 Research Article
 Open Access
 Published:
A Survey on Semilinear Differential Equations and Inclusions Involving RiemannLiouville 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 RiemannLiouville 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], ElSayed [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 ElBorai [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 righthand 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 HilleYosida 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 RiemannLiouville 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 RiemannLiouville 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 LeraySchauder 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 KrasnoselskiiScheafer 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 RiemannLiouville and Caputo fractional derivatives of order In [51, 52] some classes of semilinear functional differential equations involving the RiemannLiouville have been considered. For more details on the geometric and physical interpretation for fractional derivatives of both the RiemannLiouville 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 RiemannLiouville 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 RiemannLiouville sense. The problems considered in the survey are subject to zero data, which in this case, the RiemannLiouville 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 righthand 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 ArzeláAscoli 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 RiemannLiouville 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 LeraySchauder 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 righthand 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 ArzeláAscoli 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 righthand 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 LeraySchauder 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 wellknown 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 LeraySchauder 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 LeraySchauder.
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 LeraySchauder 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 LeraySchauder.
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 LeraySchauder 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 wellknown 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 RiemannLiouville derivative of order . The both cases of convex valued and nonconvex valued of the righthand 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 RiemannLiouville 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 righthand 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 righthand 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 ArzeláAscoli 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 righthand 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 RiemannLiouville 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 righthand 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 righthand 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 BurtonKirk 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 orderpreserving, 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 orderpreserving, 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 .
References
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, NorthHolland 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 WileyInterscience 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; 1999:217224.
Diethelm K, Ford NJ: Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 2002,265(2):229248. 10.1006/jmaa.2000.7194
ElSayed AMA: Fractional order evolution equations. Journal of Fractional Calculus 1995, 7: 89100.
ElSayed AMA: Fractionalorder diffusionwave equation. International Journal of Theoretical Physics 1996,35(2):311322. 10.1007/BF02083817
ElSayed AMA: Nonlinear functionaldifferential equations of arbitrary orders. Nonlinear Analysis: Theory, Methods & Applications 1998,33(2):181186. 10.1016/S0362546X(97)005257
Gaul L, Klein P, Kempfle S: Damping description involving fractional operators. Mechanical Systems and Signal Processing 1991,5(2):8188. 10.1016/08883270(91)90016X
Glockle WG, Nonnenmacher TF: A fractional calculus approach to selfsimilar protein dynamics. Biophysical Journal 1995,68(1):4653. 10.1016/S00063495(95)801578
Lakshmikantham V, Devi JV: Theory of fractional differential equations in a Banach space. European Journal of Pure and Applied Mathematics 2008,1(1):3845.
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; 1997:291348.
Metzler F, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. Journal of Chemical Physics 1995,103(16):71807186. 10.1063/1.470346
Momani SM, Hadid SB: Some comparison results for integrofractional differential inequalities. Journal of Fractional Calculus 2003, 24: 3744.
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):697701.
Podlubny I, Petráš I, Vinagre BM, O'Leary P, Dorčák L': Analogue realizations of fractionalorder controllers. Fractional order calculus and its applications. Nonlinear Dynamics 2002,29(1–4):281296.
Yu C, Gao G: Existence of fractional differential equations. Journal of Mathematical Analysis and Applications 2005,310(1):2629. 10.1016/j.jmaa.2004.12.015
ElBorai MM: On some fractional evolution equations with nonlocal conditions. International Journal of Pure and Applied Mathematics 2005,24(3):405413.
ElBorai MM: The fundamental solutions for fractional evolution equations of parabolic type. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(3):197211. 10.1155/S1048953304311020
Jaradat OK, AlOmari A, Momani S: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):31533159. 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, NorthHolland Mathematics Studies. Volume 108. NorthHolland, 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; 1978:331361.
Travis CC, Webb GF: Cosine families and abstract nonlinear second order differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 1978,32(12):7596. 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.: Multivalued contraction mappings in generalized metric spaces. Israel Journal of Mathematics 1970,8(1):511. 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 noncompact acyclicvalued maps. Bollettino della Unione Matematica Italiana. Serie 4 1975,11(3, supplement):7076.
Burton TA, Kirk C: A fixed point theorem of KrasnoselskiiSchaefer type. Mathematische Nachrichten 1998, 189: 2331. 10.1002/mana.19981890103
Dhage BC: Multivalued mappings and fixed points. I. Nonlinear Functional Analysis and Applications 2005,10(3):359378.
Dhage BC: Multivalued mappings and fixed points. II. Tamkang Journal of Mathematics 2006,37(1):2746.
Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):1141.
Hino Y, Murakami S, Naito T: FunctionalDifferential 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 FunctionalDifferential 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 FunctionalDifferential Equations, Mathematics and Its Applications. Volume 463. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xvi+648.
Wu J: Theory and Applications of Partial FunctionalDifferential 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(34):429440.
Belarbi A, Benchohra M, Ouahab A: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Applicable Analysis 2006,85(12):14591470. 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):3556.
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):13401350. 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: 920.
Belmekki M, Benchohra M, Górniewicz L: Functional differential equations with fractional order and infinite delay. Fixed Point Theory 2008,9(2):423439.
Heymans N, Podlubny I: Physical interpretation of initial conditions for fractional differential equations with RiemannLiouville fractional derivatives. Rheologica Acta 2006,45(5):765772. 10.1007/s0039700500435
Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002,5(4):367386.
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):452475. 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):499522. 10.1006/jmaa.1997.5899
ElSayed AMA, Ibrahim AG: Multivalued fractional differential equations. Applied Mathematics and Computation 1995,68(1):1525. 10.1016/00963003(94)00080N
Ouahab A: Some results for fractional boundary value problem of differential inclusions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):38773896. 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):181196.
Chang YK, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009,49(34):605609. 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):115.
Acknowledgment
The authors thank the referees for their comments and remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Agarwal, R., Belmekki, M. & Benchohra, M. A Survey on Semilinear Differential Equations and Inclusions Involving RiemannLiouville Fractional Derivative. Adv Differ Equ 2009, 981728 (2009). https://doi.org/10.1155/2009/981728
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/981728
Keywords
 Banach Space
 Fractional Derivative
 Existence Result
 Fixed Point Theorem
 Mild Solution