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

Abstract

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.

1. 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 [1–8]. 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∈(0,1)^𝐶𝐷^𝛼₀⁺𝐹(𝑡)=1Γ(1−𝛼)^𝑡₀(𝑡−𝑠)^−𝛼𝐹^′(𝑠)[],𝐹(𝑎)𝑑𝑠=0,𝑡∈0,1=𝐹₀,0<𝑎<1,with𝐹₀beingthevalueof𝐹at𝑎. (1) However, there is no apparent equivalence between this problem and the fractional integral representation of 𝐹 defined in Lemma 2.6 [18]; otherwise the function 𝐹(𝑡) would be constant and equal to 𝐹₀ throughout the interval [0,1] 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 ^𝐶𝐷^𝛼𝑘_𝑡+𝑘𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+𝐹𝑡,𝑢_𝑡𝑎(𝑡),ℎ(𝑡,𝑠,𝑢(𝜎(𝑠)))𝑑𝑠,_{𝑡𝑘+1𝑎}𝑘(𝑡,𝑠,𝑢(𝜏(𝑠)))𝑑𝑠,𝑡∈𝐽_𝑘𝑢(𝑎),𝑘=0,…,𝑚,=𝑢₀𝑢𝑡∈𝐸,_+𝑘=𝑢𝑡⁻_𝑘+𝐼_𝑘𝑢𝑡⁻_𝑘,𝑘=1,…,𝑚. (2) Let us first give a concrete example of such a problem in ℝ; namely; ^𝐶𝐷^𝛼₀⁺𝑢(𝑡)=𝑡^𝑝−1,𝑡∈𝐽₀=[0,1_𝐶],𝐷^𝛽₁⁺𝑢(𝑡)=(𝑡−1)^𝑞,𝑡∈𝐽₁=(],𝑢(0)𝑢11,𝑇=1,₊=𝑢(1⁻)+2, (3) where 0<𝛼<1, 0<𝛽<1, 𝑇>1, and 𝑝,𝑞∈ℝ⁺. So, we look for a piecewise continuous function 𝑢∶[0,𝑇]→ℝ satisfying (3). Solving the subproblem ^𝐶𝐷^𝛼₀⁺𝑢(𝑡)=𝑡^𝑝−1,𝑡∈𝐽₀,𝑢(0)=1, (4) we obtain 𝑢(𝑡)1=1+Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑠^𝑝−1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1Γ𝑑𝑠=1+𝑝+1Γ𝑡𝛼+𝑝+1^𝛼+𝑝−1Γ()𝑡𝛼+1^𝛼, (5) from which we get 𝑢(1)=1+(Γ(𝑝+1)/Γ(𝛼+𝑝+1))−(1/Γ(𝛼+1)).

On the other hand, the solution of the subproblem ^𝐶𝐷^𝛽₁⁺𝑢(𝑡)=(𝑡−1)^𝑞,𝑡∈𝐽₁,𝑢1⁺(1)Γ=𝑢+2=3+𝑝+1Γ−1𝛼+𝑝+1Γ(𝛼+1) (6) is given by 𝑢(𝑡)=𝑢1⁺+1Γ𝛽^𝑡₁(𝑡−𝑠)^𝛽−1(𝑠−1)^𝑞Γ𝑑𝑠=3+𝑝+1Γ−1𝛼+𝑝+1Γ()+Γ𝛼+1𝑞+1Γ()𝛽+𝑞+1𝑡−1^𝛽+𝑞. (7)

Hence, the piecewise continuous function 𝑢(𝑡)=Γ1+𝑝+1Γ𝑡𝛼+𝑝+1^𝛼+𝑝−1Γ()𝑡𝛼+1^𝛼,𝑡∈𝐽₀,Γ3+𝑝+1Γ−1𝛼+𝑝+1Γ()+Γ𝛼+1𝑞+1Γ()𝛽+𝑞+1𝑡−1^𝛽+𝑞,𝑡∈𝐽₁, (8) is a solution to the impulsive fractional problem (3).

A particular problem of (3) is as follows: ^𝐶𝐷^𝛾₀⁺𝑢(𝑡)=𝑡^𝑝−1,𝑡∈𝐽₀,_𝐶𝐷^𝛾₁⁺𝑢(𝑡)=(𝑡−1)^𝑞,𝑡∈𝐽₁,𝑢(0)𝑢1=1,₊=𝑢(1⁻)+2 (9) corresponding to the case 𝛾=𝛼=𝛽 whose solution is 𝑢(𝑡)=Γ1+𝑝+1Γ𝑡𝛾+𝑝+1^𝛾+𝑝−1Γ𝑡𝛾+1^𝛾,𝑡∈𝐽₀,Γ3+𝑝+1Γ−1𝛾+𝑝+1Γ+Γ𝛾+1𝑞+1Γ()𝛾+𝑞+1𝑡−1^𝛾+𝑞,𝑡∈𝐽₁. (10) 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.

2. 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 (𝐸,‖⋅‖)^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+𝐹𝑡,𝑢_𝑡𝑎(𝑡),ℎ(𝑡,𝑠,𝑢(𝜎(𝑠)))𝑑𝑠,_{𝑡𝑘+1𝑎}𝑘(𝑡,𝑠,𝑢(𝜏(𝑠)))𝑑𝑠,𝑡∈𝐽_𝑘𝑢(𝑎),𝑘=0,…,𝑚,=𝑢₀𝑢𝑡∈𝐸,_+𝑘=𝑢𝑡⁻_𝑘+𝐼_𝑘𝑢𝑡⁻_𝑘,𝑘=1,…,𝑚, (11) where(i)𝐽=[𝑎,𝑇] with 0≤𝑎<𝑇<∞ and 𝐽₀=[𝑎,𝑡₁], 𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1]; 𝑘=1,…,𝑚,(ii)^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘 is Caputo’s fractional derivative of order 𝛼_𝑘∈(0,1), 𝑘=0,…,𝑚,(iii)𝐴∶𝐽×𝐸→ℬ(𝐸) is a continuous operator, where ℬ(𝐸) is the Banach space of bounded linear operators on 𝐸 in itself,(iv)𝐼_𝑘∶𝐸→𝐸, 𝑡₀=𝑎<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇; 𝑢(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑢(𝑡_𝑘+𝜖) and 𝑢(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑢(𝑡_𝑘+𝜖)=𝑢(𝑡_𝑘) are, respectively, the right and left limits of 𝑢(𝑡) at the discontinuity point 𝑡=𝑡_𝑘. We set the following hypotheses:(j) the functions 𝜎,𝜏∶𝐽→𝐽 are continuous with 𝑎≤𝜎(𝑡)≤𝑡 and 𝑎≤𝜏(𝑡)≤𝑡, for every 𝑡∈𝐽, (jj) the nonlinear function 𝐹∶𝐽×𝐸×𝐸×𝐸→𝐸 is continuous, and ℎ∶𝐷×𝐸→𝐸,𝐷=(𝑡,𝑠)∈ℝ²∶𝑎≤𝑠≤𝑡≤𝑇,𝑘∶𝐷₀×𝐸→𝐸,where𝐷₀=(𝑡,𝑠)∈ℝ²∶𝑡∈𝐽,𝑎≤𝑠≤𝑇, (12)are two continuous functions over 𝐷×𝐸 and 𝐷₀×𝐸, respectively.

We will use in the sequel the following notation: 𝐻𝑢(𝑡)=^𝑡_𝑎ℎ((𝜎(𝑠𝐾𝑡,𝑠,𝑢)))𝑑𝑠,_𝑡𝑘+1𝑢(𝑡)=^𝑡𝑘+1_𝑎𝑘((𝜏(𝑠Φ𝑡,𝑠,𝑢)))𝑑𝑠,𝑘=0,1,…,𝑚,_𝑡𝑘+1(𝑡,𝑢(𝑡))=𝐹𝑡,𝑢(𝑡),𝐻𝑢(𝑡),𝐾_𝑡𝑘+1𝑢(𝑡),𝑘=0,1,…,𝑚. (13) We recall that 𝒞=𝒞(𝐽;𝐸) is the Banach space of continuous functions 𝑢∶𝐽→𝐸 endowed with the norm ‖𝑢‖_𝒞=sup_𝑡∈𝐽‖𝑢(𝑡)‖. (14) 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 𝛼∈(0,1) of a function 𝑓∶[𝑐,𝑑]→𝐸 as follows: 𝐽^𝛼_𝑐+𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑐(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝑡>𝑐. (15)
We define the left-sided fractional derivative of order 𝛼∈(0,1) of a function 𝑓∶[𝑐,𝑑]→𝐸 in the sense of Caputo by ^𝐶𝐷^𝛼_𝑐+𝑓(𝑡)=1Γ(1−𝛼)^𝑡_𝑐(𝑡−𝑠)^−𝛼𝑓^′(𝑠)𝑑𝑠,𝑡>𝑐. (16)

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 𝑓 satisfies the necessary conditions for which those integrals are well defined.

Next, we consider the linear functional space ()=𝑡𝒫𝒞𝐽;𝐸𝑢∶𝐽→𝐸,𝑢∈𝒞_𝑘,𝑡_𝑘+1,𝑡;𝐸𝑘=0,…,𝑚s.t.𝑢⁻_𝑘and𝑢𝑡⁺_𝑘existwith𝑢𝑡⁻_𝑘=𝑢𝑡_𝑘,𝑘=1,…,𝑚 (17) equipped with the norm ‖𝑢‖_𝒫𝒞=sup_𝑡∈𝐽‖𝑢(𝑡)‖. (18) We obtain a Banach space (𝒫𝒞(𝐽;𝐸),‖⋅‖_𝒫𝒞).

Now, we recall the definition of the solution of the problem (11).

Definition 3. A function 𝑢∈𝒫𝒞(𝐽;𝐸) is said to be a solution of the problem (11) if ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡) exists in 𝐽_𝑘, for 𝑘=0,…,𝑚, and satisfies(i)the equation ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+Φ_𝑡𝑘+1(𝑡,𝑢(𝑡)) in 𝐽_𝑘, 𝑘=0,…,𝑚,(ii)the initial condition 𝑢(𝑎)=𝑢₀,(iii)the impulsive conditions 𝑢(𝑡⁺_𝑘)=𝑢(𝑡⁻_𝑘)+𝐼_𝑘(𝑢(𝑡⁻_𝑘)), 𝑘=1,…,𝑚.

Lemma 4. A function 𝑢∈𝒫𝒞(𝐽;𝐸) satisfies the following nonlinear integral equation 𝑢(𝑡)=𝑢₀+1Γ𝛼₀∫^𝑡_𝑎(𝑡−𝑠)^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀∫^𝑡_𝑎(𝑡−𝑠)^𝛼0−1Φ_𝑡1(𝑠,𝑢)𝑑𝑠,𝑡∈𝑎,𝑡₁,𝑢₀+∑^𝑘_𝑖=1𝐼_𝑖𝑢𝑡⁻_𝑖+1Γ𝛼_𝑘∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴()𝑢(𝑠)+𝑠,𝑢𝑑𝑠_𝑘𝑖∑₌₁1Γ𝛼_𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼_𝑘∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1()+𝑠,𝑢𝑑𝑠_𝑘𝑖∑₌₁1Γ𝛼_𝑖−1∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑢)𝑑𝑠,𝑡∈𝐽_𝑘,𝑘=1,…,𝑚, (19) if and only if it is a solution to problem (11).

Proof. Since we have 𝑢(𝑡⁻_𝑘)=𝑢(𝑡_𝑘), then 𝑢(𝑡⁺_𝑘)=𝑢(𝑡_𝑘)+𝐼_𝑘(𝑢(𝑡⁻_𝑘)).
Now, for 𝑡∈𝐽₀=[𝑎,𝑡₁], the solution of the problem ^𝑐𝐷^𝛼0_𝑎+𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+Φ_𝑡1(𝑡,𝑢),𝑡∈𝐽₀,𝑢(𝑎)=𝑢₀∈𝐸 (20) is given by 𝑢(𝑡)=𝑢₀+1Γ𝛼₀^𝑡_𝑎(𝑡−𝑠)^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀^𝑡_𝑎(𝑡−𝑠)^𝛼0−1Φ_𝑡1(𝑠,𝑢)𝑑𝑠,𝑡∈𝐽₀. (21) We have for 𝑡=𝑡₁ the following relation𝑢(𝑡⁺₁)=𝑢(𝑡₁)+𝐼₁(𝑢(𝑡⁻₁)), and so 𝑢𝑡⁺₁=1Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1Φ_𝑡1(𝑠,𝑢)𝑑𝑠+𝑢₀+𝐼₁𝑢𝑡⁻₁. (22)
Next, for 𝑡∈𝐽₁=(𝑡₁,𝑡₂], we have 𝑢(𝑡)=𝑢𝑡⁺₁+1Γ𝛼₁^𝑡_𝑡1(𝑡−𝑠)^𝛼1−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡_𝑡1(𝑡−𝑠)^𝛼1−1Φ_𝑡2(𝑠,𝑢)𝑑𝑠=𝑢₀+1Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1Φ_𝑡1()+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡_𝑡1(𝑡−𝑠)^𝛼1−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡_𝑡1(𝑡−𝑠)^𝛼1−1Φ_𝑡2(𝑠,𝑢)𝑑𝑠+𝐼₁𝑢𝑡⁻₁ (23) from which we infer that 𝑢𝑡⁺₂=𝑢₀+1Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1Φ_𝑡1()+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡2_𝑡1𝑡₂−𝑠^𝛼1−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡2_𝑡1𝑡₂−𝑠^𝛼1−1Φ_𝑡2(𝑠,𝑢)𝑑𝑠+𝐼₁𝑢𝑡⁻₁+𝐼₂𝑢𝑡⁻₂. (24)
Arguing as before we obtain for 𝑡∈𝐽₂𝑢(𝑡)=𝑢𝑡⁺₂+1Γ𝛼₂^𝑡_𝑡2(𝑡−𝑠)^𝛼2−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₂^𝑡_𝑡2(𝑡−𝑠)^𝛼2−1Φ_𝑡3(𝑠,𝑢)𝑑𝑠=𝑢₀+1Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₀^𝑡1_𝑎𝑡₁−𝑠^𝛼0−1Φ_𝑡1()+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡2_𝑡1𝑡₂−𝑠^𝛼1−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₁^𝑡2_𝑡1𝑡₂−𝑠^𝛼1−1Φ_𝑡2()+1𝑠,𝑢𝑑𝑠Γ𝛼₂^𝑡_𝑡2(𝑡−𝑠)^𝛼2−1𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼₂^𝑡_𝑡2(𝑡−𝑠)^𝛼2−1Φ_𝑡3(𝑠,𝑢)𝑑𝑠+𝐼₁𝑢𝑡⁻₁+𝐼₂𝑢𝑡⁻₂. (25) Reasoning by induction we get, for any 𝑡∈𝐽_𝑘, 𝑘=1,…,𝑚, the general expression 𝑢(𝑡)=𝑢₀+1Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴()𝑢(𝑠)+𝑠,𝑢𝑑𝑠_𝑘𝑖=11Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1()+𝑠,𝑢𝑑𝑠_𝑘𝑖=11Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖()+𝑠,𝑢𝑑𝑠_𝑘𝑖=1𝐼_𝑖𝑢𝑡⁻_𝑖. (26) Conversely, we assume that 𝑢 satisfies (19). If 𝑡=𝑎, then 𝑢(𝑎)=𝑢₀.
Now, using the fact that Caputo’s derivative of a constant is zero, then, for every 𝑡∈𝐽_𝑘, 𝑘=0,…,𝑚, we get ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘1Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴()𝑢(𝑠)+𝑠,𝑢𝑑𝑠_𝐶𝐷^𝛼𝑘_𝑡𝑘1Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1()=𝑠,𝑢𝑑𝑠_𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝐽^𝛼𝑘_𝑡+𝑘𝐴(𝑡,𝑢)𝑢(𝑡)+^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝐽^𝛼𝑘_𝑡+𝑘Φ_𝑡𝑘+1(𝑡,𝑢). (27)
So ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+Φ_𝑡𝑘+1(𝑡,𝑢), (28) for every 𝑡∈𝐽_𝑘, 𝑘=0,…,𝑚.
Also we can easily show that 𝑢𝑡⁺_𝑘=𝑢𝑡_𝑘+𝐼_𝑘𝑢𝑡⁻_𝑘,𝑘=1,…,𝑚. (29)

We conclude this section by introducing some useful theorems which will be used in the sequel.

Theorem 5 (𝒫𝒞-type Ascoli-Arzela theorem [21]). Let 𝐸 be a Banach space and 𝒲⊂𝒫𝒞(𝐽,𝐸). If the following conditions are satisfied(i)𝒲 is a uniformly bounded subset of 𝒫𝒞(𝐽,𝐸);(ii)𝒲 is equicontinuous in (𝑡_𝑘,𝑡_𝑘+1), 𝑘=0,1,2,…,𝑚;(iii)𝒲(𝑡)={𝑢(𝑡)∶𝑢∈𝒲,𝑡∈𝐽⧵{𝑡_𝑘}}, 𝒲(𝑡⁺_𝑘)={𝑢(𝑡⁺_𝑘)∶𝑢∈𝒲}, and 𝒲(𝑡⁻_𝑘)≡{𝑢(𝑡⁻_𝑘)∶𝑢∈𝒲} are relatively compact subsets of 𝐸,  then 𝒲 is a relatively compact subset of 𝒫𝒞(𝐽,𝐸).

Theorem 6 (Schaefer’s fixed point theorem). Let 𝐸 be a Banach space and let 𝒯∶𝐸→𝐸 be a completely continuous operator. If the set 𝑋={𝑢∈𝐸∶𝑢=𝜆𝒯𝑢,𝜆∈(0,1)} (30) is bounded, then 𝒯 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: ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=𝐴(𝑡,𝑢)𝑢(𝑡)+Φ_𝑡𝑘+1(𝑡,𝑢(𝑡)),𝑡∈𝐽_𝑘𝑢(𝑎),𝑘=0,…,𝑚,=𝑢₀𝑢𝑡∈𝐸,_+𝑘=𝑢𝑡_𝑘+𝐼_𝑘𝑢𝑡⁻_𝑘,𝑘=1,…,𝑚. (31) We assume that 𝐴∶𝐽×𝐸→ℬ(𝐸) is continuous and there exists a constant 𝑀>0 such that ‖𝐴(𝑡,𝑢)−𝐴(𝑡,𝑣)‖≤𝑀‖𝑢−𝑣‖,∀𝑡∈𝐽,∀𝑢,𝑣∈𝐸. (32) We set 𝑀^′=max_𝑡∈𝐽‖𝐴(𝑡,0)‖.

It is not hard to establish the following estimates.

Lemma 7. Let the functions ℎ(𝑡,𝑠,𝑢) and 𝑘(𝑡,𝑠,𝑢) be continuous with respect to the variables 𝑠 and 𝑡, and there are two positive constants 𝐶₁ and 𝐶₂ such that ‖ℎ(𝑡,𝑠,𝑢)−ℎ(𝑡,𝑠,𝑣)‖≤𝐶₁‖‖,‖𝑘()(𝑢−𝑣∀𝑡,𝑠∈𝐽,∀𝑢,𝑣∈𝐸,𝑡,𝑠,𝑢−𝑘𝑡,𝑠,𝑣)‖≤𝐶₂‖𝑢−𝑣‖,∀𝑡,𝑠∈𝐽,∀𝑢,𝑣∈𝐸. (33) Then, there exist two positive constants 𝐶^′₁ and 𝐶^′₂ so that ‖𝐻𝑢(𝑡)‖≤(𝑇−𝑎)𝐶₁‖𝑢‖_𝒫𝒞+𝐶^′₁,‖𝐻𝑢(𝑡)−𝐻𝑣(𝑡)‖≤𝐶₁(𝑇−𝑎)‖𝑢−𝑣‖_𝒫𝒞, (34) and, for 𝑘=0,…,𝑚, one has 𝐾_𝑡𝑘+1𝑢(𝑡)≤(𝑇−𝑎)𝐶₂‖𝑢‖_𝒫𝒞+𝐶^′₂,𝐾_𝑡𝑘+1𝑢(𝑡)−𝐾_𝑡𝑘+1𝑣(𝑡)≤𝐶₂(𝑇−𝑎)‖𝑢−𝑣‖_𝒫𝒞, (35) for every 𝑢,𝑣∈𝒫𝒞(𝐽,𝐸) and 𝑡∈𝐽.

We assume the following hypotheses:(H1)𝛼₀,…,𝛼_𝑚∈(0,1). We set 𝑇^′=max_{0≤𝑖≤𝑚}{(𝑇−𝑎)^𝛼𝑖} and Γ^′=min_{0≤𝑖≤𝑚}{Γ(𝛼_𝑖+1)}.(H2) There is a positive constant 𝐿₁ such that Φ_𝑡𝑘+1(𝑡,𝑢)−Φ_𝑡𝑘+1(≤𝐿𝑡,𝑣)₁+𝐶₁+𝐶₂(𝑇−𝑎)‖𝑢−𝑣‖_𝒫𝒞,∀𝑡∈𝐽,∀𝑢,𝑣∈𝒫𝒞,𝑘=0,…,𝑚. (36) We set 𝐿=𝐿₁+(𝐶₁+𝐶₂)(𝑇−𝑎) and 𝐿₂=sup_𝑡∈𝐽‖𝐹(𝑡,0,0,0)‖.(H3) There is a positive constant 𝜇>0 such that 𝐼_𝑘(𝑢)−𝐼_𝑘(𝑣)≤𝜇‖𝑢−𝑣‖,∀𝑢,𝑣∈𝐸,𝑘=1,…,𝑚. (37)(H4) The positive real number 𝛾=𝑚𝜇+(𝑚+1)𝐿+2𝑟𝑀+𝑀^′𝑇^′Γ^′ (38) satisfies 0<𝛾<1.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 𝑢∈𝒫𝒞(𝐽,𝐸).

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_𝑟={𝑢∈𝒫𝒞(𝐽,𝐸)∶‖𝑢‖_𝒫𝒞≤𝑟} be the closed ball of 𝒫𝒞(𝐽,𝐸) centered at 0 with radius 𝑟 satisfying the following inequality: 𝜑(𝑟)∶=𝑢₀+(𝑚+1)𝑇^′Γ^′𝐿^′+(𝑚+1)𝑇^′Γ^′𝑟𝑀+𝑀^′+𝐿+𝑚𝜇𝑟≤𝑟, (39) where 𝐿^′=𝐶^′₁+𝐶^′₂(𝑇−𝑎)+𝐿₂. (40) Endowing ℬ_𝑟 with the metric 𝑑(𝑢,𝑣)=‖𝑢−𝑣‖_𝒫𝒞, for every 𝑢,𝑣∈ℬ_𝑟, we obtain a complete metric space (ℬ_𝑟,𝑑). Next, we define the operator Ψ∶ℬ_𝑟→ℬ_𝑟 by Ψ𝑢(𝑡)=𝑢₀+1Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴(𝑠,𝑢)𝑢(𝑠)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴()𝑢(𝑠)+1𝑠,𝑢𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1()+𝑠,𝑢𝑑𝑠_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (41) It is understood that the sum ∑_{𝑎<𝑡𝑖<𝑡} is zero if 𝑡∈𝐽₀.
First, we prove that if 𝑢∈𝒫𝒞(𝐽;𝐸), then Ψ𝑢∈𝒫𝒞(𝐽;𝐸).
Indeed, for each 𝑡∈(𝑡_𝑘,𝑡_𝑘+1), 𝑢∈𝒞((𝑡_𝑘,𝑡_𝑘+1),𝐸), and any sufficiently small 𝛿>0, we have ‖()(𝑡≤𝑀‖𝑢‖Ψ𝑢𝑡+𝛿−Ψ𝑢)‖_𝒫𝒞+𝑀^′+𝐿‖𝑢‖_𝒫𝒞+𝐿^′Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1−(𝑡+𝛿−𝑠)^𝛼𝑘−1+𝑀‖𝑢‖𝑑𝑠_𝒫𝒞+𝑀^′+𝐿‖𝑢‖_𝒫𝒞+𝐿^′Γ𝛼_𝑘×_𝑡^𝑡+𝛿(𝑡+𝛿−𝑠)^𝛼𝑘−1𝑑𝑠. (42) Calculating the integrals we find that ‖Ψ𝑢(𝑡+𝛿)−Ψ𝑢(𝑡)‖≤3𝑀𝑟+𝑀^′+𝐿𝑟+𝐿^′Γ^′𝛿^𝛼𝑘. (43) Thus, the right-hand side tends to zero as 𝛿→0. Likewise one gets lim_𝛿→0‖Ψ𝑢(𝑡)−Ψ𝑢(𝑡−𝛿)‖=0; this shows that Ψ𝑢 is continuous at 𝑡. Hence Ψ𝑢∈𝒞((𝑡_𝑘,𝑡_𝑘+1),𝐸).
Next, for the right endpoint 𝑡=𝑡_𝑘+1 we get for any sufficiently small 𝛿>0Ψ𝑢𝑡_𝑘+1−Ψ𝑢𝑡_𝑘+1−𝛿≤3𝑀𝑟+𝑀^′+𝐿𝑟+𝐿^′Γ^′𝛿^𝛼𝑘, (44) which shows that the right-hand side tends to zero as 𝛿→0, and accordingly, Ψ𝑢 is continuous at 𝑡_𝑘+1. Therefore Ψ𝑢∈𝒫𝒞(𝐽,𝐸).
To prove that Ψℬ_𝑟⊂ℬ_𝑟 we see that, for any 𝑢∈ℬ_𝑟 and 𝑡∈𝐽_𝑘, 𝑘=0,…,𝑚, we have ‖Ψ𝑢(𝑡)‖≤𝑢₀+1Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1‖𝐴(𝑠,𝑢)‖⋅‖𝑢(𝑠)‖𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}‖𝐴(⋅‖𝑢(𝑠+1𝑠,𝑢)‖)‖𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖. (45)
Estimating the right-hand side we find ‖(𝑡≤𝑢Ψ𝑢)‖₀+(𝑚+1)𝑇^′Γ^′×𝑀‖𝑢‖_𝒫𝒞+𝑀^′⋅‖𝑢‖_𝒫𝒞+1Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐿‖𝑢‖_𝒫𝒞+(𝑇−𝑎)𝐶^′₁+𝐶^′₂+𝐿₂+𝑑𝑠_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐿‖𝑢‖_𝒫𝒞+(𝑇−𝑎)𝐶^′₁+𝐶^′₂+𝐿₂‖𝑢‖𝑑𝑠+𝑚𝜇_𝒫𝒞. (46) So ‖(𝑡≤𝑢Ψ𝑢)‖₀+(𝑚+1)𝑇^′Γ^′𝐿^′+(𝑚+1)𝑇^′Γ^′𝑟𝑀+𝑀^′𝑟(𝑟)+𝐿+𝑚𝜇≤𝜑≤𝑟. (47) Therefore, ‖Ψ𝑢‖_𝒫𝒞≤𝑟, and consequently Ψℬ_𝑟⊂ℬ_𝑟.
Next, we prove that Ψ is a contraction mapping; indeed, for any 𝑢,𝑣∈ℬ_𝑟 and 𝑡∈𝐽_𝑘, 𝑘=0,…,𝑚, we have ‖(𝑡)(𝑡≤1Ψ𝑢−Ψ𝑣)‖Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1‖𝐴(𝑠,𝑢)𝑢(𝑠)−𝐴(𝑠,𝑣)𝑣(𝑠)‖𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}‖𝐴()𝑢(𝑠)()𝑣(𝑠+1𝑠,𝑢−𝐴𝑠,𝑣)‖𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑢)−Φ_𝑡𝑘+1(𝑠,𝑣)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑡,𝑢)−Φ_𝑡𝑖(𝑠,𝑣)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖−𝐼_𝑖𝑣𝑡⁻_𝑖. (48) Taking into account the previous assumptions we get the following estimate: ‖(𝑡)(𝑡‖‖Ψ𝑢−Ψ𝑣)‖≤𝑚𝜇𝑢−𝑣_𝒫𝒞+‖𝑢−𝑣‖_𝒫𝒞Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝑀𝑟+𝑀^′+𝑀𝑟𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}‖𝑢−𝑣‖_𝒫𝒞Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝑀𝑟+𝑀^′+𝐿‖‖+𝑀𝑟𝑑𝑠𝑢−𝑣_𝒫𝒞Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐿‖𝑢−𝑣‖_𝒫𝒞Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝑑𝑠. (49)
Thus, ‖(𝑡)(𝑡≤()Ψ𝑢−Ψ𝑣)‖𝑚𝜇+𝑚+1𝐿+2𝑟𝑀+𝑀^′𝑇^′Γ^′‖𝑢−𝑣‖_𝒫𝒞≤𝛾‖𝑢−𝑣‖_𝒫𝒞. (50) Accordingly, the mapping Ψ has a unique fixed point 𝑢=Ψ𝑢∈ℬ_𝑟, 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 (𝐸,‖⋅‖). Actually, the finite dimension requirement is due to some technical difficulties in order to prove some compactness properties. The problem is as follows: ^𝐶𝐷^𝛼𝑘_𝑡⁺_𝑘𝑢(𝑡)=𝐴(𝑡)𝑢(𝑡)+Φ_𝑡𝑘+1(𝑡,𝑢(𝑡)),𝑡∈𝐽_𝑘𝑢(𝑎)(𝑢),𝑘=0,…,𝑚,+𝑔=𝑢₀𝑢𝑡∈𝐸,_+𝑘=𝑢𝑡_𝑘+𝐼_𝑘𝑢𝑡⁻_𝑘,𝑘=1,…,𝑚. (51)

We assume that the mapping 𝐴∶𝐽→ℬ(𝐸) is continuous and we put 𝑀^′′=max_𝑡∈𝐽‖𝐴(𝑡)‖. (52) We need the following hypothesis:(H5) there exists a constant 𝐺>0 such that the mapping 𝑔∶𝒫𝒞(𝐽,𝐸)→𝐸 satisfies 𝑔(𝑢)−𝑔(𝑣)≤𝐺‖𝑢−𝑣‖_𝒫𝒞,∀𝑢,𝑣∈𝒫𝒞(𝐽,𝐸). (53)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 𝑢∈𝒫𝒞(𝐽,𝐸).

Proof. Let us define the operator 𝑄∶𝒫𝒞(𝐽,𝐸)→𝒫𝒞(𝐽,𝐸) by 𝑄𝑢(𝑡)=𝑢₀(𝑢)+1−𝑔Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴(𝑠)𝑢(𝑠)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴(𝑠)𝑢(𝑠)+1𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1()+𝑠,𝑢𝑑𝑠_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (54) First, we notice that by using the same technique as that in the proof of the Theorem 8 we can establish that if 𝑢∈𝒫𝒞(𝐽,𝐸), then 𝑄𝑢∈𝒫𝒞(𝐽,𝐸); that is, the operator 𝑄 maps the space 𝒫𝒞(𝐽,𝐸) into itself.
To prove that 𝑄 has a fixed point we use Schaefer’s fixed point theorem. We proceed in four steps.
Step  1 (𝑄 is continuous). Let {𝑢_𝑛}_𝑛≥1⊂𝒫𝒞(𝐽,𝐸) such that 𝑢_𝑛→𝑢 in 𝒫𝒞(𝐽,𝐸); then 𝑄𝑢_𝑛(𝑡)(𝑡≤𝑔𝑢−𝑄𝑢)_𝑛(𝑢+1−𝑔)Γ𝛼_𝑘×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴(𝑠)𝑢_𝑛(𝑠)(𝑠)+−𝑢𝑑𝑠_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴(𝑠)𝑢_𝑛(𝑠)(𝑠)+1−𝑢𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1𝑠,𝑢_𝑛−Φ_𝑡𝑘+1(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖𝑠,𝑢_𝑛−Φ_𝑡𝑖(+𝑠,𝑢)𝑑𝑠_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢_𝑛𝑡⁻_𝑖−𝐼_𝑖𝑢𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (55)
Taking into account the assumptions (H2)-(H3) and (H5) and using Lemma 7 we get 𝑄𝑢_𝑛(𝑡)(𝑡𝑢−𝑄𝑢)≤𝐺_𝑛−𝑢_𝒫𝒞+𝑀^′′Γ𝛼_𝑘𝑢_𝑛−𝑢_𝒫𝒞×^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝑑𝑠+𝑀^′′𝑢_𝑛−𝑢_𝒫𝒞×_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}+𝐿𝑑𝑠Γ𝛼_𝑘𝑢_𝑛−𝑢_𝒫𝒞^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝑢𝑑𝑠+𝐿_𝑛−𝑢_{𝒫𝒞𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝑢𝑑𝑠+𝑚𝜇_𝑛−𝑢_𝒫𝒞,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (56) Calculating the integrals in the right-hand side we obtain 𝑄𝑢_𝑛−𝑄𝑢_𝒫𝒞≤()𝑇𝐺+𝑚𝜇+𝑚+1^′Γ^′𝑀^′′+𝐿𝑢_𝑛−𝑢_𝒫𝒞. (57) So 𝑄𝑢_𝑛−𝑄𝑢_𝒫𝒞→0,as𝑛→∞, (58) and accordingly, 𝑄 is continuous.
Step  2. Let 𝜀>0 and 𝐵_𝜀={𝑢∈𝒫𝒞(𝐽,𝐸)∶‖𝑢‖_𝒫𝒞≤𝜀}. Define 𝒲={𝑄𝑢∶𝑢∈𝐵_𝜀}; then for any 𝑢∈𝐵_𝜀 we have ‖𝑄𝑢(𝑡)‖≤𝑢₀(𝑢+1−𝑔)Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1‖𝐴(𝑠𝑢(𝑠+)‖‖)‖𝑑𝑠_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}‖𝐴(𝑠𝑢(𝑠+1)‖‖)‖𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑢)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(+𝑠,𝑢)𝑑𝑠_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (59)
Estimating the right-hand side we obtain ‖𝑄𝑢(𝑡)‖≤𝑢₀+𝐺‖𝑢‖_𝒫𝒞+𝑔(0+()𝑇)𝑚+1^′Γ^′𝑀^′′⋅‖𝑢‖_𝒫𝒞+1Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐿‖𝑢‖_𝒫𝒞+𝐿^′𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼𝑘−1𝐿‖𝑢‖_𝒫𝒞+𝐿^′‖𝑢‖𝑑𝑠+𝑚𝜇_𝒫𝒞, (60) implying that ‖𝑄𝑢‖_𝒫𝒞≤𝑢₀+𝑔(0+()𝑇)𝑚+1^′Γ^′𝑀^′′𝜀+𝐿𝜀+𝐿^′+𝐺𝜀+𝑚𝜇𝜀∶=𝜌. (61) Hence 𝒲 is uniformly bounded.
Step  3 (we prove that 𝒲 is equicontinuous). Let 𝑢∈𝐵_𝜀; then, for any 𝑡_𝑘<𝜏₁<𝜏₂≤𝑡_𝑘+1, we have 𝑄𝑢𝜏₂−𝑄𝑢𝜏₁≤1Γ𝛼_𝑘×^𝜏1_𝑡𝑘𝜏₁−𝑠^𝛼𝑘−1−𝜏₂−𝑠^𝛼𝑘−1𝐴(𝑠𝑢(𝑠+1‖)‖‖)‖𝑑𝑠Γ𝛼_𝑘^𝜏2_𝜏1𝜏₂−𝑠^𝛼𝑘−1‖𝐴(𝑠𝑢(𝑠+1)‖‖)‖𝑑𝑠Γ𝛼_𝑘^𝜏1_𝑡𝑘𝜏₁−𝑠^𝛼𝑘−1−𝜏₂−𝑠^𝛼𝑘−1Φ_𝑡𝑘+1(+1𝑠,𝑢)𝑑𝑠Γ𝛼_𝑘^𝜏2_𝜏1𝜏₂−𝑠^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑢)𝑑𝑠. (62) So 𝑄𝑢𝜏₂−𝑄𝑢𝜏₁≤𝑀^′′+𝐿𝜀+𝐿^′Γ𝛼_𝑘^𝜏1_𝑡𝑘𝜏₁−𝑠^𝛼𝑘−1−𝜏₂−𝑠^𝛼𝑘−1+𝑀𝑑𝑠_′′+𝐿𝜀+𝐿^′Γ𝛼_𝑘^𝜏2_𝜏1𝜏₂−𝑠^𝛼𝑘−1𝑀𝑑𝑠≤3^′′+𝐿𝜀+𝐿^′Γ^′𝜏₂−𝜏₁^𝛼𝑘. (63) As 𝜏₁→𝜏₂, the right-hand side of the previous inequality tends to zero, which means that 𝒲 is equicontinuous.
We point out that the closures of the subsets 𝒲(𝑡)∶={𝑄𝑢(𝑡)∶𝑢∈𝐵_𝜀,𝑡∈𝐽⧵{𝑡_𝑘},𝑘=1,…,𝑚}, 𝒲(𝑡⁻_𝑘)∶={𝑄𝑢(𝑡⁻_𝑘)∶𝑢∈𝐵_𝜀}, and 𝒲(𝑡⁺_𝑘)∶={𝑄𝑢(𝑡⁺_𝑘)∶𝑢∈𝐵_𝜀}, 𝑘=1,…,𝑚, are bounded in 𝐸 (dim𝐸<∞); hence they are compact.
As a consequence of the previous steps and the 𝒫𝒞-type Arzela-Ascoli theorem we conclude that 𝑄 is completely continuous.
Step  4. Now, we show that the set 𝑋={𝑢∈𝒫𝒞(𝐽,𝐸)∶𝑢=𝜆𝑄𝑢,𝜆∈(0,1)} (64) is bounded.
Let 𝑢∈𝑋; then 𝑢=𝜆𝑄𝑢, for some 𝜆∈(0,1). Thus, for each 𝑡∈𝐽, ‖𝑢(𝑡)‖≤𝜆𝑢₀𝑔(𝑢+𝜆+𝜆)Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1‖𝐴(𝑠)‖⋅‖𝑢(𝑠)‖𝑑𝑠+𝜆_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1×^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}‖𝐴(𝑠⋅‖𝑢(𝑠+𝜆)‖)‖𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑢)𝑑𝑠+𝜆_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑢)𝑑𝑠+𝜆_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑢𝑡⁻_𝑖≤𝜆𝑢₀+𝑔(0+()𝑇)𝑚+1^′Γ^′×𝑀^′′𝜀+𝐿𝜀+𝐿^′+𝐺𝜀+𝑚𝜇𝜀<∞. (65) This shows that the set 𝑋 is bounded.
We conclude by Schaefer’s fixed point theorem that the operator 𝑄 has a fixed point 𝑢∈𝒫𝒞(𝐽,𝐸) such that 𝑄𝑢=𝑢, which means that 𝑢 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 𝐺+𝑚𝜇+(𝑚+1)𝐿+𝑀^′′𝑇^′Γ^′<1. (66)

Proof. Since 𝑢 is a solution to (51), then it satisfies the integral equation (19). Let 𝑣 be a solution to problem (51) with initial value 𝑣(𝑎)=𝑣₀−𝑔(𝑣). Then 𝑣(𝑡) satisfies the integral equation 𝑣(𝑡)=𝑣₀(𝑣)+1−𝑔Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1𝐴(𝑠)𝑣(𝑠)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}𝐴(𝑠)𝑣(𝑠)+1𝑑𝑠Γ𝛼_𝑘^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼𝑘−1Φ_𝑡𝑘+1(𝑠,𝑣)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}1Γ𝛼_𝑖−1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^{𝛼𝑖−1−1}Φ_𝑡𝑖(𝑠,𝑣)𝑑𝑠+_{𝑎<𝑡𝑖<𝑡}𝐼_𝑖𝑣𝑡⁻_𝑖,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (67) Estimating the difference between solutions 𝑢(𝑡) and 𝑣(𝑡) to (19) and (67), respectively, we get ‖𝑢(𝑡)−𝑣(𝑡)‖≤𝑢₀−𝑣₀+𝐺‖𝑢−𝑣‖_𝒫𝒞+𝑀^′′(𝑚+1)𝑇^′Γ^′‖𝑢−𝑣‖_𝒫𝒞+𝐿(𝑚+1)𝑇^′Γ^′‖𝑢−𝑣‖_𝒫𝒞+𝑚𝜇‖𝑢−𝑣‖_𝒫𝒞,𝑡∈𝐽_𝑘,𝑘=0,…,𝑚. (68) Taking the supremum over the interval 𝐽 we find that ‖𝑢−𝑣‖_𝒫𝒞≤1𝜌𝑢₀−𝑣₀, (69) where 𝜌=1−𝐺−𝑚𝜇−(𝑚+1)𝐿+𝑀^′′𝑇^′Γ^′, (70) which proves that the mapping 𝑢₀↦𝑢 is continuous from 𝐸→𝒫𝒞(𝐽,𝐸).

5. Example

Consider the following impulsive fractional integrodifferential problem 𝐷₀^1/2₊𝑢(𝑡)=𝑡𝑢(𝑡)(𝑡)+|𝑢(𝑡24cos𝑢)|(|𝑢(𝑡)|+2)𝑡²+12+^𝑡₀𝑒^{−|𝑢(𝑠)|/6}(𝑡+2)²𝑑𝑠+₀^1/2sin(𝑢(𝑠)/4)𝑒^𝑡+6𝑑𝑠,𝑡∈𝐽₀,𝐷₍₎^3/4_1/2+𝑢(𝑡)=𝑡𝑢(𝑡)(𝑡)+|𝑢(𝑡24cos𝑢)|(|𝑢(𝑡)|+2)𝑡²++12_𝑡0𝑒^{−|𝑢(𝑠)|/6}(𝑡+2)²𝑑𝑠+¹₀sin(𝑢(𝑠)/4)𝑒^𝑡+6𝑑𝑠,𝑡∈𝐽₁,𝑢(0)𝑢1=1,2⁺1=𝑢2⁻+|𝑢((1/2)⁻)||𝑢((1/2)⁻.)|+6 (71) We set 𝐽₀=[0,1/2], 𝐽₁=(1/2,1], and 𝐽=[0,1]. We take 𝐸=ℝ, 𝑎=𝑡₀=0, 𝑡₁=1/2, 𝑇=1, 𝛼₀=1/2, 𝛼₁=3/4, and ℬ_𝑟={𝑢∈𝒫𝒞(𝐽,𝐸),‖𝑢‖_𝒫𝒞≤𝑟}. Define 𝐴()=𝑡𝑡,𝑢()(𝑡)=24cos𝑢𝐼,𝐻𝑢^𝑡₀𝑒^{−|𝑢(𝑠)|/6}(𝑡+2)²𝐾𝑑𝑠,_𝑡𝑘+1𝑢(𝑡)=^𝑡𝑘+1₀sin(𝑢(𝑠)/4)𝑒^𝑡Φ+6𝑑𝑠,𝑘=0,1,_𝑡𝑘+1((𝑡=|𝑢(𝑡𝑡,𝑢)))|(|𝑢(𝑡)|+2)𝑡²++12^𝑡₀𝑒^{−|𝑢(𝑠)|/6}(𝑡+2)²+𝑑𝑠_𝑡𝑘+10sin(𝑢(𝑠)/4)𝑒^𝑡𝐼+6𝑑𝑠,𝑘=0,1,₁𝑢12⁻=|𝑢((1/2)⁻)||𝑢((1/2)⁻.)|+6 (72) For any 𝑢,𝑣∈ℬ_𝑟 and 𝑡∈𝐽, we have |(𝑡)(𝑡≤1𝐻𝑢−𝐻𝑣)|‖‖24𝑢−𝑣_𝒫𝒞. (73) Hence 𝐶₁=1/24. Likewise, one has, for 𝑘=0,1, 𝐾_𝑡𝑘+1𝑢(𝑡)−𝐾_𝑡𝑘+1𝑣(𝑡≤1)‖‖24𝑢−𝑣_𝒫𝒞; (74) then 𝐶₂=1/24.

By (H1), we have 𝑇^′=max(𝑇−𝑎)^𝛼0,(𝑇−𝑎)^𝛼1Γ=1,_′=minΓ𝛼₀+1,Γ𝛼₁Γ3+1=min27,Γ43=Γ2=√𝜋2. (75) On the other hand, using (H2) we obtain Φ_𝑡𝑘+1(𝑡,𝑢)−Φ_𝑡𝑘+1(≤|𝑢(𝑡𝑡,𝑣))|(|𝑢(𝑡)|+2)𝑡²−|𝑣(𝑡+12)|(|𝑣(𝑡)|+2)𝑡²+|(𝑡)(𝑡+𝐾+12𝐻𝑢−𝐻𝑣)|_𝑡𝑘+1𝑢(𝑡)−𝐾_𝑡𝑘+1𝑣(𝑡≤3)‖‖24𝑢−𝑣_𝒫𝒞. (76)

Thus 𝐿₁=1324,𝐿=.24 (77) Assumption (H3) gives 𝐼₁(𝑢)−𝐼₁(𝑣≤1)6‖𝑢−𝑣‖_𝒫𝒞, (78) so 𝜇=1/6.

Due to the definition of 𝐴(𝑡,𝑢) we have 𝑀=𝑀^′=1/24. Let us now find a threshold for the value of 𝑟 for which condition (H4) is satisfied. We should have 10<𝛾=6+23√𝜋+𝑟3√𝜋<1, (79) so 𝑟 is any positive number such that 𝑟<(5√𝜋−4)/2=2.4311. We conclude by Theorem 8 that problem (71) has a unique solution 𝑢∈𝒫𝒞([0,1],ℝ) such that ‖𝑢‖_𝒫𝒞≤𝑟.

6. 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.