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A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative
Advances in Difference Equations volume 2009, Article number: 981728 (2009)
Abstract
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.
1. Introduction
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.
2. Preliminaries
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
-
(i)
-
(ii)
for all
-
(iii)
the map
is strongly continuous, for each
, that is,
(2.6)
It is well known that the operator generates a
semigroup if
satisfies
-
(i)
-
(ii)
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
-
(i)
is measurable for each
-
(ii)
is u.s.c. for almost all
Furthermore, a Carathéodory map is said to be
-Carathéodory if
-
(iii)
for each real number
, there exists a function
such that
(2.12)
for a.e. and for all
Definition 2.6.
A multivalued operator is called
-
(a)
-Lipschitz if and only if there exists
such that
(2.13)
-
(b)
contraction if and only if it is
-Lipschitz with
-
(c)
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
-
(a)
has a fixed point, or
-
(b)
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
-
(i)
is a contraction;
-
(ii)
is completely continuous.
Then either
-
(a)
the operator equation
has a solution, or
-
(b)
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
-
(a)
is a contraction;
-
(b)
is completely continuous.
Then either
-
(i)
The operator inclusion
has a solution for
, or
-
(ii)
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:
-
(i)
is in
-
(ii)
-
(iii)
and
, and
are independent of
.
-
(i)
-
(A2) For the function
in
,
is a
-valued continuous function on
-
(A3) The space
is complete.
-
Denote by
(2.17)
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.
3. Semilinear Functional Differential Equations
3.1. Introduction
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

3.2. Existence Results for Finite Delay
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
(3.6)
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
(3.12)
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).
3.3. An Example
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
3.4. Existence Results for Infinite Delay
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
(3.39)
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
(3.48)
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).
3.5. An Example
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).
4. Semilinear Functional Differential Equations of Neutral Type
4.1. Introduction
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.
4.2. Existence Results for the Finite Delay
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
(4.3)
-
(H8) There exists a nonnegative constant
such that
(4.4)
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
(4.7) -
(H11) The function
is continuous and completely continuous, and for every bounded set
, the set
is equicontinuous in
.
-
(H12) There exists constants:
such that
(4.8)
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).
4.3. Existence Results for the Infinite Delay
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
(4.16) -
(H14) There exists a nonnegative constant
such that
(4.17)
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
(4.21) -
(H17) The function
is continuous and completely continuous, and for every bounded set
, the set
is equicontinuous in
.
-
(H18) There exists constants:
such that
(4.22)
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).
4.4. Example
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.
5. Semilinear Functional Differential Inclusions
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
(5.6)
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
(5.50)
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
(5.62)
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.
6. Perturbed Semilinear Differential Equations and Inclusions
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
(6..3) -
(H29) There exists a nonnegative constant
such that
(6..4)

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
(6..20) -
with
(6..21) -
(H33)
is Carathéodory.
-
(H34) There exist functions
such that
(6..22)
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).
7. Some Existence Results in Ordered Banach Spaces
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
-
(i)
,
-
(ii)
for
,
-
(iii)
.
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
-
(a)
is a contraction,
-
(b)
is completely continuous,
-
(c)
and
are strictly monotone increasing,
-
(d)
.
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 .
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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
Keywords
- Banach Space
- Fractional Derivative
- Existence Result
- Fixed Point Theorem
- Mild Solution