# Mathematical Analysis of Nonlocal Implicit Impulsive Problem under Caputo Fractional Boundary Conditions

Authors: Arshad Ali; Vidushi Gupta; Thabet Abdeljawad; Kamal Shah; Fahd Jarad
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper is related to frame a mathematical analysis of impulsive fractional order differential equations (IFODEs) under nonlocal Caputo fractional boundary conditions (NCFBCs). By using fixed point theorems of Schaefer and Banach, we analyze the existence and uniqueness results for the considered problem. Furthermore, we utilize the theory of stability for presenting Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam-Rassias stability results of the proposed scheme. Finally, some applications are offered to demonstrate the concept and results. The whole analysis is carried out by using Caputo fractional derivatives (CFDs).


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## Body

## 1. Introduction

It has been observed that the focus of investigation has shifted from classical integer-order models to fractional-order models. It is because of the fact that many practical systems are excellently described by using fractional-order differential equations (FODEs) instead of classical differential equations. For basic theory and some important applications of fractional-order derivatives, we refer the readers to see [1–4] and the references therein. Many researchers are devoted to work in this area and made significant contribution in this regard; we refer the readers to the recent work in [5–10].The study of implicit systems of FODEs with impulsive conditions is quite important as such systems appear in a variety of problems of applied nature, especially in biosciences, economics, engineering, etc. Such problems arise due to abrupt changes in the state of systems like earth quack, fluctuation of pendulum, etc. Here, we refer to some recent papers on impulsive problems [11–16]. The important class of FODEs known as IFODEs has been given much devotion by researchers. One of the most important aspects is investigation of problems under boundary conditions. Such problems mostly occur in engineering. Boundary and initial conditions may be local or nonlocal and both are important, and increasingly many problems have been investigated under these conditions. Replacing the local conditions by nonlocal ones produces a significant effect. This is due to the fact that the measurement computed from a nonlocal condition is usually more precise than the only one measurement given by a local condition. Therefore, the area of nonlocal boundary value problems has also attracted enough attention. In the last two decades, the area of IFODEs has been investigated from various directions including qualitative theory of existence of solution/solutions, stability, and numerical analysis. Therefore, IFODEs have also been investigated under nonlocal boundary conditions. For instance, Gupta and Dabas [17] studied the existence and uniqueness results for a class of IFODEs with nonlocal boundary conditions.(1)0cDtϱwt+ft,wt,w′t=0,ϱ∈1,2,t∈0,1,w0=0,0cDtϱw1=δ0cDtϱwρ,0<ξ<1,0<ρ≤1.By employing the fixed point technique, the authors obtained the existence and uniqueness results.This paper can be considered as generalization of the aforesaid work, in which we discuss existence, uniqueness, and various stability results for the following implicit IFODEs with three point NCFBCs of orderϱ∈1,2:(2)0cDtϱwt−ft,wt,0cDtϱwt=0,t∈0,T,Δwtq=ℐqwtq−,Δ0cDtvwtq=Jqwtq−,ν∈0,1,q=1,2,…,r,w0=0,0cDtϱwT=δ0cDtξwρ,0<δ<ξ<1,0<ρ<1.In the proposed problem, the notation0cDtϱ,0cDtv,and0cDtξ represent Caputo fractional derivatives of orders ϱ, ν, and ξ, respectively, where the points 0andt in the subscript of the differential operator D are actually the limits of the definite integral involved in the definition of CFD. The function f:0,T×ℛ×ℛ⟶ℛ is continuous, where ℛ is the set of real numbers. The impulsive functions Iq and Iq in C(R, R) are bounded. For the sequence 0=t0<t1<⋯<tr<tr+1=T, we have Δwtq=wtq+−wtq− and Δ0cDtvwtq=0cDtvwtq+−0cDtvwtq−,wtq+=limh⟶0wtq+h, and wtq−=limh⟶0wtq−h represents the right-hand and left-hand limits of wt at t=tq, respectively, with wtq−=wtq. The speciality of this proposed problem is that the nonlinear term depends not only on the unknown function but also on its fractional derivative compared with the available results in the literature. This type of study has rarely been discussed in the literature because of the complexity of fractional impulsive surfaces. The further organization of this manuscript is divided into four parts as follows: The second part of the paper demonstrates the preliminary portion in which we recall to readers the basics of used theory, notations, and definitions. The third part presents an existence result by employing Schaefer’s fixed point theorem. The fourth section is introduced to analyze and study several stability results of the considered problem, and the last section is provided to illustrate the applications of the obtained results.


## 2. Preliminaries

We takeS=0,T, S0=0,t1, and Sq=tq,tq+1. We introduce the following space of piecewise continuous functions by(3)ℬ=PCS,ℛ=w:S⟶ℛw∈CS,q=1,2,…,r,wtq+,wtq−exist forq=1,2,…,r,where ℬ is the Banach space corresponding to the norm wℬ=maxt∈Swt.Definition 1 (see [18]).
The fractional order integral of functiong∈L1a,b,ℛ+ of order ϱ∈ℛ+ is defined by(4)Iaϱgt=∫att−ηϱ−1Γϱgηdη,where Γ is the gamma function.Definition 2 (see [18]).
For a functiong given on interval a,b, the CFD of g is defined by(5)0cDtϱgt=1Γn−ϱ∫att−ηn−ϱ−1gnηdη,where n=ϱ+1.
Let there exist constantsβ>0andϵ>0 and a nondecreasing function Φ∈CS,ℛ, such that the following inequalities exist for q=1,2,…,r:(6)0cDtϱht−ft,ht,0cDtϱht≤ϵ,t∈S,Δhtq−ℐqhtq−≤ϵ,Δ0cDtϱhtq−Jqhtq−≤ϵ,(7)0cDtϱht−ft,ht,0cDtϱht≤Φt,t∈S,Δhtq−ℐqhtq−≤β,|Δ0cDtϱhtq−Jqhtq−≤β,(8)0cDtϱht−ft,ht,0cDtϱht≤ϵΦt,t∈S,Δhtq−ℐqhtq−≤ϵβ,Δ0cDtvhtq−Jqhtq−≤ϵβ.Definition 3 (see [19]).
If forϵ>0 there exists a constant C>0 such that for any solution h∈ℬ of inequality (6), there is a unique solution w∈ℬ of problem (2) which satisfies(9)ht−wt≤Cϵ,t∈S,then problem (2) is called Hyers-Ulam stable.Definition 4 (see [19]).
If forϵ>0 and set of positive real numbers ℛ+ there exists ψ∈Cℛ+,ℛ+, such that for any solution h∈ℬ of inequality (6), there is a unique solution w∈ℬ of problem (2) which satisfies(10)ht−wt≤ψε,t∈S,then problem (2) is called generalized Hyers-Ulam stable.Definition 5 (see [19]).
If forϵ>0 there exists a real number C>0, such that for any solution h∈ℬ of inequality (8), there is a unique solution w∈ℬ of problem (2) which satisfies(11)ht−wt≤CεΦt+ψ,t∈S,then problem (2) is called Hyers-Ulam-Rassias stable with respect to Φ,ψ.Definition 6 (see [19]).
If there exists constantC>0, such that for any solution h∈ℬ of inequality (7), there is a unique solution w∈ℬ of problem (2) which satisfies(12)ht−wt≤CΦt+ψ,t∈S,then problem (2) is called generalized Hyers-Ulam-Rassias stable with respect to Φ,ψ.
Here, it is to be noted that Definitions3–6 have been adopted from the paper [19].Remark 1.
The functionh∈ℬ is called a solution for inequality (6) if there exists a function ϕ∈ℬ together with a sequence ϕq, where q=1,2,…,r (which depends on h) such that(i)
ϕt≤ϵ,ϕq≤ϵ,t∈S,q=1,2,…,r(ii)
0cDtϱht=ft,ht,0cDtϱht+ϕt,t∈S,q=1,2,…,r(iii)
Δhtq=ℐqhtq−+ϕqt∈S,q=1,2,…,r(iv)
Δ0cDtνhtq=Jqhtq−+ϕq,t∈S,q=1,2,…,rRemark 2.
A functionh∈ℬ is a solution of inequality (8) if there exists a function ϕ∈ℬ and a sequence ϕq, where q=1,2,…,r (which depends on h) such that(i)
ϕt≤ϵθt,ϕq≤ϵφ,t∈S,q=1,2,…,r(ii)
0cDtϱht=ft,ht,0cDtϱht+ϕt,t∈S,q=1,2,…,r(iii)
Δhtq=ℐqhtq−+ϕq,t∈S,q=1,2,…,r(iv)
Δ0cDtvhtq=Jqhtq−+ϕq,t∈S,q=1,2,…,rLemma 1 (see [20]).
Forϱ>0, the given result holds:(13)0Itϱ0cDtϱgt=gt−∑i=0n−1citi,wheren=ϱ+1.To investigate the nonlinear IFODE2, we first consider the associated linear problem and obtain its solution.Lemma 2.
Letϱ∈1,2 and σ:0,T⟶ℛ be continuous. A function wt is a solution of the fractional integral equation:(14)wt=∫0tt−ηϱ−1Γϱσηdη−tΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξσηdη−∫0TT−ηϱ−ξ−1Γϱ−ξσηdη+∑i=1qΓ2−νti1−vJiwti−,t∈0,t1,q=1,2,…,r,∫0tt−ηϱ−1Γϱσηdη+∑i=1qℐiwti−−tΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξσηdη−∫0TT−ηϱ−ξ−1Γϱ−ξσηdη+∑i=1qΓ2−νti1−vJiwti−+∑i=1qt−tiΓ2−νti1−vJiwti−,t∈tq,tq+1,q=1,2,…,r,if and only if wt is a solution of the following BVP:(15)0cDtϱwt−σt=0,ϱ∈1,2,Δwtq=ℐqwtq−,Δ0cDtvwtq=Jqwtq−,ν∈0,1,q=1,2,…,r,w0=0,0cDtξwT=δ0cDtξwρ,0<δ<ξ<1.Proof.
Let fort∈0,t1, wt be the solution of (15), then by Lemma 1, we have(16)wt=∫0tt−ηϱ−1Γϱσηdη−c0−c1t.
Using the conditionw0=0, we get(17)c0=0.
Substitutingc0 in (16), we get(18)wt=∫0tt−ηϱ−1Γϱσηdη−c1t.
Fort∈t1,t2, we get(19)wt=∫0tt−ηϱ−1Γϱσηdη−c2−c3t.
Applying the impulsive conditionΔwt1=ℐ1wt1−, we get(20)ℐ1wt1−=−c2−c3t1+c1t1,−c2=ℐ1wt1−+c3t1−c1t1.
Substitutingc2 in (19), we get(21)wt=∫0tt−ηϱ−1Γϱσηdη+ℐ1wt1−+c3t1−c1t1−c3t,(22)wt=∫0tt−ηϱ−1Γϱσηdη+ℐ1wt1−−c1t1+c3t1−t.
From equations (18) and (22), we get(23)0cDtvwt=1Γϱ−ν∫0tt−ηϱ−ν−1σηdη−c3t1−νΓ2−ν,0cDtvwt=1Γϱ−ν∫0tt−ηϱ−ν−1σηdη−c1t1−νΓ2−ν.
Now, using the impulsive conditionΔ0cDtvwt1=J1wt1−, we get(24)J1wt1−=−c3t11−vΓ2−ν+c1t11−vΓ2−ν,c3=−Γ2−νt11−vJ1wt1−+c1.
Substitutingc3 in (22), we get(25)wt=∫0tt−ηϱ−1Γϱσηdη+ℐ1wt1−+t−t1Γ2−νt11−vJ1wt1−−c1t.
Fort∈t2,t3, we get(26)wt=∫0tt−ηϱ−1Γϱσηdη−c4−c5t.
Applying the impulsive conditionΔwt2=ℐ2wt2− in (26) and (25), we get(27)ℐ2wt2−=−c4−c5t2−ℐ1wt1−−t2−t1Γ2−νt11−vJ1wt1−+c1t2,−c4=ℐ2wt2−+c5t2+ℐ1wt1−+t2−t1Γ2−νt11−vJ1wt1−−c1t2.
Substitutingc4 in (26), we get(28)wt=∫0tt−ηϱ−1Γϱσηdη+ℐ2wt2−+c5t2+ℐ1wt1−+t2−t1Γ2−νt11−vJ1wt1−−c1t2−c5t.
By equations (25) and (28), we get(29)0cDtνwt=1Γϱ−ν∫0tt−ηϱ−ν−1σηdη−c5t1−νΓ2−ν,0cDtνwt=1Γϱ−ν∫0tt−ηϱ−ν−1σηdη+Γ2−νt11−vJ1wt1−−c1t1−νΓ2−ν.
Now, using the impulsive conditionΔ0cDtνwt2=J2wt2−, we get(30)J2wt2−=−c5t21−vΓ2−ν−Γ2−νt11−vJ1wt1−−c1t21−vΓ2−ν,c5=−Γ2−νt11−vJ1wt1−−Γ2−νt21−vJ2wt2−+c1.
Substitutingc5 in (28), we get(31)wt=∫0tt−ηϱ−1Γϱσηdη+ℐ1wt1−+ℐ2wt2−+Γ2−νt11−vJ1wt1−t−t1+Γ2−νt21−vJ2wt2−t−t2−c1t.
Similarly fort∈Sq, we get(32)wt=∫0tt−ηϱ−1Γϱσηdη+∑ı=1qℐıwtı−−c1t+∑ı=1qt−tıΓ2−νti1−vJıwtı−,0cDtνwt=∫0tt−ηϱ−ξ−1Γϱ−ξσηdη−c1t1−ξΓ2−ξ+t1−ξΓ2−ξ∑ı=1qΓ2−νti1−vJıwtı−.
Using the boundary conditions,0cDtξwT=δ0cDtξwρ, we get(33)0cDtξwT=∫0TT−ηϱ−ξ−1Γϱ−ξσηdη−c1T1−ξΓ2−ξ+T1−ξΓ2−ξ∑ı=1qΓ2−νti1−vJıwtı−,δ0cDtξwρ=δ∫0ρρ−ηϱ−ξ−1Γϱ−ξσηdη−δc1ρ1−ξΓ2−ξ+δρ1−ξΓ2−ξ∑ı=1qΓ2−νti1−vJıwtı−,c1δρ1−ξ−T1−ξΓ2−ξ=δ∫0ρρ−ηϱ−ξ−1Γϱ−ξσηdη+δρ1−ξΓ2−ξ∑ı=1qΓ2−νti1−vJıwtı−−∫0TT−ηϱ−ξ−1Γϱ−ξσηdη−T1−ξΓ2−ξ∑ı=1qΓ2−νti1−vJıwtı−,c1=Γ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξσηdη−∫0TT−ηϱ−ξ−1Γϱ−ξσηdη+∑ı=1qΓ2−νti1−vJıwtı−.
By substituting the value ofc1 and summarizing, we get the required result.
Conversely, assume thatu satisfies the impulsive fractional integral equation (8); then by direct computation, it can be seen that the solution given by (14) satisfies (15).Corollary 1.
In view of the Lemma2, problem (2) has the following solution:(34)wt=∫0tt−ηϱ−1Γϱfη,wη,0cDtϱwηdη−U,t∈0,t1,∫0tt−ηϱ−1Γϱfη,wη,0cDtϱwηdη+∑ı=1qℐıwtı−−U+∑ı=1qt−tıΓ2−νti1−vJıwtı−,t∈tq,tq+1,q=1,2,…,r,where(35)U=Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξfη,wη,0cDtϱwηdη−Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξfη,wη,0cDtϱwηdη+t∑ı=1qΓ2−νti1−vJıwtı−,q=1,2,…,r.


## 3. Existence and Uniqueness Results

In this section, we shall prove our main results. For which, we assume the following assumptions:H1 Let there exist positive constants L1,L2,L3,andL4 such that for t∈0,T and all w1,w2,h1,h2∈ℛ, the following inequalities hold:(36)ft,w1t,h1t−ft,w2t,h2t≤L1w1t−w2t+L2h1t−h2t,ℐıw1tı−−ℐıw2tı−≤L3w1tı−−w2tı−,Jıw1tı−−Jıw2tı−≤L4w1tı−−w2tı−.H2 Let the functions a1,a2,a3∈C0,T,ℛ+, which satisfy(37)ft,wt,Ywt≤a1t+a2tw+a3tYwt,fort∈S,w∈ℬ.such thata3∗=supt∈0,Ta3t<1.H3 If f,ℐı,and Jı are continuous functions such that for all w,h∈ℛ, the following inequalities hold:(38)ℐıwtı−≤Cℐıw+Mℐ,Cℐı,Mℐ>0,Jıwtı−≤CJıw+MJ,CJı,MJ>0.H4 Let there exist constants β>0andϵ>0 and a nondecreasing function Φ∈CS,ℛ, such that the following inequalities hold:(39)0ItϱΦt≤βΦt;consequently0ItϱΦt≤βΦt.We transform problem (2) into a fixed point problem. Considering an operator N:ℬ⟶ℬ, defined by(40)Nwt=∫0tt−ηϱ−1ΓϱYwηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwηdη−t∑ı=1qΓ2−νti1−vJıwtı−,t∈0,t1,q=1,2,…,r,∫0tt−ηϱ−1ΓϱYwηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwηdη−t∑ı=1qΓ2−νti1−vJıwtı−+∑ı=1qℐıwtı−+∑ı=1qt−tıΓ2−νti1−vJıwtı−,t∈tq,tq+1,q=1,2,…,r,whereYwt=ft,wt,0cDtϱwt=ft,wt,Ywt. In (40), we see that all the terms of solution w in the interval 0,t1 are contained in the solution w in interval tq,tq+1; therefore, for simplicity purpose, we will study the solution in interval tq,tq+1 only. Now, we shall prove some theorems. Our first result is based on Schaefer’s fixed theorem.Theorem 1.
If the assumptionsH1−H3 are satisfied, then problem (2) has at least one solution.Proof.
We use Schaefer’s fixed point theorem. The proof is given in the following four steps.Step 1: to show thatN is continuous, take a sequence wn such that wn⟶w∈ℬ. Then, for t∈S, we have(41)Nwnt−Nwt≤∫0tt−ηϱ−1ΓϱYwnη−Ywηdη+tΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwnη−Ywηdη+tΓ2−ξδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwnη−Ywηdη+t∑ı=1qΓ2−νti1−vJıwntı−−Jıwtı−+∑ı=1qℐıwntı−−ℐıwtı−+∑ı=1qt−tıΓ2−νti1−v×Jıwntı−−Jıwtı−,whereYwnt=ft,wnt,Ywnt and Yt=ft,wt,Ywt. Using H1, we have(42)Ywnt−Ywt=ft,wnt,Ywnt−ft,wt,Ywt≤L1wnt−wt+L2Ywnt−Ywt,which implies(43)Ywn−Ywℬ≤L11−L2wn−wℬ,Jıwntı−−Jıwtı−≤L4wntı−−wtı−.wn⟶w implies Ywnt⟶Ywt as n⟶∞. Now, since every convergent sequence is bounded, there exists a constant b>0 such that Ywnt≤b and Yw≤b for t∈S. Thus,(44)t−ηϱ−1Ywnt−Ywt≤t−ηϱ−1Ywnt+Ywt≤2bt−ηϱ−1,T−ηϱ−ξ−1Ywnt−Ywt≤T−ηϱ−ξ−1Ywnt+Ywt≤2bT−ηϱ−ξ−1,ρ−ηϱ−ξ−1Ywnt−Ywt≤ρ−ηϱ−ξ−1Ywnt+Ywt≤2bρ−ηϱ−ξ−1.The functionsη⟶2bt−ηϱ−1, η⟶2bT−ηϱ−ξ−1, and η⟶2bρ−ηϱ−ξ−1 are integrable for t∈S. Therefore, by the continuity of f,ℐ,andJ and the Lebesgue dominated convergent theorem, we conclude from (41) that Nwnt−Nwt⟶0 as n⟶∞ which implies Nwn−Nwℬ⟶0n⟶∞. This proves the continuity of N.Step 2: in this step, we will show that for eachw∈Wb=w∈ℬ:w≤b, Nwℬ≤K. For t∈S, consider(45)Nwt≤∫0tt−ηϱ−1ΓϱYwηdη+tΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwηdη+tΓ2−ξδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwηdη+t∑ı=1qΓ2−νti1−vJıwtı−+∑ı=1qℐıwtı−+∑ı=1qt−tıΓ2−νti1−vJıwtı−.UsingH2 and a1∗=supt∈Xa1t,a2∗=supt∈Xa2t,anda3∗=supt∈Xa3t<1, we have(46)Ywt=ft,wt,Ywt≤a1t+a2twt+a3tYwt.Taking the maximum value over the intervalS and simplifying, we get(47)Ywℬ≤a1∗+a2∗b1−a3∗≕K1.Using this result, (45) implies(48)Nwt≤K1∫0tt−ηϱ−1Γϱdη+tΓ2−ξK1δρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξdη+tΓ2−ξK1δρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξdη+t∑ı=1qΓ2−νti1−vCJıw+MJ+∑ı=1qCℐıw+Mℐ+∑ı=1qt−tıΓ2−νti1−vCJıw+MJ.Further simplification implies(49)Nwℬ≤K1TϱΓϱ+1+δTΓ2−ξδρϱ−ξ+Tϱ−ξδρ1−ξ−T1−ξΓϱ−ξ+1+2qTΓ2−νCJib+MJ+qCℐib+Mℐ≕K.This shows that the operatorN maps bounded sets into bounded sets.Step 3: in this step, we will show thatN is equicontinuous. Let w∈W⊆ℬ and t1,t2∈S such that t1<t2 and consider(50)Nwt2−Nwt1≤∫t1t2t2−ηϱ−1ΓϱYwηdη+t2−t1Γ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwηdη+t2−t1Γ2−ξδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwηdη+t2−t1∑0<tı<t2−t1Γ2−νti1−vJıwtı−+∑0<tı<t2−t1ℐıwtı−+∑0<tı<t2−t1t2−t1Γ2−νti1−vJıwtı−.Using assumptionsH2 and H3, we obtain(51)Nwt2−Nwt1≤K1t2−t1ϱΓϱ+1+t2−t1K1Γ2−ξδρ1−ξ−T1−ξδρϱ−ξΓϱ−ξ+1+t2−t1K1Γ2−ξδρ1−ξ−T1−ξTϱ−ξΓϱ−ξ+1+2t2−t1qΓ2−νCJib+MJ+t2−t1qΓ2−νCℐib+Mℐ.We see that ast1⟶t2, the right-hand side of inequality (51) tends to zero that is Nwt2−Nwt1⟶0 as t1⟶t2. Hence, by the Ascoli−Arzela` theorem N:ℬ⟶ℬ is completely continuous.Step 4: to complete the proof, it remains to show that the setℰ=w∈:w=ζNw,for0<ζ<1 is bounded. Let w∈ℰ, then for any t∈S, we have(52)Nwt≤ζ∫0tt−ηϱ−1ΓϱYwηdη+ζtΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYwηdη+ζtΓ2−ξδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYwηdη+ζt∑ı=1qΓ2−νti1−vJıwtı−+ζ∑ı=1qℐıwtı−+ζ∑ı=1qt−tıΓ2−νti1−vJıwtı−.Using0<ζ<1 and (47) and (49), from (52), we get the following result:(53)Nwℬ≤ζK1TϱΓϱ+1+δTΓ2−ξδρϱ−ξ+Tϱ−ξδρ1−ξ−T1−ξΓϱ−ξ+1+2qTΓ2−νCJib+MJ+qCℐib+Mℐ≕ζK≤K.This shows that the setℰ is bounded. Therefore, by Schaefer’s fixed point theorem, problem (2) has at least one solution.
The following and our second result is based on the Banach fixed point theorem.Theorem 2.
If the assumptionsH1−H3 and the inequality(54)L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ<1,are satisfied, then (2) has an unique solution.Proof.
To show that the operatorN as defined above has a unique fixed point, we consider w1,w2∈ℬ.
Fort∈tq,tq+1, we have(55)Nw1t−Nw2t≤∫0tt−ηϱ−1ΓϱYw1η−Yw2ηdη+tΓ2−ξδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYw1η−Yw2ηdη+tΓ2−ξδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYw1η−Yw2ηdη+t∑ı=1qΓ2−νtı1−νJıw1tı−−Jıw2tı−+∑ı=1qℐıw1tı−−ℐıw2tı−+∑ı=1qt−tıΓ2−νtı1−ν×Jıw1tı−−Jıw2tı−,where Yw1t=ft,w1t,Yw1 and Yw2η=ft,w2t,Yw2. Using H1, we have(56)Yw1t−Yw2t=ft,w1t,Yw1t−ft,w2t,Yw2t≤L1w1t−w2t+L2Yw1t−Yw2t,which implies(57)Yw1−Yw2ℬ≤L11−L2w1−w2ℬ,Jıw1tı−−Jıw2tı−≤L4w1tı−−w2tı−.
Thus, from (55), we have(58)Nw1t−Nw2t≤L1Tϱ1−L2Γϱ+1maxt∈tq,tq+1w1t−w2t+Γ2−ξρϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1maxt∈tq,tq+1δtw1t−w2t+L1Γ2−ξTϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1maxt∈tq,tq+1tw1t−w2t+maxt∈tq,tq+1qtΓ2−ξL4tq1−νw1t−w2t+maxt∈tq,tq+1qL3w1tı−w2tı+maxt∈tq,tq+1qt−tıΓ2−ξtı1−νL4w1tı−w2tı,q=1,2,…,r.
By further simplification, we obtain the following inequality:(59)Nw1−Nw2ℬ≤L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξw1−w2ℬ,L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ<1.
Therefore, by Banach contraction theorem, problem (2) has a unique fixed point.


## 4. Stability Analysis

In this section, we study Hyers-Ulam stability of problem (2).Theorem 3.
If assumptionsH1−H3 and the inequality(60)L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ<1,are satisfied, then problem (2) is Hyers-Ulam stable.Proof.
Leth be any solution of inequality (6), then by Remark 1, we have(61)0cDtϱht=ft,ht,0cDtϱht+ϕt,ϱ∈1,2,Δhtq=ℐqhtq−+ϕq,Δ0cDtvhtq=Jqhtq−+ϕq,ν∈0,1,q=1,2,…,r,h0=0,0cDtξhT=δ0cDtξhρ+ϕq,0<δ<ξ<1,0<ρ<1,q=1,2,…,r.
In light of Corollary1, the solution problem (61) is given by(62)ht=∫0tt−ηϱ−1ΓϱYhηdη+∫0tt−ηϱ−1Γϱϕηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYhηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξϕηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYhηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξϕηdη−t∑ı=1qΓ2−νti1−vJıhtı−−t∑ı=1qΓ2−νti1−vϕq,t∈0,t1,q=1,2,…,r,∫0tt−ηϱ−1ΓϱYhηdη+∫0tt−ηϱ−1Γϱϕηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYhηdη−Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξϕηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYhηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξϕηdη−t∑ı=1qΓ2−νti1−vJıhtı−−t∑ı=1qΓ2−νti1−vϕq+∑ı=1qℐhtı−ı+∑ı=1qϕq+∑ı=1qt−tıΓ2−νti1−vJıhtı−+∑ı=1qt−tıΓ2−νti1−vϕq,t∈tq,tq+1,q=1,2,…,r,where Yht=ft,ht,Yht. Now if w is a unique solution of problem (2), then for t∈S, consider(63)ht−wt≤∫0tt−ηϱ−1ΓϱYhη−Ywηdη+∫0tt−ηϱ−1Γϱϕηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYhη−Ywηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξϕηdη+Γ2−ξtδρ1−ξ−T1−ξ∫0TT−ηϱ−ξ−1Γϱ−ξYhη−Ywηdη∫0TT−ηϱ−ξ−1Γϱ−ξϕηdη+t∑ı=1qΓ2−νtı1−νJıhtı−−Jıwtı−+t∑ı=1qΓ2−νtı1−νϕq+∑ı=1qℐıhtı−−ℐıwtı−+∑ı=1qϕq+∑ı=1qt−tıΓ2−νtı1−νJıhtı−−Jıwtı−+∑ı=1qt−tıΓ2−νtı1−νϕq,t∈tq,tq+1,q=1,2,…,r,where Ywt=ft,wt,Ywt. By using assumption H1−H3, Remark 1, we have from (63)(64)h−wℬ≤L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξh−wℬ+TϱΓϱ+1+δTΓ2−ξδρϱ−ξ+Tϱ−ξδρ1−ξ−T1−ξΓϱ−ξ+1+q+2qTΓ2−ξϵ.
Let(65)ℰ1=L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ,ℰ2=TϱΓϱ+1+δTΓ2−ξδρϱ−ξ+Tϱ−ξδρ1−ξ−T1−ξΓϱ−ξ+1+q+2qTΓ2−ξ,then(66)h−wℬ≤ℰ21−ℰ1ϵ=Cϵ,where C=ℰ2/1−ℰ1; ℰ1<1. This shows that problem (2) is Hyers-Ulam stable. Moreover, by setting ψϵ=Cϵandψ0=0, then the solution of problem (2) is generalized Hyers-Ulam stable.Theorem 4.
If assumptionsH1−H4 and the inequality(67)ℰ1=L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ<1,are satisfied, then problem (2) is Hyers-Ulam-Rassias stable.Proof.
From the proof of Theorem3, we can write(68)ht−wt≤∫0tt−ηϱ−1ΓϱYhη−Ywηdη+∫0tt−ηϱ−1Γϱϕηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξYhη−Ywηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0ρρ−ηϱ−ξ−1Γϱ−ξϕηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0TT−ηϱ−ξ−1Γϱ−ξYhη−Ywηdη+Γ2−ξtδρ1−ξ−T1−ξδ∫0TT−ηϱ−ξ−1Γϱ−ξϕηdη+t∑ı=1qΓ2−νti1−vJıhtı−−Jıwtı−+t∑ı=1qΓ2−νti1−vϕq+∑ı=1qℐıhtı−−ℐıwtı−+∑ı=1qϕq+∑ı=1qt−tıΓ2−νti1−vJıhtı−−Jıwtı−+∑ı=1qt−tıΓ2−νti1−vϕq,t∈tq,tq+1,q=1,2,…,r.
By using assumptionH1−H4, Remark 2, we have from (68)(69)h−wℬ≤ℰ1h−wℬ+ϵβ+δTβΓ2−ξδρ1−ξ−T1−ξ+TβΓ2−ξδρ1−ξ−T1−ξθt+ϵq+2qTΓ2−νφ≤ℰ1h−wℬ+ℰ3θt+φϵ,which implies that(70)h−wℬ≤ϵℰ3θt+φ1−ℰ1,h−wℬ≤Cϵθt+φ,where(71)ℰ1=L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ+qL3+2L4TΓ2−ξ<11−L2δρ1−ξ−T1−ξΓϱ−ξ+1,ℰ3=βδρ1−ξ−T1−ξ+δTβΓ2−ξ+TβΓ2−ξδρ1−ξ−T1−ξ+q+2qTΓ2−ν.
Therefore, problem (2) is Hyers-Ulam-Rassias stable.


## 5. Example

Consider the following implicit fractional system with given impulsive conditions.Example 1.
(72)0cDt3/2wt=wt26t+1/21+wt+sin0cDt3/2wt26+t3,t∈S=0,1,t≠t1,q=r=1,w0=0,0cDt1/2w1=1412+wρ+0cDt1/2wρ72eρ+31+wρ+0cDt1/2wρρ=1/5,Δwt1=wt1−40+wt1−,Δ0cDt1/2wt1=wt1−32+wt1−.
Obviously, the given functionf is continuous. We see ϱ=5/4, ξ=ν=1/2, T=1, δ=1/4,andρ=1/5. For w1,w2∈ℬandYw1,Yw2∈CS,ℛandt∈S,(73)ft,w1t,Yw1−ft,w2t,Yw2≤126w1t−w2t+Yw1−Yw2,which satisfies H1 with L1=L2=1/26. Further, for t1=1/5, let(74)ℐ1wt1−=wt1−40+wt1−,J1wt1−=wt1−32+wt1−,wherew1∈ℬ.
For anyw1,w2∈ℬ, we have(75)ℐw1t1−−ℐw2t1−=wt1−40+wt1−−wt1−40+wt1−≤140w1t1−−w2t1−,ℐw1t1−−ℐw2t1−=wt1−32+wt1−−wt1−32+wt1−≤132w1t1−−w2t1−,which satisfy the 2nd and 3rd inequalities of H1 with L3=1/40andL4=1/32.
Forϱ=1/5, ξ=ν=1/2, T=1, δ=1/4,ρ=1/4,L1=L2=1/26,L3=1/40,andL4=1/32, we have(76)ℰ1=L1Tϱ1−L2Γϱ+1+L1δTΓ2−ξδρϱ−ξ+Tϱ−ξ1−L2δρ1−ξ−T1−ξΓϱ−ξ+1+qL3+2L4TΓ2−ξ=0.301+0.009+0.08=0.39<1.
Therefore, by Theorem2, problem (72) has a unique solution. By Theorem 4, problem (52) is Hyers-Ulam stable and consequently generalized Hyers-Ulam stable.
Further, assumingΦt=1, we have(77)I13/20Φt=1Γ3/2∫011−η3/2−1ηdη≤13Π,Consequently,(78)I13/2−10Φt=1Γ3/2−1∫011−η3/2−2ηdη≤13Π,which satisfies H4 with β=1/3Π and Φt=1; therefore, by Theorem 4, the solution of (72) is Hyers-Ulam-Rassias stable corresponding to Φ,ψ. With the help of Matlab, we plot the result by using the RK4 method in Figure 1.Figure 1
Graphical presentation of result of Example1.


## 6. Conclusion

In this paper, using Schaefer’s fixed point theorem, we have derived sufficient conditions for at least one solution to a class of IFODEs under NCFBC 2. Similarly, using the Banach contraction theorem, we obtained conditions under which problem 2 has unique solution. Moreover, by the application of qualitative theory and nonlinear functional analysis, we established results concerning to various kinds of Hyers-Ulam stability. The concerned results have been verified by a suitable example.



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