Open Access

A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative

Advances in Difference Equations20092009:981728

DOI: 10.1155/2009/981728

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 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 [911], 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
(2.1)
For the norm of is defined by
(2.2)
For the norm of is defined by
(2.3)
denotes the Banach space of bounded linear operators from into with norm
(2.4)
denotes the Banach space of measurable functions which are Bochner integrable normed by
(2.5)

Definition 2.1.

A semigroup of class is a one parameter family satisfying the conditions

  1. (i)

     
  2. (ii)

    for all

     
  3. (iii)
    the map is strongly continuous, for each , that is,
    (2.6)
     

It is well known that the operator generates a semigroup if satisfies

  1. (i)

     
  2. (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 by
(2.7)

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

Definition 2.3 (see [4, 5]).

The Riemann-Liouville fractional derivative of order of a continuous function is defined by
(2.8)
Let be a metric space. We use the notations
(2.9)
Consider given by
(2.10)

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
(2.11)

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

  1. (i)

    is measurable for each

     
  2. (ii)

    is u.s.c. for almost all

     

Furthermore, a Carathéodory map is said to be -Carathéodory if

  1. (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

  1. (a)
    -Lipschitz if and only if there exists such that
    (2.13)
     
  1. (b)

    contraction if and only if it is -Lipschitz with

     
  2. (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
(2.14)
then there exists a constant such that
(2.15)

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
  1. (a)

    has a fixed point, or

     
  2. (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
(2.16)

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
  1. (i)

    is a contraction;

     
  2. (ii)

    is completely continuous.

     
Then either
  1. (a)

    the operator equation has a solution, or

     
  2. (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
  1. (a)

    is a contraction;

     
  2. (b)

    is completely continuous.

     
Then either
  1. (i)

    The operator inclusion has a solution for , or

     
  2. (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:
    1. (i)

      is in

       
    2. (ii)

       
    3. (iii)

      and , and are independent of .

       
  • (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
(2.18)
(2.19)

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
(2.20)

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
(2.21)
endowed with the following norm
(2.22)

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 [4750]), 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
(3.1)
(3.2)
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
(3.3)

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
(3.4)

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
(3.5)

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
(3.7)
Let us define the iterates of operator by
(3.8)
It will be sufficient to prove that is a contraction operator for sufficiently large. For every we have
(3.9)
Indeed,
(3.10)
Therefore (3.9) is proved for . Assuming by induction that (3.9) is valid for , then
(3.11)

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
(3.13)
Since is a continuous function, then we have
(3.14)
Thus is continuous. Now for any , and each we have for each
(3.15)

Thus maps bounded sets into bounded sets in .

Now, let , Thus if and we have for any
(3.16)

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
(3.17)
Since is a compact operator for , the set
(3.18)
is precompact in for every Moreover
(3.19)
Therefore, the set is precompact in . Hence the operator is completely continuous. Now, it remains to show that the set
(3.20)

is bounded.

Let be any element. Then, for each ,
(3.21)
Then
(3.22)
We consider the function defined by
(3.23)
Let such that , if then by (3.22) we have, for (note )
(3.24)

If then and the previous inequality holds.

By Lemma 2.7 we have
(3.25)
Hence
(3.26)

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
(3.27)

where is continuous and is a given function.

Let
(3.28)
Take and define by with domain
(3.29)
Then
(3.30)
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
(3.31)
Since the analytic semigroup is compact, there exists a constant such that
(3.32)
Also assume that there exist continuous functions such that
(3.33)

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
(3.34)
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
(3.35)
Consider the following space:
(3.36)
where is the restriction of to Let be the seminorm in defined by
(3.37)

Definition 3.4.

One says that a function is a mild solution of problem (3.34) if and
(3.38)

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
(3.40)
For , we define the function
(3.41)
Then . Set
(3.42)
It is obvious that satisfies (3.38) if and only if satisfies and
(3.43)
Let
(3.44)
For any , we have
(3.45)
Thus is a Banach space. Let the operator defined by
(3.46)
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:
(3.47)

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
(3.49)
Since is a continuous function, then we have
(3.50)
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
(3.51)
Then we have for each
(3.52)
Taking the supremum over we have
(3.53)
Now let , thus if and we have for each
(3.54)
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
(3.55)
Since is a compact operator for , the set
(3.56)
is precompact in for every Moreover
(3.57)
Therefore, the set is precompact in . Hence the operator is completely continuous. Now, it remains to show that the set
(3.58)
is bounded. Let be any element. Then, for each ,
(3.59)
Then
(3.60)
but
(3.61)
Take the right-hand side of the above inequality as , then by (3.60) we have
(3.62)
Using the above inequality and the definition of we have
(3.63)
By Lemma 2.7, there exists a constant such that we have
(3.64)

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:
(3.65)

where are continuous functions.

Consider and define by with domain
(3.66)

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:
(3.67)
If we put for and
(3.68)

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
(4.1)

Definition 4.1.

One says that a function is a mild solution of problem (4.1) if and
(4.2)

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
(4.5)
As in Theorem 3.2, we show by induction that satisfy for any , the following inequality:
(4.6)

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
(4.9)

is continuous and completely continuous. This can be done following the proof of Theorem 3.3.

Now, it remains to show that the set
(4.10)
is bounded. Let be any element. Then, for each ,
(4.11)
We consider the function defined by
(4.12)
Let such that , If then we have, for (note )
(4.13)

If then and the previous inequality holds.

By Lemma 2.7 there exists such that
(4.14)

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:
(4.15)

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
(4.18)
In analogy to Theorem 3.2, we consider the operator defined by
(4.19)
As in Theorem 3.2, we show by induction that satisfy for any , the following inequality:
(4.20)

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
(4.23)
is continuous and completely continuous. Now, it remains to show that the set
(4.24)

is bounded.

Let be any element. Then, for each ,
(4.25)
Denote as in Theorem 3.6. Then
(4.26)
Then
(4.27)
By Lemma 2.7 there exists a constant such that
(4.28)
where
(4.29)

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:
(4.30)
where and and are continuous functions. Let and consider the operator
(4.31)
defined by
(4.32)
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:
(4.33)
If we put for and
(4.34)

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
(5.1)
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
(5.2)
For the norm of is defined by
(5.3)
Recall that for each the set
(5.4)

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
(5.5)

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
(5.7)
defined by such that
(5.8)

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
(5.9)
Using we have for each ,
(5.10)

Then for each we have .

Now let for , and let , If and we have
(5.11)
where . Using the following semigroup identities
(5.12)
we get
(5.13)
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
(5.14)
where . Since is a compact operator, the set
(5.15)
is precompact in for every Moreover, for every we have
(5.16)

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
(5.17)
We must show that there exists such that, for each
(5.18)
Since has compact values, there exists a subsequence such that
(5.19)
Since is u.s.c., then for every , there exist such that for every , we have
(5.20)
and hence,
(5.21)
Then for each
(5.22)
Hence,
(5.23)
Now it remains to show that the set
(5.24)
is bounded. Let be any element, then there exists such that
(5.25)
Then by (H20) and (H21) for each we have
(5.26)
Consider the function defined by
(5.27)
Let such that , If then we have, for (note )
(5.28)

If then and the previous inequality holds.

By Lemma 2.7 we have
(5.29)
Taking the supremum over we get
(5.30)
Hence
(5.31)

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
(5.32)
with
(5.33)
If
(5.34)

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
(5.35)
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
(5.36)
then there exists a subsequence in such that converges strongly to in Then for each ,
(5.37)

So,

Now Let and . Then there exists such that
(5.38)
Then from there is such that
(5.39)
Consider the multivalued operator defined by
(5.40)
Since the multivalued operator is measurable (see [64, proposition III4]) there exists a measurable selection for . So, and
(5.41)
Let us define for each
(5.42)
Then we have
(5.43)
For , the previous inequality is satisfied. Taking the supremum over we get
(5.44)
By analogous relation, obtained by interchanging the roles of and , it follows that
(5.45)

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:
(5.46)
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:
(5.47)
where is the restriction of to Let be the seminorm in defined by
(5.48)

Definition 5.4.

One says that a function is a mild solution of problem (5.46) if and there exists such that
(5.49)

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
(5.51)
defined by
(5.52)

where .

For , we define the function
(5.53)
Then . Set
(5.54)
It is obvious that satisfies (5.49) if and only if satisfies and
(5.55)
Let
(5.56)
For any , we have
(5.57)
Thus is a Banach space. Let the operator defined by
(5.58)

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
(5.59)

is bounded.

Let be any element, then there exists a selection such that
(5.60)
Then for each we have
(5.61)

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
(5.63)
If
(5.64)

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
(6..1)

Definition 6.1.

One says that a continuous function is a mild solution of problem (6.1) if and
(6..2)

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)
(6..5)

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
(6..6)
defined by
(6..7)

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
(6..8)
Taking the supremum over ,
(6..9)
which implies by (6.5) that is a contraction. Now, it remains to show that the set
(6..10)

is bounded.

Let be any element. Then, for each ,
(6..11)
Then
(6..12)
where
(6..13)
We consider the function defined by
(6..14)
Let such that . If then by the previous inequality we have, for (note )
(6..15)

If then and the previous inequality holds.

By Lemma 2.7, there exists a constant such that we have
(6..16)
Hence,
(6..17)

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
(6..18)

Definition 6.3.

One says that a continuous function is a mild solution of problem (6.18) if and there exist and such that
(6..19)

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
(6..23)
defined by such that
(6..24)
defined by such that
(6..25)

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
(6..26)
Let Then, for each , we have
(6..27)
Since has convex values, one has
(6..28)

and hence is convex for each

Let be any element. Then, there exists such that
(6..29)
By (H31), we have for all
(6..30)

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
(6..31)

is bounded.

Let be any element. Then there exists and such that for each ,
(6..32)
Then
(6..33)
where
(6..34)
We consider the function defined by
(6..35)
Let such that . If then by the previous inequality we have, for (note )
(6..36)

If then and the previous inequality holds.

By Lemma 2.7, there exists a constant such that we have
(6..37)
Hence
(6..38)

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

  1. (i)

    ,

     
  2. (ii)

    for ,

     
  3. (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
(7.1)

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
(7.2)

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
(7.3)
Therefore is increasing on . Finally, let be any element. By (H37), we deduce that
(7.4)

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
  1. (a)

    is a contraction,

     
  2. (b)

    is completely continuous,

     
  3. (c)

    and are strictly monotone increasing,

     
  4. (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
(7.5)

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
(7.6)

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 .

Declarations

Acknowledgment

The authors thank the referees for their comments and remarks.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Laboratoire de Mathématiques, Université de Sidi Bel Abbès

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