A Survey on Semilinear Differential Equations and Inclusions Involving RiemannLiouville Fractional Derivative
 RaviP Agarwal^{1}Email author,
 Mohammed Belmekki^{2} and
 Mouffak Benchohra^{2}
DOI: 10.1155/2009/981728
© Copyright © 2009 2009
Received: 16 July 2008
Accepted: 5 February 2009
Published: 8 March 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
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.
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
where Then is a metric space and is a generalized metric space (see [30]).
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.
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.
 (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.
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.
 (i)
is a contraction;
 (ii)
is completely continuous.
 (a)
the operator equation has a solution, or
 (b)
the set is unbounded for .
Recently Dhage states the multivalued version of the previous theorem.
 (a)
is a contraction;
 (b)
is completely continuous.
 (i)
The operator inclusion has a solution for , or
 (ii)
the set is unbounded.

(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
We have that the spaces , and satisfy conditions . satisfies but is not satisfied.
Example 2.14.
The spaces , and .
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 .
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].
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.
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]).
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.

(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.
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.

(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.
Thus maps bounded sets into bounded sets in .
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 .
is bounded.
If then and the previous inequality holds.
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
where is continuous and is a given function.
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
Definition 3.4.
The first existence result is based on Banach's contraction principle.
Theorem 3.5.

(H4) There exists a nonnegative constant such that(3.39)
Then there exists a unique mild solution of problem (3.34) on
Proof.
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.

(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.
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
where are continuous functions.
Then generates a semigroup (see [29]).
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
Definition 4.1.
Our first existence result is based on the Banach's contraction principle.
Theorem 4.2.

(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.
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.

(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.
is continuous and completely continuous. This can be done following the proof of Theorem 3.3.
If then and the previous inequality holds.
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
Our first existence result is based on the Banach's contraction principle.
Theorem 4.4.

(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.
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.

(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.
is bounded.
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
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.
is known as the set of selections of the multivalued .
Definition 5.1.
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.

(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.
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.
Then for each 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 .
If then and the previous inequality holds.
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.
then the problem (5.1) has at least one mild solution on
Proof.
So,
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).
Definition 5.4.
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.

(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.
where .
where .
is bounded.
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.

(H26) There exists such that(5.62)
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).
Definition 6.1.
Our first main result in this section reads as follows.
Theorem 6.2.

(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.
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.
is bounded.
If then and the previous inequality holds.
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).
Definition 6.3.
Theorem 6.4.

(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.
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
and hence is convex for each
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.
is bounded.
If then and the previous inequality holds.
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)
.
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.
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.

(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.
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]).
 (a)
is a contraction,
 (b)
is completely continuous,
 (c)
and are strictly monotone increasing,
 (d)
.
We need the following definitions in the sequel.
Definition 7.8.
Similarly an upper mild solution of IVP (6.1) is defined by reversing the order.
Theorem 7.9.

(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 .
Declarations
Acknowledgment
The authors thank the referees for their comments and remarks.
Authors’ Affiliations
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