Research Article | Open Access
An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space
This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem.
It is well known that the theory of fractional calculus deals with the concepts of differentiation and integration of arbitrary orders, real and complex. Actually, the real importance of fractional derivatives lies in their nonlocal character which gives rise to a long memory effect and thus to a better insight into the modelled processes. On the other hand, since models using classical derivatives are just a special case of those using fractional derivatives, then most of the investigators in different areas such as electronics, viscoelasticity, satellite guidance, medicine, anomalous diffusion, signal processing, and many other branches of science and technology have revisited some classical dynamic systems in the framework of fractional derivatives to get better results; see the references [1–8]. We point out that most of dynamic systems are naturally governed by fractional differential equations; for further applications of fractional derivatives in other areas and useful backgrounds we refer the reader to the works [1–5, 7–12].
As far as we are concerned with impulsive fractional differential equations, we intend to improve and correct in this paper some existence results established earlier in [4, 13–18] for impulsive fractional differential equations. There have been in the last couple of years several concepts of solutions satisfying some fractional equations subjected to impulsive conditions, see [13, 14, 18, 19], while the authors of  claimed that their new concept is the more realistic than the existing ones. Actually, we believe that nobody holds all the truth about this subject and a lot of dark sides of these approaches are not yet well elucidated.
Regarding the concept of a solution for impulsive fractional equations introduced by  we point out that Lemma 2.6 which has been used by the authors to obtain the equivalence between an impulsive fractional problem and an integral equation is false as we see in the following counterexample.
In the famous book of Nagy and Riesz [20, page 48], there is an example of monotonic continuous function which is not constant in any subinterval of and satisfies , almost everywhere in . So, in terms of Caputo’s derivative we would have formally for any However, there is no apparent equivalence between this problem and the fractional integral representation of defined in Lemma 2.6 ; otherwise the function would be constant and equal to throughout the interval which is a contradiction! Moreover, since in the same work Lemma 2.7 is based on the latter lemma then it is not correct and may lead to apparent contradiction.
Our main contribution in this paper is the study of new fractional problems of several orders in a Banach space subjected to some impulsive conditions of the form Let us first give a concrete example of such a problem in ; namely; where , , , and . So, we look for a piecewise continuous function satisfying (3). Solving the subproblem we obtain from which we get .
On the other hand, the solution of the subproblem is given by
Hence, the piecewise continuous function is a solution to the impulsive fractional problem (3).
A particular problem of (3) is as follows: corresponding to the case whose solution is The paper is organized as follows. We present in Section 2 our problem as we establish some equivalence between the the given problem and a nonlinear integral equation. Next, we state a piecewise-continuous type of the Ascoli-Arzela theorem as well as Schaefer’s fixed point theorem in order to apply them subsequently in our proofs. In Section 3 we use the Banach contraction theorem to establish an existence and uniqueness theorem of a quasilinear impulsive fractional problem in an abstract Banach space. In Section 4 we apply Schaefer’s fixed point theorem to some semilinear impulsive fractional problem in a finite dimensional Banach space to obtain the existence of a piecewise continuous solution; on the other hand we prove the stability of the obtained solution with respect to the initial value. Finally, we conclude the paper by a concrete example illustrating one of our results.
The main purpose of this paper is the investigation of the existence and uniqueness of solution corresponding to the following impulsive fractional integrodifferential equation in a Banach space where(i) with and , ; ,(ii) is Caputo’s fractional derivative of order , ,(iii) is a continuous operator, where is the Banach space of bounded linear operators on in itself,(iv), ; and are, respectively, the right and left limits of at the discontinuity point . We set the following hypotheses:(j) the functions are continuous with and , for every , (jj) the nonlinear function is continuous, and are two continuous functions over and , respectively.
We will use in the sequel the following notation: We recall that is the Banach space of continuous functions endowed with the norm Next, we introduce the definition of the fractional derivative in the sense of Caputo. We have the following.
Definition 1. We define the left-sided fractional Riemann-Liouville integral of order of a function as follows:
We define the left-sided fractional derivative of order of a function in the sense of Caputo by
Remark 2. (1) We point out that the previous integrals are understood in the sense of Bochner.
(2) We assume of course that the function satisfies the necessary conditions for which those integrals are well defined.
Next, we consider the linear functional space equipped with the norm We obtain a Banach space .
Now, we recall the definition of the solution of the problem (11).
Definition 3. A function is said to be a solution of the problem (11) if exists in , for , and satisfies(i)the equation in , ,(ii)the initial condition ,(iii)the impulsive conditions , .
Lemma 4. A function satisfies the following nonlinear integral equation if and only if it is a solution to problem (11).
Proof. Since we have , then .
Now, for , the solution of the problem is given by We have for the following relation, and so
Next, for , we have from which we infer that
Arguing as before we obtain for Reasoning by induction we get, for any , , the general expression Conversely, we assume that satisfies (19). If , then .
Now, using the fact that Caputo’s derivative of a constant is zero, then, for every , , we get
So for every , .
Also we can easily show that
We conclude this section by introducing some useful theorems which will be used in the sequel.
Theorem 5 (-type Ascoli-Arzela theorem ). Let be a Banach space and . If the following conditions are satisfied(i) is a uniformly bounded subset of ;(ii) is equicontinuous in , ;(iii), , and are relatively compact subsets of , then is a relatively compact subset of .
Theorem 6 (Schaefer’s fixed point theorem). Let be a Banach space and let be a completely continuous operator. If the set is bounded, then has at least a fixed point.
3. A Quasilinear Impulsive Fractional Problem
We begin our investigation by the following result which ensures the existence and the uniqueness of the solution of the following impulsive quasilinear problem: We assume that is continuous and there exists a constant such that We set .
It is not hard to establish the following estimates.
Lemma 7. Let the functions and be continuous with respect to the variables and , and there are two positive constants and such that Then, there exist two positive constants and so that and, for , one has for every and .
We assume the following hypotheses:(H1). We set and .(H2) There is a positive constant such that We set and .(H3) There is a positive constant such that (H4) The positive real number satisfies .Next, we state and prove the existence and uniqueness result for the quasilinear integrodifferential problem (31); we have the following.
Theorem 8. If the assumptions (H1)–(H4) are satisfied, then problem (31) has one and only one solution .
Proof. Since we are concerned with the existence and uniqueness of the solution of (31) then, it is wise to use the Banach contraction principle in order to establish such results.
Let be the closed ball of centered at with radius satisfying the following inequality: where Endowing with the metric , for every , we obtain a complete metric space . Next, we define the operator by It is understood that the sum is zero if .
First, we prove that if , then .
Indeed, for each , , and any sufficiently small , we have Calculating the integrals we find that Thus, the right-hand side tends to zero as . Likewise one gets ; this shows that is continuous at . Hence .
Next, for the right endpoint we get for any sufficiently small which shows that the right-hand side tends to zero as , and accordingly, is continuous at . Therefore .
To prove that we see that, for any and , , we have
Estimating the right-hand side we find So Therefore, , and consequently .
Next, we prove that is a contraction mapping; indeed, for any and , , we have Taking into account the previous assumptions we get the following estimate:
Thus, Accordingly, the mapping has a unique fixed point , which completes the proof.
4. A Semilinear Impulsive Fractional Problem
In this section we consider a semilinear impulsive fractional integrodifferential problem subjected to a nonlocal condition in a finite dimensional normed space . Actually, the finite dimension requirement is due to some technical difficulties in order to prove some compactness properties. The problem is as follows:
We assume that the mapping is continuous and we put We need the following hypothesis:(H5) there exists a constant such that the mapping satisfies Now, we are ready to state and prove the following result.
Theorem 9. If the assumptions (H1)–(H3) and (H5) are satisfied, then problem (51) has at least one solution .
Proof. Let us define the operator by
First, we notice that by using the same technique as that in the proof of the Theorem 8 we can establish that if , then ; that is, the operator maps the space into itself.
To prove that has a fixed point we use Schaefer’s fixed point theorem. We proceed in four steps.
Step 1 ( is continuous). Let such that in ; then
Taking into account the assumptions (H2)-(H3) and (H5) and using Lemma 7 we get Calculating the integrals in the right-hand side we obtain So and accordingly, is continuous.
Step 2. Let and . Define ; then for any we have
Estimating the right-hand side we obtain implying that Hence is uniformly bounded.
Step 3 (we prove that is equicontinuous). Let ; then, for any , we have So As , the right-hand side of the previous inequality tends to zero, which means that is equicontinuous.
We point out that the closures of the subsets , , and , , are bounded in (); hence they are compact.
As a consequence of the previous steps and the -type Arzela-Ascoli theorem we conclude that is completely continuous.
Step 4. Now, we show that the set is bounded.
Let ; then , for some . Thus, for each ,