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.
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 [
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. [
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
Motivated by the researches mentioned previously, we will study the following nonlocal impulsive differential inclusions of fractional order
To study the theory of abstract impulsive differential inclusions with fractional order, the first step is how to define the mild solution. Mophou [
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 [
In Section
The following are some simple examples for operators that generate a noncompact semigroup. The ordinary differential operator The ordinary differential operator
In Section
Most previous controllability works contained the assumption of the compactness of the operator semigroup. However, Hernández and O’Regan [
The present paper is organized as follows. In Section
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.
Let
Let
Let
A sequence it is integrally bounded; that is, there is the set
We recall one fundamental result which follows from Dunford-Pettis Theorem.
Every semi-compact sequence in
For more about multifunctions, we refer to [
Let
A measure of noncompactness monotone if nonsingular if regular if invariant with respect to union with compact sets if for any compact subset algebraic semiadditive if semiadditive if the Lipschitz property let
Note that the property (vii) implies the continuity property of
Let
If
Let
Let
Let
For
The fractional integral of order
The Caputo derivative of order
Note that the integrals appearing in the two previous definitions are taken in Bochner’ sense and
Let
In the following we recall the properties of
For any fixed For If For any fixed If
Let
As in [
Of course
By using the concept of mild solutions of impulsive fractional evolution equation in Ouahab [
By a mild solution for (
The above definition of piecewise continuous mild solutions comes from Ouahab [
If
The following fixed point theorem for contraction multivalued is proved by Covitz and Nadler [
Let
Let The For every There exists a function There exists a function The function For every
Then the problem (
In view of (HF1) for every
In view of (
Note that, by Lemma
Therefore, by passing to the limit as
This proves that
To prove that, let
Similarly, we obtain
Therefore,
We consider the following cases.
We only need to check
For
For
According the definition of
Arguing as in the first case, we can see that
From (
Now for every
By Lemma
In the following step we prove (
From the definition of
Now, since
Furthermore, condition (HI) implies, for every
In order to estimate the quantity
Using Hölder’s inequality to obtain for any
We observe that, from (H5) it holds that for a.e.
Note that
Using (
Then, by (
This inequality with (
By means of a finite number of steps, we can write
At this point, we are in position to apply the generalized Cantor's intersection property (Lemma
Indeed,
Consider a sequence
Observe that for every
Hence, the sequence
Therefore,
As a result of Steps 1–5, the multivalued
To end this section, we prove that the set of mild solutions of (
If the function
Note that by Theorem
By arguing as in Step 2 of Theorem
In this section we use the methods in the above section to discuss the controllability of (
Now, we suppose that
A function
The system (
Let For every There is a function for every for every There is a positive constant For any The linear bounded operator
Then (
By (HF4) and (HF5)(i), we conclude from Lemma
We will show that, when using this control, the multivalued function
Since
Define
Obviously
Then, by (
This inequality with (
Similarly, for any
By interchanging the role of
Next, we give another controllability result.
Let
We assume the conditions (HF4), (HF5), (Hg)*, and
Then, (
As in the proof of Theorem
As in the proof of Theorem
This inequality with (Hg)* and (
Similarly for
By interchanging the role of
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.
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.