3.1) and (3.2) is defined by reversing the order.
The following fixed point theorem is crucial for our existence result.
Theorem 7.5 (see [66]).
Let 
be a normal cone in a partially ordered Banach space 
. Let 
be increasing on the interval 
and transform 
into itself, that is, 
and 
. Assume further that 
is continuous and completely continuous. Then 
has at least one fixed point 
.
Our first main result reads as follows.
Theorem 7.6.
Assume that assumptions (H2)-(H3) hold. Assume moreover that
(H35) The function 
is increasing in 
for each 
.
(H36) 
is order-preserving, that is, 
whenever 
(H37) The IVP (3.1) and (3.2) has a lower mild solution 
and an upper mild solution 
with 
.
Then IVP (3.1) and (3.2) has at least one mild solution 
on 
with 
Proof.
It can be shown, as in the proof of Theorem 3.2, that 
is continuous and completely continuous on 
. We will show that 
is increasing on 
. Let 
be such that 
Then by (H35),(H36), we have for each 
Therefore 
is increasing on 
. Finally, let 
be any element. By (H37), we deduce that
which shows that 
for all 
. Thus, the functions 
satisfies all conditions of Theorem 7.5, and hence IVP (3.1) and (3.2) has a mild solution on 
belonging to the interval 
.
Now reconsider the perturbed initial value problem (6.1). To state our second main result in this section we use the following fixed point theorem due to Dhage and Henderson.
Theorem 7.7 (see [67]).
Let 
be an order interval in a Banach space and let 
be two functions satisfying
is a contraction,
is completely continuous,
and 
are strictly monotone increasing,
.
Further if the cone 
in 
is normal, then the equation 
has at least fixed point 
and a greatest fixed point 
. Moreover 
and 
, where 
and 
are the sequences in 
defined by
We need the following definitions in the sequel.
Definition 7.8.
One says that a continuous function 
is a lower mild solution of problem (6.1) 
and
Similarly an upper mild solution 
of IVP (6.1) is defined by reversing the order.
Theorem 7.9.
Assume that assumptions (H27)–(H29) hold. Suppose moreover that
(H38) The functions 
and 
are increasing in 
for each 
.
(H39) 
is order-preserving, that is, 
whenever 
(H40) The IVP (6.1) has a lower mild solution 
and an upper mild solution 
with 
.
Then IVP (6.1) has a minimal and a maximal mild solutions on 
.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.
Kiryakova V: Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series. Volume 301. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1994:x+388.
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.
Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, Switzerland; 1993:xxxvi+976.
Agarwal RP, Benchohra M, Hamani S: Boundary value problems for fractional differential equations. to appear in Georgian Mathematical Journal
Diethelm K, Freed AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties. Edited by: Keil F, Mackens W, Voß H, Werther J. Springer, Heidelberg, Germany; 1999:217-224.
Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002,265(2):229-248. 10.1006/jmaa.2000.7194
El-Sayed AMA: Fractional order evolution equations. Journal of Fractional Calculus 1995, 7: 89-100.
El-Sayed AMA: Fractional-order diffusion-wave equation. International Journal of Theoretical Physics 1996,35(2):311-322. 10.1007/BF02083817
El-Sayed AMA: Nonlinear functional-differential equations of arbitrary orders. Nonlinear Analysis: Theory, Methods & Applications 1998,33(2):181-186. 10.1016/S0362-546X(97)00525-7
Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mechanical Systems and Signal Processing 1991,5(2):81-88. 10.1016/0888-3270(91)90016-X
Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophysical Journal 1995,68(1):46-53. 10.1016/S0006-3495(95)80157-8
Lakshmikantham V, Devi JV: Theory of fractional differential equations in a Banach space. European Journal of Pure and Applied Mathematics 2008,1(1):38-45.
Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanis. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainard F. Springer, Vienna, Austria; 1997:291-348.
Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. Journal of Chemical Physics 1995,103(16):7180-7186. 10.1063/1.470346
Momani SM, Hadid SB: Some comparison results for integro-fractional differential inequalities. Journal of Fractional Calculus 2003, 24: 37-44.
Momani SM, Hadid SB, Alawenh ZM: Some analytical properties of solutions of differential equations of noninteger order. International Journal of Mathematics and Mathematical Sciences 2004,2004(13–16):697-701.
Podlubny I, Petráš I, Vinagre BM, O'Leary P, Dorčák L': Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dynamics 2002,29(1–4):281-296.
Yu C, Gao G: Existence of fractional differential equations. Journal of Mathematical Analysis and Applications 2005,310(1):26-29. 10.1016/j.jmaa.2004.12.015
El-Borai MM: On some fractional evolution equations with nonlocal conditions. International Journal of Pure and Applied Mathematics 2005,24(3):405-413.
El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(3):197-211. 10.1155/S1048953304311020
Jaradat OK, Al-Omari A, Momani S: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):3153-3159. 10.1016/j.na.2007.09.008
Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245.
Fattorini HO: Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985:xiii+314.
Travis CC, Webb GF: Second order differential equations in Banach spaces. In Nonlinear Equations in Abstract Spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex., 1977). Academic Press, New York, NY, USA; 1978:331-361.
Travis CC, Webb GF: Cosine families and abstract nonlinear second order differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 1978,32(1-2):75-96. 10.1007/BF01902205
Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA; 1991:x+282.
Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.
Kisielewicz M: Differential Inclusions and Optimal Control, Mathematics and Its Applications. Volume 44. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xx+240.
Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.
Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications. Volume 495. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:x+399.
Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.
Henry D: Geometric Theory of Semilinear Parabolic Partial Differential Equations. Springer, Berlin, Germany; 1989.
Covitz H, Nadler SB Jr.: Multi-valued contraction mappings in generalized metric spaces. Israel Journal of Mathematics 1970,8(1):5-11. 10.1007/BF02771543
Granas A, Dugundji J: Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2003:xvi+690.
Martelli M: A Rothe's type theorem for non-compact acyclic-valued maps. Bollettino della Unione Matematica Italiana. Serie 4 1975,11(3, supplement):70-76.
Burton TA, Kirk C: A fixed point theorem of Krasnoselskii-Schaefer type. Mathematische Nachrichten 1998, 189: 23-31. 10.1002/mana.19981890103
Dhage BC: Multi-valued mappings and fixed points. I. Nonlinear Functional Analysis and Applications 2005,10(3):359-378.
Dhage BC: Multi-valued mappings and fixed points. II. Tamkang Journal of Mathematics 2006,37(1):27-46.
Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11-41.
Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin, Germany; 1991:x+317.
Hale JK: Theory of Functional Differential Equations, Applied Mathematical Sciences. Volume 3. 2nd edition. Springer, New York, NY, USA; 1977:x+365.
Hale JK, Verduyn Lunel SM: Introduction to Functional-Differential Equations, Applied Mathematical Sciences. Volume 99. Springer, New York, NY, USA; 1993:x+447.
Kolmanovskii V, Myshkis A: Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xvi+648.
Wu J: Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences. Volume 119. Springer, New York, NY, USA; 1996.
Belarbi A, Benchohra M, Hamani S, Ntouyas SK: Perturbed functional differential equations with fractional order. Communications in Applied Analysis 2007,11(3-4):429-440.
Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Applicable Analysis 2006,85(12):1459-1470. 10.1080/00036810601066350
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Fractional Calculus & Applied Analysis 2008,11(1):35-56.
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. Journal of Mathematical Analysis and Applications 2008,338(2):1340-1350. 10.1016/j.jmaa.2007.06.021
Belmekki M, Benchohra M: Existence results for fractional order semilinear functional differential equations. Proceedings of A. Razmadze Mathematical Institute 2008, 146: 9-20.
Belmekki M, Benchohra M, Górniewicz L: Functional differential equations with fractional order and infinite delay. Fixed Point Theory 2008,9(2):423-439.
Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta 2006,45(5):765-772. 10.1007/s00397-005-0043-5
Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002,5(4):367-386.
Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366.
Hilfe R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463.
Hernández E, Henríquez HR: Existence results for partial neutral functional differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):452-475. 10.1006/jmaa.1997.5875
Hernández E, Henríquez HR: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):499-522. 10.1006/jmaa.1997.5899
El-Sayed AMA, Ibrahim A-G: Multivalued fractional differential equations. Applied Mathematics and Computation 1995,68(1):15-25. 10.1016/0096-3003(94)00080-N
Ouahab A: Some results for fractional boundary value problem of differential inclusions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3877-3896. 10.1016/j.na.2007.10.021
Agarwal RP, Benchohra M, Hamani S: Boundary value problems for differential inclusions with fractional order. Advanced Studies in Contemporary Mathematics 2008,16(2):181-196.
Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009,49(3-4):605-609. 10.1016/j.mcm.2008.03.014
Yosida K: Functional Analysis, Grundlehren der Mathematischen Wissenschaften. Volume 123. 6th edition. Springer, Berlin, Germany; 1980:xii+501.
Castaing C, Valadier M: Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics. Volume 580. Springer, Berlin, Germany; 1977:vii+278.
Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York, NY, USA; 1994:xii+514.
Joshi MC, Bose RK: Some Topics in Nonlinear Functional Analysis, A Halsted Press Book. John Wiley & Sons, New York, NY, USA; 1985:viii+311.
Dhage BC, Henderson J: Existence theory for nonlinear functional boundary value problems. Electronic Journal of Qualitative Theory of Differential Equations 2004,2004(1):1-15.
The authors thank the referees for their comments and remarks.
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.
Agarwal, R., Belmekki, M. & Benchohra, M. A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative. Adv Differ Equ 2009, 981728 (2009). https://doi.org/10.1155/2009/981728
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DOI: https://doi.org/10.1155/2009/981728