# On Atangana–Baleanu-Type Nonlocal Boundary Fractional Differential Equations

Authors: Mohammed A. Almalahi; Satish K. Panchal; Mohammed S. Abdo; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This research paper is devoted to investigating two classes of boundary value problems for nonlinear Atangana–Baleanu-type fractional differential equations with Atangana–Baleanu fractional integral conditions. The applied fractional derivatives work as the nonlocal and nonsingular kernel. Upon using Krasnoselskii’s and Banach’s fixed point techniques, we establish the existence and uniqueness of solutions for proposed problems. Moreover, the Ulam–Hyers stability theory is constructed by using nonlinear analysis. Eventually, we provide two interesting examples to illustrate the effectiveness of our acquired results.


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## Body

## 1. Introduction

Fractional calculus and its potential applications have become a powerful tool to model many complex phenomena in apparently wide-ranging fields of science and technology [1–8]. To meet the needs of modeling many practical problems in different fields of science and engineering, some researchers have realized the necessity development of the concept of fractional calculus by searching for new fractional derivatives with different singular or nonsingular kernels. From this perspective, new fractional operators have turned into the best effective tool of numerous specialists and researchers with their contribution to physical phenomena and their performance in applying to real-world problems. Until 2015, all fractional derivatives had only singular kernels. Therefore, it is difficult to use these singularities to simulate physical phenomena. In 2015, Caputo and Fabrizio [9] studied a new kind of FD in the exponential kernel. Some properties of this new type had been discussed by Losada and Nieto in [10]. A new type and interesting FD with Mittag-Leffler kernels has been investigated by Atangana and Baleanu (A-B) in [11]. Abdeljawad in [12] extended the kind investigated by A-B from order between zero and one to higher arbitrary order and formulated their associated integral operators. Also, he proved some properties such as the existence and uniqueness theorems for two classes of fractional derivative, Riemann type (ABR) and Caputo type (ABC), for initial value problems in higher arbitrary order and proved a Lyapunov-type inequality in the frame of Mittag-Leffler kernels for the ABR fractional boundary value problems of order 2<α≤3. Abdeljawad and Baleanu, in [13, 14], deliberated the discrete versions of those new operators. For some theoretical works on Atangana–Baleanu FDEs, we refer the reader to a series of papers [15–17]. For important applications and mathematical modeling of the ABC fractional operator, see [18–21]. On the contrary, there are some important numerical approaches regarding nonsingular kernels; for example, in [22], via a spectral collocation method based on the shifted Legendre polynomials with extending the unknown functions and their derivatives using the shifted Legendre basis, Tuan et al. solved fractional rheological models and Newell–Whitehead–Segel equations with the nonlocal and nonsingular kernels. Nikan et al. in [23] developed the solution of the two-dimensional time-fractional evolution model using a finite difference scheme derived from the radial basis function (RBF-FD) method. Ganji et al. in [24] studied the multivariable-order differential equations (MVODEs) with the nonlocal and nonsingular kernels in the Atangana and Baleanu sense of variable orders. They used the fifth-kind Chebyshev polynomials as basic functions to obtain operational matrices and used them with the collocation method to transfer the original equations to a system of algebraic equations. Firoozjaee et al. in [25] studied the Fokker–Planck equation (FPE) with CF-FD being considered. They presented a numerical approach that is based on the Ritz method with known basis functions to transform this equation into an optimization problem. Baleanu et al. in [26] showed that four fractional integrodifferential inclusions have solutions. Also, they showed that the dimension of the set of solutions for the second fractional integrodifferential inclusion problem is infinite-dimensional under some different conditions. By using the fractional CF derivative, Aydogan et al. [27] introduced two types of new high-order derivations called CFD and DCF. Also, they studied the existence of solutions for two such types of high-order fractional integrodifferential equations. Guoa et al. in [28, 29] studied the existence and HU stability of the FDEs with impulse and fractional Brownian motion under nonlocal conditions based on the semigroups of operators method and Mönch fixed point method.There are some researchers in various areas who investigated the existence and Ulam–Hyers (UH) stability of solutions for fractional differential equations (FDEs) of some problems with initial (or boundary, nonlocal) conditions; more details can be found within research series [30–35].Recently, Abdo et al. [36] considered ABC-type pantograph FDEs with nonlocal conditions:(1)Da+pABCϑσ=fσ,ϑσ,ϑγσ,σ∈a,T,p∈0,1,ϑa=∑k=1mckϑϰk,ϰk∈a,T.In this regard, the ABC-type implicit FDEs(2)Da+pABCϑσ=fσ,ϑσ,Da+pABCϑσ,σ∈a,T,with nonlinear integral conditions(3)ϑa−ϑ′a=∫aTg(s,ϑs)ds,whenp∈(0,1],ϑa=0,ϑT=∫aTg(s,ϑs)ds,whenp∈(1,2],had been studied by Abdo et al. [37].Motivated by the above argumentations, we intend to investigate and develop the sufficient conditions for existence, uniqueness, and UH stability results of two new classes for the ABR and ABC fractional differential problems described as follows:(4)Da+pABRϑσ=fσ,ϑσ,σ∈a,b,p∈2,3,ϑa=0,ϑb=Ia+δABϑζ,ζ∈a,b,(5)Da+pABCϑσ=fσ,ϑσ,σ∈a,b,p∈1,2,ϑa=0,ϑb=Ia+δABϑζ,ζ∈a,b,where Da+pABR, Da+pABC are the ABR and ABC fractional derivatives of order p∈(2,3] and p∈(1,2], respectively, δ∈(0,1], ζ∈a,b, and f:a,b×ℝ⟶ℝ is a continuous function.To the best of our understanding, this is the first work that transacts with ABC and ABR fractional derivatives of a high order, especially with AB integral conditions. Also, we studied existence and uniqueness without using the semigroup property. In consequence, our results will be a useful contribution to the existing literature on these interesting operators.The paper is organized as follows. In Section2, we present some notations and some preliminary facts which are used throughout the paper. In Section 3, we derive the formula of an equivalent integral equation for ABR-type FDEs (4). Section 4 discusses the existence and uniqueness results for ABR-type FDEs (4) and ABC-type FDEs (5). In Section 4, we discuss the stability results in the frame of UH. Section 5 provides two examples to illustrate the validity of our results. Concluding remarks about our results are in the last section.


## 2. Preliminaries

LetJ=a,b⊂ℝ and CJ,ℝ be the space of all continuous functions ϑ:J⟶ℝ with the norm ϑ=maxϑσ:σ∈J. Then, CJ,ℝ,⋅ is a Banach space.Definition 1.
(see [11]). Let 0<p≤1. Then, the left-sided ABC and ABR fractional derivatives of order p for a function ϑ are defined by(6)Da+pABCϑσ=Bp1−p∫aσEppp−1σ−θpϑ′θdθ,σ>a,Da+pABRϑσ=Bp1−pddσ∫aσEppp−1σ−θpϑ′θdθ,σ>a,respectively, where Bp is the normalization function such that B0=B1=1 and Ep is the Mittag-Leffler function defined by(7)Epϑ=∑i=0∞ϑiΓip+1,Rep>0,ϑ∈ℂ.
The analogous AB fractional integral is given by(8)Ia+pABϑσ=1−pBpϑσ+pBpΓp∫aσσ−sp−1ϑsds.Lemma 1 (see [13]).
Let0<p≤1. If the ABC fractional derivative exists, then we have(9)Ia+pABDa+pABCϑσ=ϑσ−ϑa.Definition 2.
(see [12]). The relation between the ABR and ABC fractional derivatives is given by(10)Da+pABCϑσ=Da+pABRϑσ−Bp1−pϑaEppp−1σ−ap.Remark 1.
By applyingIa+pABϑσ on both sides of the equation defined in Definition 2 and using Lemma 1, we get(11)Ia+pABDa+pABRϑσ=ϑσ.Lemma 2 (see [11]).
Letϑ>0. Then, Ia+pAB is bounded from CJ,ℝ into CJ,ℝ.Definition 3 (see Definition 3.1 of [12]).
Letn<p≤n+1 and ϑ be such that ϑn∈H1a,b. Set β=p−n. Then, 0<β≤1, and we define(12)Da+pABCϑσ=Da+βABCϑnσ,Da+pABRϑσ=Da+βABRϑnσ.
The correspondent AB fractional integral is given by(13)Ia+pABϑσ=Ia+nIa+βABϑσ.Lemma 3 (see Proposition 3.1 of [12]).
Forϑσ defined on J and n<p≤n+1, for some n∈ℕ0, we have(i)
Da+pABCIa+pABϑσ=ϑσ(ii)
Ia+pABDa+pABRϑσ=ϑσ−∑i=0n−1ϑia/i!σ−ai(iii)
Ia+pABDa+pABCϑσ=ϑσ−∑i=0n−1ϑia/i!σ−aiLemma 4.
Letn<p≤n+1. Then, Da+pABRσ−am=0, for m=1,2,…,n−1.Proof.
Letϑσ=σ−am. Then, by Definition 3, we have(14)Da+pABCϑσ=Da+βABRϑnσ=Da+βABRσ−amn=Da+βABRddσnσ−am.
Sincem<n∈ℕ, we have d/dσnσ−am=0. Consequently, Da+pABRσ−am=0.Theorem 1 (see [38]).
Assume thatK is a closed subspace from a Banach space X, and G is a contraction defined as Q:K⟶K, i.e., Qx−Qy≤Mx−y for some 0<M<1 and all x,y∈K. Then, Q has a fixed point in K.Theorem 2 (see [39]).
LetX be a Banach space, and let a set K⊂X be nonempty, closed, convex, and bounded. If there are two operators Φ1,Φ2 such that (i) Φ1u+Φ2v∈X, for all, u,v∈XΦ1 is compact and continuous, and (iii) Φ2 is a contraction mapping, then there exists a function z∈K such that z=Φ1z+Φ2z.Lemma 5 (see Lemma 5.1 of [12]).
Letp∈(2,3] and h∈CJ,ℝ. Then, the solution of the following linear problem,(15)Da+pABRϑσ=hσ,ϑa=c1,ϑb=c2,is given by(16)ϑσ=c1+c2σ−a+Ia+pABhσ=c1+c2σ−a+Ia+2Ia+βABhσ,0<β=p−2≤1,where(17)Ia+βABhσ=1−βBβhσ+βBβΓβ∫aσσ−sβ−1hsds=3−pBp−2hσ+p−2Bp−2Γp−2∫aσσ−sp−3hsds,(18)Ia+pABhσ=Ia+2Ia+δABhσ=3−pBp−2∫aσσ−shsds+p−2Bp−2Γp−2∫aσσ−sp−1hsds.Lemma 6 (see Example 3.3 of [12]).
Letp∈(1,2] and h∈CJ,ℝ. Then, the solution of the following linear problem,(19)Da+pABCϑσ=hσ,ϑa=c1,ϑ′a=c2,is given by(20)ϑσ=c1+c2σ−a+Ia+pABhσ,where(21)Ia+pABhσ=2−pBp−1∫aσhsds+p−1Bp−1Γp∫aσσ−sp−1hsds.


## 3. Equivalent Integral Equation for Problem (4)

Lemma 7.
Letp∈(2,3],δ∈(0,1], Λ=b−a−Ia+δABζ−a≠0, and h1∈CJ,ℝ. A function ϑ is a solution of the following ABR problem,(22)Da+pABRϑσ=h1σ,σ∈J,ϑa=0,ϑb=Ia+δABϑζ,ζ∈a,b,if and only if ϑ satisfies the following fractional integral equation:(23)ϑσ=σ−aΛIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+pABh1σ.Proof.
Assume thatϑ is the solution to the first equation of (22). Then, via Lemma 5, we have(24)ϑσ=c1+c2σ−a+Ia+pABh1σ.
Now, by condition(ϑa=0), we get c1=0, and hence, equation (24) reduces to(25)ϑσ=c2σ−a+Ia+pABh1σ.
Taking the operatorIa+δAB on both sides of equation (25), we have(26)Ia+δABϑσ=Ia+δABc2σ−a+Ia+δABIa+pABh1σ.
Now, replacingσ by ζ, we get(27)Ia+δABϑζ=Ia+δABc2ζ−a+Ia+δABIa+pABh1ζ.
By the second condition(ϑb=Ia+δABϑζ) together with equations (25) and (27), we get(28)c2=1ΛIa+δABIa+pABh1ζ−Ia+pABh1b.
Substitutingc1 and c2 in (24), we get (23).
Conversely, assume thatϑ satisfies (23). Then, by applying the operator Da+pABR on both sides of (23) and using Lemmas 3 and 4, we get(29)Da+pABRϑσ=Da+pABRσ−aΛIa+δABIa+pABh1ζ−Ia+pABh1b+Da+pABRIa+pABh1σ=h1σ.
Next, takingσ⟶a in (23), we get ϑa=0. On the contrary, applying Ia+δAB on both sides of (23), we get(30)Ia+δABϑσ=Ia+δABσ−aΛIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+δABIa+pABh1σ.
Replacingσ by ζ, we get(31)Ia+δABϑζ=Ia+δABζ−aΛIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+δABIa+pABh1ζ.
By the definition ofΛ, we can rewrite (31) as(32)Ia+δABϑζ=b−a−b−a−Ia+δABζ−ab−a−Ia+δABζ−aIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+δABIa+pABh1ζ=b−ab−a−Ia+δABζ−aIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+pABh1b=b−aΛIa+δABIa+pABh1ζ−Ia+pABh1b+Ia+pABh1b=ϑb.
Thus, the Atangana–Baleanu fractional integral conditions are satisfied.As a result of Lemma7, we get the following theorem.Theorem 3.
Letp∈(2,3],δ∈(0,1],Λ=(b−a)−Ia+δAB(ζ−a)≠0, and f:J×ℝ⟶ℝ be a continuous function. Then, ABR-type FDEs (4) are equivalent to the following FIE:(33)ϑσ=σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑsds−A5∫abb−sfs,ϑsds−A6Γ(p−2)∫abb−sp−1fs,ϑsds]+A5∫aσσ−sfs,ϑsds+A6Γ(p−2)∫aσσ−sp−1fs,ϑsds,where(34)A1=1−δ3−pBδBp−2,A2=1−δp−2BδBp−2,A3=δ3−pBδBp−2,A4=δp−2BδBp−2,A5=3−pBp−2,A6=p−2Bp−2.Proof.
In view of Lemma7, we have(35)ϑσ=σ−aΛIa+δABIa+pABfζ,ϑζ−Ia+pABfb,ϑb+Ia+pABfσ,ϑσ.
By using definitions ofIa+δAB in the case δ∈0,1 defined in (8) and Ia+pAB in the case p∈(2,3] defined in (18), we can rewrite equation (35) as follows:(36)ϑσ=σ−aΛ[1−δBδ3−pBp−2∫aζζ−sfs,ϑsds+p−2Bp−2Γp−2∫aζζ−sp−1fs,ϑsds+δBδΓδ∫aζζ−sδ−13−pBp−2∫ass−ηfη,ϑηdη+p−2Bp−2Γp−2∫ass−ηp−1fη,ϑηdηds−3−pBp−2∫abb−sfs,ϑsds−p−2Bp−2Γp−2∫abb−sp−1fs,ϑsds]+3−pBp−2∫aσσ−sfs,ϑsds+p−2Bp−2Γp−2∫aσσ−sp−1fs,ϑsds=σ−aΛ[1−δBδ3−pBp−2∫aζζ−sfs,ϑsds+1−δBδp−2Bp−2Γp−2∫aζζ−sp−1fs,ϑsds+3−pBp−2δBδΓδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+p−2Bp−2Γp−2δBδΓδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds−3−pBp−2∫abb−sfs,ϑsds−p−2Bp−2Γp−2∫abb−sp−1fs,ϑsds]+3−pBp−2∫aσσ−sfs,ϑsds+p−2Bp−2Γp−2∫aσσ−sp−1fs,ϑsds=σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑsds−A5∫abb−sfs,ϑsds−A6Γp−2∫abb−sp−1fs,ϑsds]+A5∫aσσ−sfs,ϑsds+A6Γp−2∫aσσ−sp−1fs,ϑsds.For simplicity, we set(37)ρ1=[A1ζ−a22+A2ζ−apΓp−1+A3ζ−aδ+1Γδ+1+A4ζ−aδ+pΓδ+p−1+A5b−a22+A6b−apΓp−1],ρ2=A5b−a22+A6b−apΓp−1,ρ3=[A7ζ−a+A8ζ−apΓp+1+A9ζ−aδ+1Γδ+1+A10ζ−aδ+pΓδ+p+1+A11b−a+A12b−apΓp+1],ρ4=A11b−a+A12b−apΓp+1.


## 4. Existence and Uniqueness of Solutions for Problem (4)

In this section, we devote our intention to prove the existence and uniqueness of solutions for ABR-type FDEs (4).Theorem 4.
Suppose thatf:J×ℝ⟶ℝ is a continuous function. Moreover, we assume that(38)fσ,u−fσ,u¯≤Lfu−u¯,Lf>0,u,u¯∈ℝ.
If(39)Z=Lfρ1b−aΛ+ρ2<1,then ABR-type FDEs (4) have a unique solution in J.Proof.
Consider the operatorΞ:CJ,ℝ⟶CJ,ℝ defined by(40)Ξϑσ=σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑsds−A5∫abb−sfs,ϑsds−A6Γ(p−2)∫abb−sp−1fs,ϑsds]+A5∫aσσ−sfs,ϑsds+A6Γ(p−2)∫aσσ−sp−1fs,ϑsds.
Define a closed ballΠφ as(41)Πφ=ϑ∈CJ,ℝ:ϑ≤φ,with radius φ≥Z1/1−Z, where(42)Z1=b−aΛρ1+ρ2ωf,where ωf=supσ∈Jfσ,0. Now, we show that ΞΠφ⊂Πφ. For all ϑ∈Πφ and σ∈J, we have(43)Ξϑσ≤σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γ(δ)∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γ(δ+p−2)∫aζζ−sδ+p−1fs,ϑsds+A5∫abb−sfs,ϑsds+A6Γ(p−2)∫abb−sp−1fs,ϑsds]+A5∫aσσ−sfs,ϑsds+A6Γ(p−2)∫aσσ−sp−1fs,ϑsds.
By (38), we have(44)fσ,ϑσ=fσ,ϑσ−fσ,0+fσ,0≤Lfϑσ+fσ,0≤Lfφ+ωf.
Hence,(45)Ξϑ≤Lfφ+ωfσ−aΛρ1+Lfφ+ωfρ2≤Lfb−aΛρ1+ρ2φ+b−aΛρ1+ρ2ωf=Zφ+Z1<φ.
Thus,ΞΠφ⊂Πφ. Next, we will show that Ξ is a contraction mapping. Let ϑ,ϑ^∈CJ,ℝ and σ∈J. Then, we have(46)Ξϑσ−Ξϑ^σ≤σ−aΛ[A1∫aζζ−sfs,ϑs−fs,ϑ^sds+A2Γp−2∫aζζ−sp−1fs,ϑs−fs,ϑ^sds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑη−fη,ϑ^ηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑs−fs,ϑ^sds+A5∫abb−sfs,ϑs−fs,ϑ^sds+A6Γp−2∫abb−sp−1fs,ϑs−fs,ϑ^sds]+A5∫aσσ−sfs,ϑs−fs,ϑ^sds+A6Γp−2∫aσσ−sp−1fs,ϑs−fs,ϑ^sds.
From our assumptions, we obtain(47)fs,ϑs−fs,ϑ^s≤Lfϑs−ϑ^s≤Lfϑ−ϑ^.
Hence,(48)Ξϑ−Ξϑ^≤Lfσ−aΛ[A1ζ−a22+A2ζ−apΓp−1+A3ζ−aδ+1Γδ+1+A4ζ−aδ+pΓδ+p−1+A5b−a22+A6b−apΓp−1]ϑ−ϑ^+LfA5σ−a22+A6σ−apΓp−1ϑ−ϑ^≤Lfρ1b−aΛ+Lfρ2ϑ−ϑ^=Zϑ−ϑ^.
Due toZ<1, we conclude that Ξ is a contraction operator. Hence, Theorem 1 implies that Ξ has a unique fixed point in J.Theorem 5.
Assume that the conditions in Theorem4 are satisfied. Then, ABR-type FDEs (4) have at least one solution in CJ,ℝ.Proof.
Let us consider the operatorΞ defined in Theorem 4. Now, we divide the operator Ξ into two operators Ξ1 and Ξ2 such that(49)Ξϑσ=Ξ1ϑσ+Ξ2ϑσ,where(50)Ξ1ϑσ=σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑsds−A5∫abb−sfs,ϑsds−A6Γp−2∫abb−sp−1fs,ϑsds],Ξ2ϑσ=A5∫aσσ−sfs,ϑsds+A6Γp−2∫aσσ−sp−1fs,ϑsds.
Let us consider a closed ballΠφ defined in Theorem 4. In order to apply Theorem 2, we will divide the proof into the following steps.Step 1.
We show thatΞ1ϑ+Ξ2ϑ˜∈Πφ for all ϑ,ϑ˜∈Πφ. For operator Ξ1, we have(51)Ξ1ϑσ≤σ−aΛ[A1∫aζζ−sfs,ϑsds+A2Γp−2∫aζζ−sp−1fs,ϑsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑsds+A5∫abb−sfs,ϑsds+A6Γ(p−2)∫abb−sp−1fs,ϑsds].
By (44), we get(52)Ξ1ϑ≤Lfφ+ωfb−aΛρ1.
In the same manner, one can prove that(53)Ξ2ϑ˜=Lfφ+ωfρ2.
Inequalities (52) and (53) give(54)Ξ1ϑ+Ξ2ϑ˜≤Ξ1ϑ+Ξ2ϑ≤Lfφ+ωfb−aΛρ1+Lfφ+ωfρ2≤Lfb−aΛρ1+ρ2φ+b−aΛρ1+ρ2ωf=Zφ+Z1<φ.
Thus,Ξ1ϑ+Ξ2ϑ˜∈Πφ.Step 2.
Ξ1 is a contraction map. Due to the operator Ξ being a contraction map, we conclude that Ξ1 is contraction too.Step 3.
Ξ2 is continuous and compact. Since f is continuous, Ξ2 is continuous too. Also, by (53), Ξ2 is uniformly bounded on Πφ. Now, we show that Ξ2Πφ is equicontinuous. For this purpose, let ϑ∈Πφanda≤σ1<σ2≤b.
Then, by using (44), we have(55)Ξ2ϑσ2−Ξ2ϑσ1≤A5∫aσ1σ2−s−σ1−sfs,ϑsds+∫σ1σ2σ2−sfs,ϑsds+A6Γp−2∫aσ1σ2−sp−1−σ1−sp−1fs,ϑsds+A6Γp−2∫σ1σ2σ2−sp−1fs,ϑsds≤Lfφ+ωfA5σ2−σ122+A6Γp−1σ2−ap−σ1−ap.
Thus,(56)Ξ2ϑσ2−Ξ2ϑσ1⟶0asσ2⟶σ1.
According to the above steps and Arzela–Ascoli theorem, we understand thatΞ2Πφ is relatively compact. Consequently, Ξ2 is completely continuous. Thus, by Theorem 2, we infer that ABR-type FDEs (4) have at least one solution in J.


## 5. Existence of a Unique Solution for Problem (5)

In this section, we devote our intention to proving the existence of a unique solution for ABC-type FDEs (5).Lemma 8.
Letp∈(1,2],δ∈(0,1], Λ=b−a−Ia+δABζ−a≠0, and h2∈CJ,ℝ. A function ϑ is a solution of the following ABC problem,(57)Da+pABCϑσ=h2σ,σ∈a,b,ϑa=0,ϑb=Ia+δABϑζ,ζ∈a,b,if and only if ϑ satisfies the following fractional integral equation:(58)ϑσ=σ−aΛIa+δABIa+pABh2ζ−Ia+pABh2b+Ia+pABh2σ,where Ia+pAB was defined in (21).Proof.
Assume thatϑ is the solution to the first equation of (57). Then, via Lemma 6 and by the same way in Lemma 7, one can prove that the solution of equation (57) is given as (58), where Ia+pABh2σ is defined in (21).Remark 2.
The AB fractional integralIa+pAB used in Lemma 7 was defined in (18), while the AB fractional integral Ia+pAB used in Lemma 7 was defined in (21).Theorem 6.
Letp∈(1,2],δ∈(0,1],Λ=(b−a)−Ia+δAB(ζ−a)≠0, and f:J×ℝ⟶ℝ⟶ be a continuous function. Then, in the light of Lemma 6, ABC-type FDEs (5) are equivalent to the following FIE:(59)ϑσ=σ−aΛ[A7∫aζfs,ϑsds+A8Γp∫aζζ−sp−1s,ϑsds+A9Γδ∫aζζ−sδ−1∫asfη,ϑηdηds+A10Γ(δ+p)∫aζζ−sδ+p−1s,ϑsds−A11∫abfs,ϑsds−A12Γp∫abb−sp−1s,ϑsds]+A11∫aσfs,ϑsds+A12Γp∫aσσ−sp−1s,ϑsds,where(60)A7=1−δ2−pBδBp−1,A8=1−δp−1BδBp−1,A9=δ2−pBδBp−1,A10=δp−1BδBp−1,A11=2−pBp−1,A12=p−1Bp−1.Proof.
In view of Lemma8, we have(61)ϑσ=σ−aΛIa+δABIa+pABfζ,ϑζ−Ia+pABfb,ϑb+Ia+pABfσ,ϑσ.
By using definitions ofIa+δAB in the case δ∈0,1 defined in (8) and Ia+pAB in the case p∈(1,2] defined in (21), we can rewrite equation (61) as follows:(62)ϑσ=σ−aΛ[1−δBδ2−pBp−1∫aζfs,ϑsds+p−1Bp−2Γp∫aζζ−sp−1s,ϑsds+δBδΓδ∫aζζ−sδ−12−pBp−1∫asfη,ϑηdη+p−1Bp−2Γp∫ass−ηp−1fη,ϑηdηds−2−pBp−1∫abfs,ϑsds−p−1Bp−2Γp∫abb−sp−1s,ϑsds]+2−pBp−1∫aσfs,ϑsds+p−1Bp−2Γp∫aσσ−sp−1s,ϑsds=σ−aΛ[A7∫aζfs,ϑsds+A8Γp∫aζζ−sp−1s,ϑsds+A9Γδ∫aζζ−sδ−1∫asfη,ϑηdηds+A10Γ(δ+p)∫aζζ−sδ+p−1s,ϑsds−A11∫abfs,ϑsds−A12Γp∫abb−sp−1s,ϑsds]+A11∫aσfs,ϑsds+A12Γp∫aσσ−sp−1s,ϑsds.Theorem 7.
Suppose thatf:J×ℝ⟶ℝ is a continuous function and there exists a constant number Lf>0 such that fσ,u−fσ,u¯≤Lfu−u¯ for any u,u¯∈J. If(63)Z3=Lfρ3b−aΛ+ρ4<1,then ABC-type FDEs (5) have a unique solution in J.Proof.
Define the operatorϜ:CJ,ℝ⟶CJ,ℝ by(64)Ϝϑσ=σ−aΛ[A7∫aζfs,ϑsds+A8Γp∫aζζ−sp−1s,ϑsds+A9Γδ∫aζζ−sδ−1∫asfη,ϑηdηds+A10Γδ+p∫aζζ−sδ+p−1s,ϑsds−A11∫abfs,ϑsds−A12Γp∫abb−sp−1s,ϑsds]+A11∫aσfs,ϑsds+A12Γp∫aσσ−sp−1s,ϑsds.
Define a closed ballΘϱ as(65)Θϱ=ϑ∈CJ,ℝ:ϑ≤ϱ,with radius ϱ≥Z2/1−Z3, where(66)Z2=b−aΛρ3+ρ4ωf.
Now, we will show thatΞΘϱ⊂Θϱ. For all ϑ∈Θϱ and σ∈J and by (47), we have(67)Ϝϑ≤Lfϱ+ωfσ−aΛρ3+Lfϱ+ωfρ4≤Lfb−aΛρ3+ρ4ϱ+b−aΛρ3+ρ4ωf=Z3ϱ+Z2<ϱ.
Thus,ϜΘϱ⊂Θϱ. Next, we need to prove that Ϝ is a contraction map. Let ϑ,ϑ^∈CJ,ℝ and σ∈J. Then,(68)Ϝϑ(σ)−Ϝϑ^σ≤σ−aΛ[A7∫aζfs,ϑs−fs,ϑ^sds+A8Γp∫aζζ−sp−1fs,ϑs−fs,ϑ^sds+A9Γδ∫aζζ−sδ−1∫asfη,ϑη−fη,ϑ^ηdηds+A10Γδ+p∫aζζ−sδ+p−1fs,ϑs−fs,ϑ^sds−A11∫abfs,ϑs−fs,ϑ^sds−A12Γp∫abb−sp−1fs,ϑs−fs,ϑ^sds]+A11∫aσfs,ϑs−fs,ϑ^sds+A12Γp∫aσσ−sp−1fs,ϑs−fs,ϑ^sds.
From our assumption, we obtain(69)fs,ϑs−fs,ϑ^s≤Lfϑs−ϑ^s≤Lfϑ−ϑ^.
Hence,(70)Ϝϑ−Ϝϑ^≤Lfσ−aΛ[A7ζ−a+A8ζ−apΓp+1+A9ζ−aδ+1Γδ+1+A10ζ−aδ+pΓδ+p+1−A11b−a+A12b−apΓp+1]ϑ−ϑ^+LfA11σ−a+A12σ−apΓp+1ϑ−ϑ^≤Lfρ3b−aΛ+Lfρ4ϑ−ϑ^=Z3ϑ−ϑ^.
Due toZ3<1, we conclude that Ϝ is a contraction operator. Hence, by Theorem 1, we conclude that Ϝ has a unique fixed point in J. Consequently, ABC-type FDEs (5) have a unique solution in J.


## 6. Ulam–Hyers Stability

In this part, we will discuss two kinds of stability results for ABR-type FDEs (4) and ABC-type FDEs (5), namely, UH and GUH stabilities. Before that, for ε>0, we consider the following inequality:(71)Da+pABRϑ^σ−Fϑ^σ≤ε,σ∈J.Definition 4.
The problem ABR-type FDEs (4) is Ulam–Hyers stable if there exists a real number Cf>0 such that, for each ε>0 and for each solution ϑ^∈CJ,ℝ of inequality (71), there exists a unique solution ϑ∈CJ,ℝ of ABR-type FDEs (4) with(72)ϑ^σ−ϑσ≤Cfε.
Also, ABR-type FDEs (4) have generalized Ulam–Hyers stability if we can find φf:ℝ+⟶ℝ+ with φf0=0 such that(73)ϑ^σ−ϑσ≤φfε.Remark 3.
Letϑ^∈CJ,ℝ be the solution of inequality (71) if and only if we have a function h∈CJ,ℝ which depends on ϑ such that(i)
hσ≤ε for all σ∈J(ii)
Da+pABRϑ^σ=fσ,ϑ^σ+hσ, σ∈JLemma 9.
Ifϑ∈CJ,ℝ is a solution to inequality (71), then ϑ satisfies the following inequality:(74)ϑ^σ−Ψϑ^−A5∫aσσ−sfs,ϑ^sds−A6Γp−2∫aσσ−sp−1fs,ϑ^sds≤ερ1b−aΛ+ρ2,where(75)Ψϑ^=σ−aΛ[A1∫aζζ−sfs,ϑ^sds+A2Γp−2∫aζζ−sp−1fs,ϑ^sds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑ^ηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑ^sds−A5∫abb−sfs,ϑ^sds−A6Γp−2∫abb−sp−1fs,ϑ^sds].Proof.
In view of Remark3, we have(76)Da+pABRϑ^σ=fs,ϑ^s+hσ,ϑ^a=0,ϑ^b=Ia+δABϑ^ζ.
Then, by Theorem3, we get(77)ϑ^σ=σ−aΛ[A1∫aζζ−sfs,ϑ^s+hsds+A2Γp−2∫aζζ−sp−1fs,ϑ^s+hsds+A3Γδ∫aζζ−sδ−1∫ass−ηfη,ϑ^η+hηdηds+A4Γδ+p−2∫aζζ−sδ+p−1fs,ϑ^s+hsds−A5∫abb−sfs,ϑ^s+hsds−A6Γp−2∫abb−sp−1fs,ϑ^s+hsds]+A5∫aσσ−sfs,ϑ^s+hsds+A6Γp−2∫aσσ−sp−1fs,ϑ^s+hsds,which implies(78)ϑ^σ−Ψϑ^−A5∫aσσ−sfs,ϑ^sds−A6Γp−2∫aσσ−sp−1fs,ϑ^sds≤σ−aΛ[A1∫aζζ−shsds+A2Γ(p−2)∫aζζ−sp−1hsds+A3Γδ∫aζζ−sδ−1∫ass−ηhηdηds+A4Γδ+p−2∫aζζ−sδ+p−1hsds+A5∫abb−shsds+A6Γp−2∫abb−sp−1hsds]+A5∫aσσ−shsds+A6Γ(p−2)∫aσσ−sp−1hsds≤ερ1b−aΛ+ρ2.Theorem 8.
Suppose that (38) holds. If(79)LfA5b−a22+A6b−apΓp−1<1,then ABR-type FDEs (4) are Ulam–Hyers stable.Proof.
Letε>0 and ϑ^∈CJ,ℝ be a function which satisfies inequality (71), and let ϑ∈CJ,ℝ be a unique solution of the following problem:(80)Da+pABRϑ^σ=fσ,ϑ^σ,σ∈a,b,p∈(2,3]ϑa=ϑ^a=0,ϑb=ϑ^b=Ia+δABϑ^ζ.
Then, by Theorem3, we get(81)ϑ^σ=Ψϑ^+A5∫aσσ−sfs,ϑ^sds−A6Γp−2∫aσσ−sp−1fs,ϑ^sds.
Sinceϑa=ϑ^a=0 and ϑb=ϑ^b=Ia+δABϑ^ζ, Ψϑ^=Ψϑ, and hence, by Lemma 9 and (47), we have(82)ϑ^σ−ϑσ≤ϑ^σ−Ψϑ^−A5∫aσσ−sfs,ϑ^sds−A6Γp−2∫aσσ−sp−1fs,ϑ^sds+A5∫aσσ−sfs,ϑ^s−fs,ϑsds+A6Γp−2∫aσσ−sp−1fs,ϑ^s−fs,ϑ^sds≤ερ1b−aΛ+ρ2+LfA5b−a22+A6b−apΓp−1ϑ−ϑ^.
Thus,(83)ϑ−ϑ^≤Cfε,where(84)Cf=ρ1b−a2/Λ+ρ21−LfA5b−a2/2+A6b−ap/Γp−1.
Now, by choosingφfε=Cfε such that φf0=0, ABR-type FDEs (4) have generalized Ulam–Hyers stability.Theorem 9.
Suppose that (38) holds. If(85)LfA11b−a+A12b−apΓp+1<1,then ABC-type FDEs (5) are Ulam–Hyers stable.Proof.
By the same technique of Theorem8, one can prove that(86)ϑ−ϑ^≤Cf∗ε,where(87)Cf∗=ρ3b−a/Λ+ρ41−LfA11b−a+A12b−ap/Γp+1.
Now, by choosingφfε=Cf∗ε such that φf0=0, ABC-type FDEs (5) have generalized Ulam–Hyers stability.


## 7. An Example

In this section, we justify the validity of Theorems4, 5, and 7–9 through an example.Example 1.
Forp∈(2,3], we consider the following ABR fractional problem:(88)D0+7/2ABRϑσ=σ210eσ−1ϑσ1+ϑσ,σ∈0,1,ϑ0=0,ϑ1=I0+1/2ABϑ12.
Here,p=(7/2)∈(2,3],a=0,b=1,δ=(1/2),ζ=(1/2), and fσ,ϑσ=σ2/10eσ−1ϑσ/(1+|ϑσ|). Let σ∈0,1,u,u¯∈ℝ. Then,(89)fσ,u−fσ,u¯≤110u−u¯.
Therefore, hypothesisH1 holds with Lf=1/10. Also, ρ1≃1.23,ρ2≃2,Λ≃0.66, and Z≃0.38<1. Then, all conditions in Theorem 4 are satisfied, and hence, ABR-type FDEs (4) have a unique solution. For every ε=maxε1,ε2>0 and each ϑ^∈CJ,ℝ satisfying the inequality(90)Da+pABRϑ^σ−Fϑ^σ≤ε,there exists a solution ϑ∈CJ,ℝ of ABR-type FDEs (4) with(91)ϑ^−ϑ1≤Cfε,where(92)Cf=ρ1b−a/Λ+ρ21−LfA5b−a2/2+A6σ−ap/Γp−1>0.
Therefore, all conditions in Theorem9 are satisfied, and hence, ABR-type FDEs (4) are UH stable.Example 2.
Forp∈(1,2], we consider the following ABC fractional problem:(93)D0+3/2ABCϑσ=σ210eσ−1ϑσ1+ϑσ,σ∈0,1,ϑ0=0,ϑ1=I0+1/2ABϑ12.
Here,p=(3/2)∈(1,2],a=0,b=1,δ=(1/2),ζ=(1/2), and fσ,ϑσ=σ2/10eσ−1ϑσ/(1+|ϑσ|). Let σ∈0,1,u,u¯∈ℝ. Then,(94)fσ,u−fσ,u¯=σ210eσ−1u1+u−u¯1+u¯≤110u−u¯.
Therefore, hypothesisH1 holds with Lf=1/10. Also, ρ3≃1.4,ρ4≃2.5,Λ≃0.66, and(95)Z3=1101.40.66+2.5≈0.47<1.
Then, all conditions in Theorem7 are satisfied, and hence, ABC-type FDEs (5) have a unique solution. For every ε=maxε1,ε2>0 and each ϑ^∈CJ,ℝ satisfying(96)Da+pABCϑ^σ−Fϑ^σ≤ε,there exists a solution ϑ∈CJ,ℝ of ABC-type FDEs (5) with(97)ϑ^−ϑ1≤Cf∗ε,where(98)Cf∗=ρ3b−a/Λ+ρ41−LfA11b−a+A12σ−ap/Γp+1>0.
Therefore, all conditions in Theorem9 are satisfied, and hence, ABC-type FDEs (5) are UH stable.


## 8. Conclusion Remarks

In recent interest, the theory of fractional operators in the frame of Atangana and Baleanu is novel and significant; thus, there are some researchers who studied and developed some qualitative properties of solutions of FDEs involving such operators. On the contrary, there are some important numerical approaches regarding nonsingular kernels that are explained in the introduction. For this purpose, the sufficient conditions of the existence and uniqueness of solutions for two classes of nonlinear Atangana–Baleanu FDEs on the intervala,b, subjected to integral conditions, have been developed and investigated. Furthermore, the stability results through mathematical analysis techniques have been analyzed. Two examples are provided to justify our results.Observe that our approach used in this work is new because we prove the existence, uniqueness, and UH stability results without using the semigroup property and relies on a minimum number of hypotheses. Our approach was based on the reduction of the proposed problem into the fractional integral equation and using some standard fixed point theorems due to Banach type and Krasnoselskii type. Furthermore, through mathematical analysis techniques, we analyzed the stability results in the UH sense. An example was provided to justify the main results. In fact, our outcomes extended those in [36, 37]. The supposed problem with given integroderivative boundary conditions can describe some mathematical models of real and physical processes in which some parameters are frequently acclimated to appropriate circumstances. So, the value of these parameters can change the impacts of fractional integrals and derivatives. The main results are illustrated with a numerical example. In future works, we study the new numerical results regarding this operator with the higher order.



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