An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space

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 themodelled processes. On the other hand, sincemodels 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 [18] 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 [18] 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 F : [0, 1] → R which is not constant in any subinterval of [0, 1] and satisfies F󸀠 = 0, almost everywhere in [0, 1]. So, in terms of Caputo’s derivative we would have formally for any α ∈


Introduction
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][2][3][4][5][6][7][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][2][3][4][5][7][8][9][10][11][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][14][15][16][17][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 [18] 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 [18] 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.
However, there is no apparent equivalence between this problem and the fractional integral representation of  defined in Lemma 2.6 [18]; otherwise the function () would be constant and equal to  0 throughout the interval [0, 1] International Journal of Differential Equations 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.
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.

Preliminaries
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 (, ‖ ⋅ ‖) are two continuous functions over  ×  and  0 × , respectively.We will use in the sequel the following notation: We recall that C = C(; ) 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  ∈ (0, 1) of a function  : [, ] →  as follows: We define the left-sided fractional derivative of order  ∈ (0, 1) 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
equipped with the norm We obtain a Banach space (PC(; ), ‖ ⋅ ‖ PC ).Now, we recall the definition of the solution of the problem (11).
if and only if it is a solution to problem (11).
is given by We have for  =  1 the following relation ( + 1 ) = ( 1 ) +  1 (( − 1 )), and so Next, for  ∈  1 = ( 1 ,  2 ], we have from which we infer that Arguing as before we obtain for  ∈  2 Reasoning by induction we get, for any  ∈   ,  = 1, . . ., , the general expression Conversely, we assume that  satisfies (19).If  = , then () =  0 .Now, using the fact that Caputo's derivative of a constant is zero, then, for every  ∈   ,  = 0, . . ., , we get So for every  ∈   ,  = 0, . . ., .Also we can easily show that We conclude this section by introducing some useful theorems which will be used in the sequel.

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  :  ×  → B() is continuous and there exists a constant  > 0 such that We set   = max ∈ ‖(, 0)‖.
It is not hard to establish the following estimates.
Lemma 7. Let the functions ℎ(, , ) and (, , ) be continuous with respect to the variables  and , and there are two positive constants  1 and  2 such that Then, there exist two positive constants   1 and   2 so that and, for  = 0, . . ., , one has for every , V ∈ PC(, ) and  ∈ .
Next, we state and prove the existence and uniqueness result for the quasilinear integrodifferential problem (31); we have the following.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 B  = { ∈ PC(, ) : ‖‖ PC ≤ } be the closed ball of PC(, ) centered at 0 with radius  satisfying the following inequality: where Endowing B  with the metric (, V) = ‖ − V‖ PC , for every , V ∈ B  , we obtain a complete metric space (B  , ).Next, we define the operator Ψ : B  → B  by It is understood that the sum ∑ <  < is zero if  ∈  0 .First, we prove that if  ∈ PC(; ), then Ψ ∈ PC(; ).Indeed, for each  ∈ (  ,  +1 ),  ∈ C((  ,  +1 ), ), and any sufficiently small  > 0, we have Calculating the integrals we find that Thus, the right-hand side tends to zero as  → 0. Likewise one gets lim  → 0 ‖Ψ() − Ψ( − )‖ = 0; this shows that Ψ is continuous at . Hence Ψ ∈ C((  ,  +1 ), ).
Next, for the right endpoint  =  +1 we get for any sufficiently small  > 0 which shows that the right-hand side tends to zero as  → 0, and accordingly, Ψ is continuous at  +1 .Therefore Ψ ∈ PC(, ).
To prove that ΨB  ⊂ B  we see that, for any  ∈ B  and  ∈   ,  = 0, . . ., , we have Estimating the right-hand side we find Therefore, ‖Ψ‖ PC ≤ , and consequently ΨB  ⊂ B  .
Next, we prove that Ψ is a contraction mapping; indeed, for any , V ∈ B  and  ∈   ,  = 0, . . ., , we have Taking into account the previous assumptions we get the following estimate: Thus, Accordingly, the mapping Ψ has a unique fixed point  = Ψ ∈ B  , which completes the proof.

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  :  → B() is continuous and we put We need the following hypothesis: (H5) there exists a constant  > 0 such that the mapping Now, we are ready to state and prove the following result.Proof.Let us define the operator  : PC(, ) → PC(, ) by First, we notice that by using the same technique as that in the proof of the Theorem 8 we can establish that if  ∈ PC(, ), then  ∈ PC(, ); that is, the operator  maps the space PC(, ) 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 {  } ≥1 ⊂ PC(, ) such that   →  in PC(, ); then Taking into account the assumptions (H2)-(H3) and (H5) and using Lemma 7 we get International Journal of Differential Equations 9 Calculating the integrals in the right-hand side we obtain So and accordingly,  is continuous. Step Hence W is uniformly bounded.
As a consequence of the previous steps and the PCtype Arzela-Ascoli theorem we conclude that  is completely continuous.
This shows that the set  is bounded.We conclude by Schaefer's fixed point theorem that the operator  has a fixed point  ∈ PC(, ) such that  = , which means that  is a solution to problem (51).
Next, we establish the continuous dependence of the solution upon the initial value.We have the following.

Proposition 10. Under the hypotheses (H1)-(H3) and (H5) the solution of problem (51) depends continuously upon its initial value if
Proof.Since  is a solution to (51), then it satisfies the integral equation (19).Let V be a solution to problem (51) with initial value V() = V 0 −(V).Then V() satisfies the integral equation Estimating the difference between solutions () and V() to ( 19) and (67), respectively, we get Taking the supremum over the interval  we find that where which proves that the mapping  0  →  is continuous from  → PC(, ).

Example
Consider the following impulsive fractional integrodifferential problem (75) On the other hand, using (H2) we obtain so  = 1/6.Due to the definition of (, ) we have  =   = 1/24.Let us now find a threshold for the value of  for which condition (H4) is satisfied.We should have so  is any positive number such that  < (5√ − 4)/2 = 2.4311.We conclude by Theorem 8 that problem (71) has a unique solution  ∈ PC([0, 1], R) such that ‖‖ PC ≤ .

Concluding Remarks
In this work we have first noticed that most of the published papers dealing with impulsive differential equations of fractional orders are not mathematically correct, so we have proved through a concrete counterexample that the concept of solution proposed recently by some authors is not realistic.On the other hand, we introduced a new class of impulsive fractional problems with several fractional orders and we established an equivalence with some integral equation.Moreover, we derived two existence results by using two different fixed point theorems as we proved the stability of the solution of the given problem with respect to the initial value.Finally, we illustrated our first theorem of existence and uniqueness by a concrete example in R.

Lemma 4 .
A function  ∈ PC(; ) satisfies the following nonlinear integral equation