ABSTRACT

In this paper, we give various existence results concerning the existence of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense in Banach space. We consider the case when the values of the orient field are convex as well as nonconvex. Our obtained results improve and generalize many results proved in recent papers.

1. Introduction

During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering. For some of these applications, one can see [1-4] and the references therein. El Sayed et al. [5] initiated the study of fractional multi-valued differential inclusions. Recently, some basic theory for initial—value problems for fractional differential equations and inclusions was discussed by [6-14].

The theory of impulsive differential equations and impulsive differential inclusions has been an object interest because of its wide applications in physics, biology, engineering, medical fields, industry and technology. The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot described using the classical differential problems. For some of these applications we refer to [15-17]. During the last ten years, impulsive differential inclusions with different conditions have intensely student by many mathematicians. At present, the foundations of the general theory of impulsive differential equations and inclusions are already laid, and many of them are investigated in details in the book of Benchohra et al. [18].

Moreover, a strong motivation for investigating the nonlocal Cauchy problems, which is a generalization for the classical Cauchy problems with initial condition, comes from physical problems. For example, it used to determine the unknown physical parameters in some inverse heat condition problems. The nonlocal condition can be applied in physics with better effect than the classical initial condition For example, may be given by

where are given constants and . For the applications of nonlocal conditions problems we refer to [19,20]. In the few past years, several papers have been devoted to study the existence of solutions for differential equations or differential inclusions with nonlocal conditions [21-23]. For impulsive differential equation or inclusions with nonlocal conditions of order one we refer to [22,23]. For impulsive differential equation or inclusions of fractional order we refer to [10,24-27] and the references therein.

In this paper we are concerned with the existence of mild solution to the following nonlocal impulsive semilinear differential inclusions with delay and of order of the type

(1.1)

where, is the Caputo derivative of order is the infinitesimal generator of a semigroup on a real separable Banach space, be a multi-function, is a given continuous function, is a nonlinear function related to the nonlocal condition at the origin, impulsive functions which characterize the jump of the solutions at impulse points, and are the right and left limits of at the point respectively. Finally, for any defined by

where and will define in the next section.

To study the theory of abstract impulsive differential inclusions with fractional order, the first step is how to define the mild solution. Mophou [24] firstly introduced a concept on a mild solution which was inspired by Jaradat et al. [25]. However, it does not incorporate the memory effects involved in fractional calculus and impulsive conditions. Wang et al. [10] introduced a new concept of PC-mild solutions for (1.1) without delay and derived existence and uniqueness results concerning the PC-mild solutions for (1.1) when is a Lipschitz single-valued function or continuous and maps bounded sets into bounded sets and is compact.

In order to do a comparison between our obtained results in this paper and the known recent results in the same domain, we refer to: Ouahab [9] proved a version of Fillippov’s theorem for (1.1) without impulse, without delay and is an almost sectorial operator, Wang et al. [11] proved existence and controllability results for (1.1) without impulse, without delay and with local condition, Zhang et al. [12] considered the problem (1.1) without impulse, without delay, is a single-valued function and is strongly equicontinuous C₀-semigroup, Zhou et al. [13,14] introduced a suitable definition of mild solution for (1.1) based on Laplace transformation and probability density functions for (1.1) when is single-valued function and without impulse, Cardinali et al. [22] proved the existence of mild solutions to the problem (1.1) without delay, when andthe multivalued function satisfies the lower Scorza-Dragoni property and is a family of linear operator, generating a strongly continuous evolution operators, Fan [23] studied a nonlocal Cauchy problem in the presence of impulses, governed by autonomous semilinear differential equation, Dads et al. [26] and Henderson et al. [27] considered the problem (1.1) when Among the previous works, little is concerned with nonlocal fractional differential inclusions with impulses and with delay.

In Section 3 in this paper, motivated by the works mentioned above, we derive various existence results of mild solutions for (1.1) when the values of the orient field are convex as well as non-convex.

The paper is organized as follows: In Section 2, we collect some background material and lemmas to be used later. In Section 3, we prove three existence results for (1.1). We adopt the definition of mild solution introduced by Wang et al. [10]. Our basic tools are the properties of multi-functions, methods and results for semilinear differential inclusions, and fixed point techniques.

2. Preliminaries and Notations

Let the space of -valued continuous functions on with the uniform norm

the space of E-valued Bochner integrable functions on with the norm

, = {: B is nonempty and bounded}, = {: B is nonempty and closed}, = {: B is nonempty and compact}, = {: B is nonempty, closed and convex}, = {: B is nonempty, convex and compact}, (respectively, ) be the convex hull (respectively, convex closed hull in) of a subset

Definition 1 ([28]). A semigroup of bounded linear operators on a Banach space is said to be 1) uniformly continuous if

where is the identity operator.

2) strongly continuous if

A strongly continuous semigroup of bounded linear operators on will be called a semigroup of class or simply a -semigroup. It is known that if is a -semigroup, then there exist constants and such that

A semigroup is called compact if for every is compact. It is known that ([28], Theorem 3.2) every compact semigroup is uniformly continuous.

Definition 2 ([28]). Let be a semigroup of bounded linear operators on a Banach space The linear operator defined by

and

is called the infinitesimal generator of the semigroup is the domain of

Definition 3 ([29-33]). Let and be two topological spaces. A multifunction is said to be upper semicontinuous (u.s.c.) if

is an open subset of

for every open. is said to be lower semicontinuous if is an open subset of for every open is called closed if its graph

is closed subset of the topological space. is said to be completely continuous if is relatively compact for every bounded subset of If the multifunction is completely continuous with non empty compact values, then is u.s.c. if and only if is closed.

Lemma 1 ([29] Theorem 8.2.8). Let be a complete finite measure space, a complete separable metric space and be a measurable multivalued function with non empty closed images. Consider a multivalued function from to is a complete separable metric space such that for every the multivalued function is measurable and for every the multivalued function is continuous.

Then the multivalued function is measurable. In particular for every measurable singlevalued function the multivalued function is measurable and for every Caratheodory single-valued function the multivalued function is measurable.

Definition 4 A nonempty subset is said to be decomposable provided for every and each Lebesgue measurableset in

where is the characteristic function of the set

Definition 5 A sequence is said to be semi-compact if:

1) It is integrably bounded, i.e. there is such that

2)The set is relatively compact in

We recall one fundamental result which follows from Dunford-Pettis Theorem.

Lemma 2 ([33]). Every semi-compact sequence in is weakly compact in

For more about multifunctions we refer to [29-33].

Lemma 3 ([11], lemma 2.10). For and

, we have.

Definition 6 According to the Riemann -Liouville approach, the fractional integral of order of a function is defined by

provided the right side is defined on, where is the Euler gamma function defined by

Definition 7 The Caputo derivative of order of a continuously differentiable function is defined by

Note that the integrals appear in the two previous definitions are taken in Bochner’ sense and for all For more informations about the fractional calculus we refer to [2,4].

Definition 8 ([14], Lemma 3.1 and Definition 3.1, see also [11-13]). Let A function is said to be a mild solution of the following system:

(2.1)

if it satisfies the following integral equation

(2.2)

where

and is a probability density function defined on that is Note that the function must be chosen such that the integral appears in (2.2) is well be defined.

Remark 1 Since are associated with the numbrer there are no analogue of the semigroup property, i.e.

In the following we recall the properties of .

Lemma 4 ([14], Lemma 3.2, Lemma 3.3 and Lemma 3.5)

1) For any fixed are linear bounded operators.

2) For

3) If thenfor any

and

4) For any fixed are strongly continuous.

5) If is compact, then and are compact.

In order to define the concept of mild solution of (1.1), let and consider the set of functions:

and

It is easy to check that are are Banach spaces endowed with the norms

and

For any and any the element of defined by

Here represents the history of the state time up the present time For any subset and for any let

Of course

Let us recall the concept of mild solutions, introduced by Wang et al. [10], for the impulsive fractional evolution equation:

(2.3)

where.

At first Wang et al. [10] considered the following nonhomogeneous impulsive fractional equation

(2.4)

where and It is easily observe that can be decomposed to where is the continuous mild solution for

(2.5)

and is the mild solution for the impulsive evolution equation

(2.6)

Indeed, by adding together (2.5) with (2.6), it follows (2.4). Note is continuous, so

. On the other hand, any solution of (2.4) can be decomposed to (2.5) and (2.6). By Definition 9, a mild solution of (2.5) is given by

(2.7)

Now we rewrite system (2.6) in the equivalent integral equation

(2.8)

The above equation can be expressed as

(2.9)

where

We apply the Laplace transform for (2.8) to get (see, [25])

which implies

Note that the Laplace transform for is

Thus we can derive the mild solution of (2.6) as

(2.10)

By (2.7) and (2.10), the mild solution of (2.4) is given by

By using the above results, we can write the following definition of mild solution of the system (2.3).

Definition 9 ([10], Definition 3.1). By a mild solution of the system (2.3) we mean a function which satisfies the following integral equation

Now we can give the concept of mild solution for our considered problem (1.1).

Definition 10 By a mild solution for (1.1), we mean a function which satisfies the following integral equation

(2.11)

where and is an integrable selection for.

Remark 2 It is easily to see that the solution given by (2.11) satisfies the relation

.

Remark 3 If for all and if there is no delay then Formula (2.11) will take the form

This means that when there is no neither impulse nor delay in the problem (1.1), its solution is equal to the formula given in (2.2).

Theorem 1 ([34]). Let be a nonempty subset of a Banach space, which is bounded, closed and convex. Suppose is u.s.c. with closed, convex values, and such that and is compact. Then has a fixed point The following fixed point theorem for contraction multivalued is proved by Govitz and Nadler [35].

Theorem 2 Let be a complete metric space. If is contraction, then has a fixed point.

Theorem 3 ([36], Corollary 3.3.1) (Schauder fixed point theorem). Let be a Banach space, a nonempty, convex, closed and bounded subset of and be continuous. If is compact or is compact, then has a fixed point.

3. Existence Results for the Problem (1.1)

In this section, we give the main results of mild solutions of (1.1).

3.1. Convex Case

In the following Theorem we derive the first existence result concerning the mild solution for the problem (1.1).

Theorem 4 Let be a multifunction. Assume the following conditions:

(H₁) A is the infinitesimal generator of a semigroup and is compact.

(H₂) For every is measurable, for almost is upper semi-continuous and for each the set

is nonempty.

(H₃) There exist a function, such that for any

(3.1)

(H₄) is continuous, compact and there exist two positive numbers such that

(3.2)

(H₅) For every, is continuous and compact and there exists a positive constant such that

(3.3)

Then, for a given continuous function the problem (1.1) has a mild solution provided that there is such that

(3.4)

where, such that,

and

Proof. In view of (H₂), for each the set

is nonempty. So, we can define a multifunction as follows: if and only if

where Obviously, every fixed point for

is a mild solution for the problem (1.1). So, our goal is to apply Theorem 1. The proof will be given in several steps.

Step 1. The values of are convex and closed subset in

Since the values of are convex, it is easily to see that the values of are convex. In order to prove that the values of are closed, let and be a sequence in such that in Then, according to the definition of there is a sequence in such that for any

(3.5)

Not that, from (3.1), for any for almost

This show that the set is integrably bounded. Moreover, because

for a.e. the set

is relativity compact in for a.e. Therefore, the set is semi-compact and then, by Lemma 2 it is weakly compact in So, without loss of generality we can assume that converges weakly to a function From Mazur’s lemma, there is a sequence such that and

converges strongly to. Since, the values of are convex, and hence, by the compactness of

Moreover, for every and for every

Therefore, by passing to the limit as in (3.5), we obtain from the Lebesgue dominated convergence theorem that, for every

Then

Step 2. We claim that where

and. To prove that, let

, and If then by (3.2)

(3.6)

For. By using Lemma 4(3), (3.1), (3.2) and (3.5) we get

(3.7)

Similarly, by using Lemma 4(3), (3.1), (3.2),(3.3) and (3.5) we have for,

(3.8)

Therefore, from (3.4).(3.6).(3.7) and (3.8), we conclude that.

Step 3. Let We claim that is equicontinuous, let and. According to the definition of we have

where By the continuity of we can see easily that if then

To show that it suffices to verify that is equicontinuous for every, where

We consider the following cases:

Case 1. Let In view of Holder’s inequality we get

(3.9)

Since is compact, is also, (see, Lemma 4(v)), and hence, is uniformly continuous on (see [28]). Therefore, the last inequality tends to zero as independently of

Case 2. Let be two points in, then

(3.10)

where

and

We only need to check as for every. At first, we note that, as we mention above the operators are uniformly continuous on

So, independently of

For by the Holder inequality we have

(3.11)

independently of

For we note that then for

we have By applying Lemma 3 and taking into account we get

Then

This leads to

Therefore,

(3.12)

independently of

For by using (H₁) and the Lebesgue dominated convergence theorem, we get

(3.13)

By the uniform continuity of we conclude that independently of

Case 3. When, let be two points in Invoking to the definition of we have

Arguing as in the first case we get

(3.14)

Case 4. When, , let be such that and such that

, then we have

According the definition of we get

Arguing as in the first case we can see that

(3.15)

From (3.9) ® (3.15) we conclude that is equicontinuous for every.

Step 4. Our aim in this step is to show that for any, the set

is relatively compact in.

Let us introduce the following maps:

where

and if and only if

where Obviously,

Because is a bounded subset in and

is compact, the set

is relatively compact in. Also, since the functions are compact, the set

is relatively compact in It remains to show that the set

is relatively compact in For each arbitrary and, we define

Note that we can rewrite in the form

Since the operator is compact and is compact on the set

is relatively compact in. Moreover, by using (H₃) and (H₄) we get

Using Hölder’s inequality to get

Obviously, by Lemma 4(2), the right hand side of the previous inequality tend to zero as Hence, there exists a relatively compact set that can be arbitrary close to the set Hence, this set is relatively compact in. Hence, is relatively compact As a consequence of Steps 3 and 4 with Arzela-Ascoli theorem we conclude that is relatively compact.

Step 5. has a closed graph on.

Let in and

with in We will show that By recalling the definition of for any there exists such that

(3.16)

Let us show that the sequence is semicompact. From the uniform convergence of towards for any

Moreover is upper semicontinuous with compact values, then for every there exists a natural number such that for every

where Then, the compactness of implies that the set is relatively compact for. In addition, assumption (H₃) implies

Then, by Lemma 2, is semicompact, hence weakly compact. Arguing as in Step 1 from Mazur’s theorem, there is a sequence such that

and converges strongly to. Since, the values of are convex, and hence,

. By passing to the limit in (3.16), with taking into account that is continuous, we obtain

This proves that the graph of is closed.

Now, as a consequence of Step 1 to Step 5, we conclude that the multifunction of is a compact multivalued function, u.s.c with convex compact values. By applying Theorem 1, we can deduce that has a fixed point which is a mild solution of Problem (1.1).

In the following Theorems we give another version for an existence result for (1.1).

Theorem 5 Let be a multifunction, A is the infinitesimal generator of a semigroup and We suppose the following assumptions:

(H₆) For every is measurable.

(H₇) There is a function such that For every

(H₈) There is a positive constant such that

(H₉) For each there is such that

(H₁₀)

where, and

Then (1.1) has a mild solution.

Proof. For set

By Lemma 1, (H₆) and (H₇), is measurable. Since its values are closed, it has a measurable selection (see [29], Theorem 8.1.3) which, by hypothesis

(H₇), belongs to Thus is nonempty. Let us transform the problem into a fixed point problem. Consider the multifunction map,

defined as follows: for is the set of all functions such that for each

where It is easy to see that any fixed point for is a mild solution for (1.1). So, we shall show that satisfies the assumptions of Theorem 2. The proof will be given in two steps.

Step1. The values of are nonempty and closed.

Since is non-empty, the values of are non-empty. In order to prove the values of are closed, let and be a sequence in such that in Then, according to the definition of there is a sequence in such that for any

(3.17)

Since is closed, for any there is such that In view of (H₉), for every and for a.e.

This show that the set is integrably bounded. Using the fact that has compact values, the set is relativity compact in for a.e. Therefore, the set is semi-compact and in Then, by Lemma 2, it is weakly compact. So, we may pass to a subsequence if necessary to get that converges weakly to a function From Mazur’s theorem, there is a sequence such that

and converges strongly to Since, the values of

are convex, and hence, by the compactness of Note that for every and for every

Therefore, by means of the Lebesgue dominated convergence Theorem and the continuity of we obtain from (3.17)

So,

Step 2. is contraction. Let and Then, there is such that for any

(3.18)

Consider the multifunction defined by

For each is nonempty. Indeed, let from (H₇), we have

Hence, there exists such that

Since the functions are measurable, Proposition III.4 in [30], tells us that the multifunction is measurable. Because its values are nonempty and closed there is with

(3.19)

Let us define

(3.20)

Obviously, and if then by (H₈)

If we get from and (H₈)

(3.21)

Similarly, if we get from, (H₈) and (H₉)

(3.22)

By interchanging the role of and, we obtain from (3.21), (3.22) and (H₁₀)

Therefore, is contraction and thus by Theorem 2 has a fixed point which is a mild solution for (1.1).

3.2. Nonconvex Case

In the following Theorem we give nonconvex version for Theorem 4. Our hypothesis on the orient field is the following:

(H₁₁) is a multifunction such that 1) is graph measurable and

is lower semicontinuous.

2) There exists a function, such that for any

Theorem 6 If the hypotheses (H₁), (H₄), (H₅) and (H₁₁), then the problem (1.1) has a mild solution provided that there is such that the condition (3.4) is satisfied.

Proof. Consider the multivalued Nemitsky operator

defined by

We shall prove that has a nonempty closed decomposable value and l.s.c. Since has closed values, is closed ([37]). Because is integrably bounded, is nonempty (see, Theorem 3.2 [37]). It is readily verified, is decomposable. To check the lower semi-continuity of, we need to show that, for every is upper semicontinuous. To this end from Theorem 2.2 [37], we have

(3.23)

We shall show that, for any the set

is closed. For this purpose, let and assume that in. Then, for all in By virtue of (H₁₁)(1) the function

is u.s.c. So, via the Fatou Lemma, and (3.23) we have

Therefore, and this proves the lower semicontinuity of This allows us to apply Theorem 3 of [38] and obtain a continuous map such that for every Then,

Consider a map defined by

Arguing as in the proof of Theorem 4, we can show that satiesfies all the conditions of Theorem 3 (Schauder fixed point theorem). Thus, there is such that This means that is a mild solution for (1.1).

Remark 4 The condition (3.4) will be satisfied if

Indeed, condition (3.4) can be written as

4. Conclusion

In this paper, existence problems of nonlocal fractionalorder impulsive semi-linear differential inclusions with delay have been considered. We have been considered the case when the values of the orient field are convex as well as non-convex. Some sufficient conditions have been obtained, as pointed in the first section, theses conditions are strictly weaker than the most of the existing ones. In addition, our technique allows us to discuss some fractional differential inclusions with delay.

REFERENCES