Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 518306, 16 pages

http://dx.doi.org/10.1155/2013/518306

Research Article

Existence and Controllability Results for Nonlocal Fractional Impulsive Differential Inclusions in Banach Spaces

JinRong Wang1,2 and Ahmed G. Ibrahim³

¹School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, China

²Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

³Department of Mathematical, Faculty of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received 24 May 2013; Accepted 5 August 2013

Academic Editor: Henryk Hudzik

Copyright © 2013 JinRong Wang and Ahmed G. Ibrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We firstly deal with the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions involving Caputo derivative in Banach spaces in the case when the linear part is the infinitesimal generator of a semigroup not necessarily compact. Meanwhile, we prove the compactness property of the set of solutions. Secondly, we establish two cases of sufficient conditions for the controllability of the considered control problems.

1. Introduction

During the past two decades, fractional differential equations and 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 [1–3] and the references therein. El-Sayed and Ibrahim [4] initiated the study of fractional multivalued differential inclusions. Recently, some basic theory for initial value problems for fractional differential equations and inclusions was discussed in [5–13].

The theory of impulsive differential equations and inclusions has been an object of 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 be described using the classical differential problems. During the last ten years, impulsive differential inclusions with different conditions have been intensely studented 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. [14].

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 . In the few past years, several papers have been devoted to study the existence of solutions for differential equations or inclusions with nonlocal conditions [15–17]. For impulsive differential equations or inclusions with nonlocal conditions of order we refer to [16, 17]. For impulsive differential equations or inclusions of fractional order we refer to [9, 18–21] and the references therein.

Motivated by the researches mentioned previously, we will study the following nonlocal impulsive differential inclusions of fractional order of the type:

where is the Caputo derivative of order with the lower limit zero, is the infinitesimal generator of a -semigroup on a real Banach space which in not necessarily compact, is a multifunction, , impulsive functions which characterize the jump of the solutions at impulsive points, is a nonlinear function related to the nonlocal condition at the origin, and , are the right and left limits of at the point , respectively.

To study the theory of abstract impulsive differential inclusions with fractional order, the first step is how to define the mild solution. Mophou [18] firstly introduced a concept on a mild solution which was inspired by Jaradat et al. [19]. However, it does not incorporate the memory effects involved in fractional calculus and impulsive conditions. Wang et al. [9] introduced a new concept of -mild solutions for (1) and derived existence and uniqueness results concerning the -mild solutions for (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 the following: Ouahab [8] proved a version of Filippov’s Theorem for (1) without impulse and is an almost sectorial operator, Cardinali and Rubbioni [16] proved the existence of mild solutions to (1) when and the multivalued function satisfies the lower Scorza-Dragoni property, and is a family of linear operator, generating a strongly continuous evolution operators, Fan [17] studied a nonlocal Cauchy problem in the presence of impulses, governed by autonomous semilinear differential equation, Dads et al. [20] and Henderson and Ouahab [21] considered (1) when , and Zhou and Jiao [12, 13] introduced a suitable definition of mild solution for (1) based on Laplace transformation and probability density functions for (1) when is single-valued function and without impulse. More recently, Wang and Zhou [10] proved existence and controllability results for (1) without impulse and with local conditions. Very recently, Zhang and Liu [11] considered (1) without impulse, is a single-valued function, and is strongly equicontinuous -semigroup. Ibrahim and Al Sarori [22] gave some existence results of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense when the semigroup is compact. Among the previous works, little is concerned with nonlocal impulsive fractional differential inclusions via noncompact semigroup and the techniques of the measure of noncompactness.

In Section 3, we apply the methods and techniques to derive some sufficient conditions for existence results for (1) when the values of the multivalued function are nonempty convex and compact. At the end of this section, we prove that the set of mild solutions of (1) is compact. We adopt the definition of mild solution introduced by Wang et al. [9]. Unlike the papers [9, 10, 12, 13, 22], we do not assume is a compact semigroup, and instead we assume that satisfies a compactness condition involving the Hausdorff measure of noncompactness.

The following are some simple examples for operators that generate a noncompact semigroup.

(i)The ordinary differential operator on the normed space with domain generates a noncompact semigroup , , defined by

(ii)The ordinary differential operator on the normed space with domain generates a noncompact semigroup , , defined by

In Section 4, we will consider the following two nonlocal semilinear impulsive evolution system of order of the type

where the control function is given in , a Banach space of admissible control functions with being a Banach space, and is a bounded linear operator from into and , .

Most previous controllability works contained the assumption of the compactness of the operator semigroup. However, Hernández and O’Regan [23] and Obukhovski and Zecca [24] pointed that if the operator semigroup was compact, then the assumption (HW) (see Section 4) was valid if and only if was finite dimensional. Controllability of (4) when was discussed by Guo et al. [25]. We refer to, in recent years, the problem of controllability for various kinds of fractional differential and integro-differential equations and inclusions have been discussed in [26–28].

The present paper is organized as follows. In Section 2, we collect some background material and basic results from multivalued analysis and fractional calculus to be used later. In Section 3, we prove existence results for (1); we also show that the set of solutions is compact. In Section 4, we present two interesting controllability results of (4) and (5).

Our basic tools are the methods and results for semilinear differential inclusions, the properties of noncompact measure, compactness criterion in the piecewise continuous functions of space, and fixed point techniques.

2. Preliminaries and Notation

Let the space of -valued continuous functions on with the uniform norm , the space of -valued Bochner integrable functions on with the norm , = { is nonempty and bounded}, = { is nonempty and closed}, = { is nonempty and compact}, = { is nonempty, closed, and convex}, and = { is nonempty, convex, and compact}, and (resp., ) be the convex hull (resp., convex closed hull in ) of a subset .

Definition 1. Let and be two topological spaces. A multifunction is said to be upper 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 nonempty compact values, then is . if and only if is closed.

Lemma 2 (see [29, Theorem 8.2.8]). Let be a complete -finite measure space, a complete separable metric space, and a measurable multivalued function with nonempty 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 single-valued function the multivalued function is measurable, and for every Caratheodory single-valued function the multivalued function is measurable.

Definition 3. A sequence is said to be semicompact if

(i)it is integrally bounded; that is, there is such that

(ii)the set is relatively compact in a.e., .

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

Lemma 4 (see [30]). Every semi-compact sequence in is weakly compact in .

For more about multifunctions, we refer to [29, 31–34].

Let be a partially ordered set. A function is called a measure of noncompactness (MNC) in if , for every .

Definition 5 (see [35]). A measure of noncompactness is called

(i)monotone if , , implies ;(ii)nonsingular if , for every , ;(iii)regular if is equivalent to the relative compactness of ;(iv)invariant with respect to union with compact sets if for any compact subset in and any , ;(v)algebraic semiadditive if , for every , , where ;(vi)semiadditive if , for every , ; the Hausdorff measure of noncompactness which is defined as

possesses all the properties (i)−(vi) and the following additive properties:(vii)the Lipschitz property , for every , , where is the Hausdorff distance;(viii), for every , ;(ix)let be a bounded linear operator. Then , for every .

Note that the property (vii) implies the continuity property of with respect to the Hausdorff metric. For more information about the measure of noncompactness, we refer to [29, 34–36].

Lemma 6 (see [29, Lemma 1.1.9]). Let be a sequence of subsets where is a compact in the separable Banach space . Then

Lemma 7 (see [31], generalized Cantor’s intersection). If is a decreasing sequence of nonempty, bounded, and closed subsets of and , then is nonempty and compact.

Lemma 8 (see [35]). Let be the Hausdorff measure of noncompactness on . If is bounded, then for every,

where . Furthermore, if is equicontinuous on , then the map is continuous on and .

Lemma 9 (see [36, Page 125]). Let be a bounded set in . Then for every , there is a sequence in such that

Lemma 10 (see [37, Lemma 4]). Let , be an integrably bounded sequence such that

where . Then for each , there exists a compact , a measurable set , with measure less than , and a sequence of functions such that , for all and

Lemma 11 (see [10, Lemma 2.10]). For and , one has .

Definition 12. 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 13. The Caputo derivative of order of a continuously differentiable function is defined by

Note that the integrals appearing in the two previous definitions are taken in Bochner’ sense and for all . For more information about the fractional calculus we refer to [2, 3].

Definition 14 (see [13, Lemma 3.1, and Definition 3.1]; see also [10–12]). Let . A function is said to be a mild solution of the following system:

if it satisfies the following integral equation:

where

, and are a probability density function defined on ; that is, .

In the following we recall the properties of , .

Lemma 15 (see [13, Lemma 3.2, Lemma 3.3, and Lemma 3.5]).

(i) For any fixed , , are linear bounded operators.(ii) For , .(iii) If for all , then for any , and for all .(iv) For any fixed , , are strongly continuous.(v) If , is compact, then and are compacts.

Let and , . To give the concept of mild solution of (1), we consider the set of functions:

It is easy to check that is a Banach space endowed with the norm:

As in [16] we consider the map defined by

where is the Hausdorff measure of noncompactness on the Banach space and

Of course . It is easily seen that is the Hausdorff measure of noncompactness on the Banach space .

By using the concept of mild solutions of impulsive fractional evolution equation in Ouahab [8], we can give the concept of mild solution for our considered problem (1).

Definition 16. By a mild solution for (1), we mean a function which satisfies the following integral equation:

where , , and is an integrable selection for .

Remark 17. The above definition of piecewise continuous mild solutions comes from Ouahab [8] which is more suitable than the corresponding definition of piecewise continuous mild solutions for impulsive semilinear fractional evolution equations in Shu et al. [38]. In fact, if , then (22) reduces to

which is the standard formula of -mild solutions of impulsive differential inclusions:

However, one cannot expect to obtain the above coincide formula by using the definition of mild solutions in Shu et al. [38]. For more discussion on the formula of solutions to impulsive fractional differential equations, the reader can refer to Fckan et al. [39].

Theorem 18 (see [31, Corollary 3.3.1, and Proposition 3.5.1]). If is a closed convex subset of a Banach space and is closed and -condensing, where is a nonsingular measure of noncompactness; then has a fixed point. If the set of fixed points for is a bounded subset of , then it is compact.

The following fixed point theorem for contraction multivalued is proved by Covitz and Nadler [30].

Theorem 19. Let be a complete metric space. If is contraction, then has a fixed point.

3. Existence of Mild Solutions for the Problem (1)

Theorem 20. Let be the infinitesimal generator of a -semigroup , a multifunction, , and . We assume the following conditions.

(HA) The -semigroup is equicontinuous; that is, for any , , independently of .(HF1) For every , is measurable, for every , is upper semicontinuous, and for every , the set is nonempty.(HF2)There exists a function , and a nondecreasing continuous function such that for any

(HF3)There exists a function , satisfying

and for every bounded subset , , for a.e. , where is the Hausdorff measure of noncompactness in , and .(Hg) The function is continuous, compact, and there are two positive constants , such that

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

Then the problem (1) has a mild solution provided that there is such that

where , and are such that .

Proof. In view of (HF1) for every , the set is nonempty. So, we can introduce the multifunction which is defined as let . A function if and only if for each ,

where . It is easy to see that any fixed point for is a mild solution for (1). So our goal is to prove, by using Theorem 19, that has a fixed point. The proof will be given in several steps.

Step 1 (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 , ,

In view of (25) for every and for a.e. ,

This shows that the set is integrably bounded. Moreover, because , for a.e. , the set is relativity compact in for a.e. . Therefore, the set is semicompact and then, by Lemma 6 it is weakly compact in . So, without loss of generality we can assume that converges weakly to a function . From Mazur’s lemma, for every natural number there is a natural number and a sequence of nonnegative real numbers , such that , and the sequence of convex combinations , converges strongly to in as . Since takes convex values, using Lemma 6, we obtain for a.e. ,

Note that, by Lemma 15(iii), for every , and every ,

Therefore, by passing to the limit as in (31), we obtain from the Lebesgue dominated convergence theorem that, for every ,

This proves that is closed.

Step 2. Let . Obviously, is a bounded, closed and convex subset of . We claim that .

To prove that, let and . For . By using Lemma 15, (25), (Hg), and Hölder’s inequality we get for ,

Similarly, we obtain

Therefore, .

Step 3. Let . We claim that the set is equicontinuous for every , where

Let . Then there is with . By recalling the definition of , there is such that

We consider the following cases.

Case 1. When , let , be two points in ; then

where

We only need to check as for every . By the equicontinuity of we have

independently of .

For and , one can repeat the same procedure in Theorem 4 of [22] to obtain

independently of .

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

independently of and .

Case 2. When , let , be two points in . Invoking to the definition of , we have

Arguing as in the first case, we get

Case 3. 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

From (40)–(49) we conclude that is equicontinuous for every .

Now for every , set . From Step 1, is a nonempty, closed, and convex subset of . Moreover, . Also . By induction, the sequence , is a decreasing sequence of nonempty, closed, and bounded subsets of . Our goal is to show that the subset is nonempty and compact in .

By Lemma 7 (generalized Cantor’s intersection), it is enough to show that

where is the Hausdorff measure of noncompactness on defined in Section 2.

In the following step we prove (50).

Step 4. Our aim in this step is to show that Relation (50) is satisfied. Let be a fixed natural number and . In view of Lemma 9, there exists a sequence , in such that

From the definition of , the above inequality becomes

where and is the Hausdorff measure of noncompactness on . Arguing as in the previous step we can show that , , is equicontinuity. Then, by applying Lemma 8, we obtain

where is the Hausdorff measure of noncompactness on . Therefore, by using the nonsingularity of , (52) becomes

Now, since , , there is such that , . By recalling the definition of for every there is such that for every ,

Since is compact, the set is relatively compact. Hence,for every , we have

Furthermore, condition (HI) implies, for every and every ,

In order to estimate the quantity , we consider the linear continuous map

where

Using Hölder’s inequality to obtain for any , and any ,

Furthermore, by (H₄), for almost , . Consequently, , .

We observe that, from (H₅) it holds that for a.e. ,

Note that . Then, by virtue of Lemma 10, there exists a compact , a measurable set , with measure less than and a sequence of functions such that for all , , and

Using (60) and (62), to obtain for all and all

By taking into account that is arbitrary, we get for all and all

Therefore, for all ,

Then, by (55)–(57) and (65) for every ,

This inequality with (54) and with the fact that is arbitrary, implies

By means of a finite number of steps, we can write

Since this inequality is true for every , by (26) and by passing to the limit as , we obtain (50), and so our aim in this step is verified.

At this point, we are in position to apply the generalized Cantor's intersection property (Lemma 7) and claim that the set is a nonempty and compact subset of . Moreover, for every being bounded, closed, and convex, is also bounded, closed, and convex.

Step 5. Let us verify that .

Indeed, , for every . Therefore, . On the other hand for every . So, .

Step 6. The graph of the multivalued function is closed.

Consider a sequence in with in and let with in . We have to show that . By recalling the definition of , for any , there is such that

Observe that for every and for a.e. ,

This shows that the set is integrably bounded. In addition, the set is relatively compact for a.e. because assumption (HF3) both with the convergence of implies that

Hence, the sequence is semicompact; hence, by Lemma 4, it is weakly compact in . So, without loss of generality we can assume that converges weakly to a function . From Mazur’s lemma, for every natural number , there are a natural number and a sequence of nonnegative real numbers , such that and the sequence of convex combinations , , converges strongly to in as . Since takes convex values, using Lemma 6, we get for a.e. ,

Since is upper semicontinuous with compact values, . Because the values of are convex, we conclude that , a.e. . Then, by the continuity of , , , and by the same arguments used in Step 1, we conclude from relation (69) that

Therefore, . This shows that the graph of is closed.

As a result of Steps 1–5, the multivalued is closed and -condensing, with nonempty convex compact values. Applying Theorem 18, there is such that . Clearly is a mild solution of (1).

To end this section, we prove that the set of mild solutions of (1) is compact.

Theorem 21. If the function in (HF2) is given of the form , then under the assumptions (HA), (HF1), (HF3), (Hg), and (HI) of Theorem 20, the set of mild solutions of (1) is compact in provided that

Proof. Note that by Theorem 20, the set of solutions of (1) is nonempty. Assume that is a mild solution of (1). Then

where is an integrable selection for .

By arguing as in Step 2 of Theorem 20 we get

Therefore,

This inequality with (74) yields that the set of solutions is bounded subset of . Hence, by Theorem 18, it is compact.

4. Application to Controllability

In this section we use the methods in the above section to discuss the controllability of (4) and (5).

Now, we suppose that is a real separable Banach space and is a real Banach space.

Definition 22. A function is said to be a mild solution for (4) if there exists an integrable selection for such that for each ,

Definition 23. The system (4) is said to be nonlocally controllable on the interval if for every , , there exists a control function such that a mild solution of (4) satisfies , .

Theorem 24. Let , be a bounded linear operator, the infinitesimal generator of a strongly semigroup , a multifunction, , and . We assume the following conditions.

(HF4) For every , is measurable.(HF5) There is a function , such that

(i) for every ,

(ii) for every

(Hg)* There is a positive constant such that

(HI)*For any , there is such that

(HW) The linear bounded operator defined by

has an invertible , and there exists a positive constant such that and .

Then (4) is controllable on provided

where .

Proof. By (HF4) and (HF5)(i), we conclude from Lemma 2 that for any , there is an integrable selection for . Then, we can consider a multivalued as: let , a function if and only if for each ,

where is an integrable selection for . Also, for any and any integrable selection for , we define the control function by

We will show that, when using this control, the multivalued function has a fixed point. This fixed point is the desired mild solution. In fact, if is a fixed point for , then from (85) and (86) we get

In the following two steps, we show that satisfies the assumptions of Theorem 19.

Step 1. The values of are closed. In view of (HF5)(ii), for any , and , we have

Since is a linear bounded operator, we can follow the same arguments used in Step 1 of Theorem 20 to prove that the values of are closed subset in .

Step 2 ( is contraction). Take , , and let . Then, there is such that for any ,

Consider the multifunction , . In view of (HF5)(i), for any , is nonempty. Moreover, since the functions, , , are measurable, the multifunction is measurable with closed values (see [38]). Thus, there is with

Define

Obviously . To simplify the calculation, we set

Then, by (90) and (Hg)* we obtain

This inequality with (89)-(90) yields for

Similarly, for any , we get from (93) and conditions (Hg)* and (HI),

By interchanging the role of and in both (94) and (95) we obtain

Therefore, is contraction, and thus by Theorem 19, has a fixed point which is a mild solution for (4).

Next, we give another controllability result.

Theorem 25. Let , , be a bounded linear operator, is the infinitesimal generator of a strongly semigroup , be a multifunction, and .

We assume the conditions (HF4), (HF5), (Hg)*, and and the following condition:

(HΥ) the linear bounded operator defined by

has an invertible , and there exists, a positive constant such that and .

Then, (5) is controllable on provided that

Proof. As in the proof of Theorem 21, for every we define a multivalued as if and only if for each ,

where is an integrable selection for and is the control function defined in (86). Arguing as in the proof of Theorem 24, we can show that, when using this control any fixed point of is the desired mild solution. We will show that satisfies the assumptions of Theorem 19. We only give the main different steps from the proof of Theorem 24. Take , , and let . Then, there is such that for any , ,

As in the proof of Theorem 24 we can find such that

where and satisfying the relation (90). Arguing as in (93) we get

This inequality with (Hg)* and (98) yields for

Similarly for , ,

By interchanging the role of and in both (103) and (104) we get

Therefore, is contraction, and thus by Theorem 19, has a fixed point which is a mild solution for the problem (1).

5. Conclusion

In this paper, existence and controllability problems of fractional order impulsive semilinear differential inclusions with nonlocal condition have been considered. Some sufficient conditions have been obtained, as pointed in the first section; these conditions are strictly weaker than most of the existing ones. In addition, our technique allows us to discuss some fractional inclusions which contains a linear operator that generates a noncompact semigroup.

Acknowledgments

The authors thank the referees for their valuable comments and suggestions which improved their paper. This work is supported by the National Natural Science Foundation of China (11201091), Key projects of Science and Technology Research in the Chinese Ministry of Education (211169), Key Support Subject (Applied Mathematics) and Key project on the reforms of teaching contents and course system of Guizhou Normal College.

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