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 normFor the norm of is defined byFor 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,

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.

Definition 2.2 (see [4, 5]). The Riemann-Liouville fractional primitive of order of a function of order is defined byprovided 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

Definition 2.3 (see [4, 5]). 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 bywhere 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 byis 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 thatfor a.e. and for all

Definition 2.6. A multivalued operator is called
(a)-Lipschitz if and only if there exists such that(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.

Theorem 2.12 (see [39, 40]). 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 .(A2)For the function in , is a -valued continuous function on (A3)The space is complete. Denote byHereafter 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 bysendowed with the following normThen 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 formwhere 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 byHere 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 byBefore 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.
There exists a nonnegative constant such that Then there exists a unique mild solution of problem (3.1) and (3.2) on

Proof. Transform the IVP (3.1) and (3.2) into a fixed point problem. Consider the operator defined byLet us define the iterates of operator byIt will be sufficient to prove that is a contraction operator for sufficiently large. For every we haveIndeed,Therefore (3.9) is proved for . Assuming by induction that (3.9) is valid for , thenand 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.
The semigroup is compact for .There exist functions such that Then the problem (3.1) and (3.2) has at least one mild solution on

Proof. Transform the IVP (3.1) and (3.2) into a fixed point problem. Consider the operator as defined in Theorem 3.2. To show that is continuous, let us consider a sequence such that in . ThenSince is a continuous function, then we haveThus 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 defineSince is a compact operator for , the setis precompact in for every MoreoverTherefore, the set is precompact in . Hence the operator is completely continuous. Now, it remains to show that the setis bounded.
Let be any element. Then, for each ,ThenWe consider the function defined byLet such that , if then by (3.22) we have, for (note )If then and the previous inequality holds.
By Lemma 2.7 we haveHenceThis 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 formwhere is continuous and is a given function.

Let Take and define by with domainThenwhere 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 bySince the analytic semigroup is compact, there exists a constant such thatAlso assume that there exist continuous functions such thatWe 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 problemwhere 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 byConsider 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.
There exists a nonnegative constant such that Then there exists a unique mild solution of problem (3.34) on

Proof. Transform the IVP (3.34) into a fixed point problem. Consider the operator defined byFor , we define the functionThen . SetIt is obvious that satisfies (3.38) if and only if satisfies andLetFor any , we haveThus is a Banach space. Let the operator defined byIt 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.
The semigroup is compact for .(