# A Study onψ-Caputo-Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions

Authors: Fouad Fredj; Hadda Hammouche; Mohammed S. Abdo; Wedad Albalawi; Abdulrazak H. Almaliki
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this research paper, we investigate the existence, uniqueness, and Ulam–Hyers stability of hybrid sequential fractional differential equations with multiple fractional derivatives ofψ-Caputo with different orders. Using an advantageous generalization of Krasnoselskii’s fixed point theorem, we establish results of at least one solution, whereas the uniqueness of solution is derived via Banach’s fixed point theorem. Besides, the Ulam–Hyers stability for the proposed problem is investigated by applying the techniques of nonlinear functional analysis. In the end, we provide an example to illustrate the applicability of our results.


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## Body

## 1. Introduction

Over the last couple of years, the concept of fractional differential equations (FDEs) has been validated as being an effective and powerful gadget to model complex and real world phenomena due to their wide range of applications in several fields of sciences and engineering; see [1–5] and the references therein. Recently, FDEs gained the attention of mathematicians and researchers working in different disciplines of science and technology, which resulted in plenty of research papers that have been carried out on FDEs. That made a valuable contribution ranging from the qualitative theory of the solutions of FDEs, such as existence, uniqueness, stability, and controllability to the numerical analysis. Speaking in this context, the stability analysis of functional and differential equations is important in many applications, such as optimization and numerical analysis, where computing the exact solution is rather hard. There are various kinds of stability; one of those types has recently received considerable attention from many mathematicians, so-called Ulam–Hyers (U-H) stability. The source of U-H stability goes back to 1940 by Ulam [6], next by Hyers [7]. A variety of works have been done by many authors in regard of the U-H stability of FDEs; for example, the authors in [8] studied the existence and stability results for implicit FDE. Some recent developments in Ulam’s type stability are discussed by Belluot et al. [9]. Ibrahim, in [10], obtained the generalized U-H stability for FDEs. Some approximate analytical methods for solving FDEs can be found in [11–14]; also, computational analyses of some fractional dynamical and biological models were investigated recently, see [15, 16].On the contrary, quadratic perturbation of nonlinear differentials, also known as the hybrid differential equations, had rapid progress over the last years; this is due to its importance, which lies in the fact that they include perturbations that facilitate the study of such equations by using the perturbation techniques. These equations are also considered as a particular case in dynamic systems. The starting point for this field is when Dhage and Lakshemikantham [17] formulated a hybrid differential equation, where they investigated the existence and uniqueness of the solutions to the following hybrid equation:(1)ddθϰθgθ,ϰθ=fθ,ϰθ,a.e.θ∈θ0,θ0+℧,θ0∈ℝ,℧>0,ϰθ0=ϰ0,ϰ0∈ℝ.Their results were based on the fixed point theorem (FPT) for the product of two operators in Banach algebra.In 2011, Zhao et al. [18] extended Dhage’s work [17] to fractional order and studied the existence of solutions to the following Riemann–Liouville (RL)-type hybrid FDEs:(2)Dpϰθhθ,ϰθ=fθ,ϰθ,a.e.θ∈0,℧,0<p≤1,ϰ0=0.After several years, Sitho et al. [19] derived a new existence result for the following hybrid sequential integro-differential equations:(3)DpDqϰθ−∑i=1nIηigiθ,ϰθhθ,ϰθ=fθ,ϰθ,Iγϰθ,θ∈0,℧,0<p,q≤1,ϰ0=0,Dqϰ0=0.In [20], Baitiche et al. studied the existence of solutions for the following hybrid sequential FDEs:(4)Dp+1c+λDpcϰθhθ,ϰθ=fθ,ϰθ,a.e.θ∈0,1,0<p≤2,ϰ0=ϰη=ϰ1=0,0<η<1.They generalized Darbo’s FPT for the product of two operators associated with measures of noncompactness.Some existence results forψ-Caputo-type hybrid FDEs are obtained in [21, 22]. For recent analysis techniques in FDEs involving generalized Caputo FD, we refer to [23, 24]. Just recently, Boutiara et al. [25] discussed some qualitative analyses to the following fractional hybrid system:(5)Dpi,ψϰiθ−∑k=1nIηik,ψgikθ,ϰθhiθ,ϰθ=fiθ,ϰ1θ,ϰ2θ,θ∈a,b,0<pi≤1,ηi>0,ϰia=0,i=1,2.The above findings motivated us to study the existence, uniqueness, and U-H stability of solutions for the followingψ-Caputo hybrid fractional sequential integro-differential equation (for short, ψ-Caputo HFSIDE):(6)LψpDq;ψcϰθ−∑i=1mIηi;ψgiθ,ϰθhθ,ϰθ=fθ,ϰθ,δIγ;ψϰθ,θ∈ℑ=0,℧,endowed with the hybrid fractional integral boundary conditions:(7)ϰ0=0,cDq;ψϰ0=0,Dq;ψcϰθ−∑i=1mIηi;ψgiθ,ϰθhθ,ϰθθ=℧=ρDq;ψcϰθ−∑i=1mIηi;ψgiθ,ϰθhθ,ϰθθ=ξ,0<ρ,ξ<℧,with(8)Lψp=Dp;ψc+λDp−1;ψc,where 1<p≤2, 0<q≤1, Dp;ψc and Dq;ψc denote the ψ-Caputo FD of order p,q, respectively, and Iηi;ψ and Iγ;ψ are the ψ-RL fractional integral of order ηi>0,i=1,…,m and γ>0, respectively, h∈Cℑ×ℝ,ℝ∖0, f∈Cℑ×ℝ2,ℝ, gi∈Cℑ×ℝ,ℝ with gi0,0=0, and λ is appropriate positive real constants.The modernity of our proposed problem in contrast with past problems is that, in this work, we consider a kind of general case of boundary value problems in a setup of aψ-Caputo HFSIDE delineated by (6) and (7). To be sure, the advantage of this work is that the applied FD has the freedom of choice of the kernel ψ which makes it conceivable to bring together and cover most of the preceding results on hybrid FDEs.The research paper is organized as follows. Section2 presents some basic mathematical pieces of knowledge required throughout the paper. The main results for ψ-Caputo HFSIDE (6), (7) are proved in Section 3. In Section 4, we give an illustrative numerical example, and Section 5 is related to a brief conclusion.


## 2. Preliminaries

LetE=Cℑ,R and E∗=Cℑ,ℝ∖0 be Banach spaces of continuous real-valued functions defined on ℑ. Clearly, E is a Banach algebra with the norm(9)ϰ=supθ∈ℑϰθ,and multiplication(10)ϰϰ¯θ=ϰθϰ¯θ,∀θ∈ℑ.Let an increasing functionψ:ℑ⟶ℝ satisfy ψ′θ≠0, for all θ∈ℑ. For effortlessness, we set Ψrθ,s≔ψ′sψθ−ψsr and Ψ0rθ=ψθ−ψ0r.Definition 1 (see [3]).
Theψ-RL fractional integral of orderp>0of an integrable functionϰ:a,∞⟶ℝis defined by(11)Ip;ψϰθ=1Γp∫aθΨp−1θ,sϰsds,a<θ.Definition 2 (see [26]).
Theψ-Caputo FD of orderpn−1<p<n∈ℕof a functionϰ∈Cn0,∞is defined by(12)Dp;ψcϰθ=1Γp−n∫aθΨp−n−1θ,sDψnϰsds,a<θ,where n=p+1 and Dψn=1/ψ′θd/dθn.
In case, if1<p≤2, we have(13)Dp;ψcϰθ=1Γp−2∫aθΨp−3θ,sDψ2ϰsds,a<θ.As a special case, ifψθ=θ, then the above definitions reduce to the well-known classical fractional definitions; for more details, see [1,3].Lemma 1 (see [26]).
Letp>0. The following holds:(i)
Ifϰ∈E, then(14)Dp;ψcIp;ψϰθ=ϰθ,θ∈ℑ.(ii)
Ifϰ∈En,n−1<p<n, then(15)Ip;ψDp;ψcϰθ=ϰθ−∑k=0n−1ckΨ0kθ,θ∈ℑ,where ck=Dψkϰ0/k!.Concerning the applied FPTs, we will suffice with indication to [27, 28].


## 3. Main Results

Lemma 2.
Let1<p≤2and0<q≤1. For any functionsF∈E,H∈E∗, andGi∈EwithGi0=0,i=1,…,2, the following linear fractional BVP,(16)LψpDq;ψcϰθ−∑i=1mIηi;ψGiθHθ=Fθ,θ∈ℑ,supplemented with the conditions,(17)ϰ0=0,cDq;ψϰ0=0,Dq;ψcϰθ−∑i=1mIηi;ψGiθHθθ=℧=ρDq;ψcϰθ−∑i=1mIηi;ψGiθHθθ=ξ,0<ρ,ξ<℧,has a unique solution, that is,(18)ϰθ=Iq;ψHs∫0sIp−1;ψFτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψFτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψFτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψHs1−e−λΨ0sθ+∑i=1mIηi+q;ψGiθ,where Δ=1+ρe−λΨ0ξ−e−λΨ0℧−ρ≠0, λ∈ℝ+.proof.
Applying theψ-RL fractional integral of order p−1 to both sides of (16) and using Lemma 1, we obtain(19)D1;ψcDq;ψcϰθ−∑i=1mIηi;ψGiθHθ+λDq;ψcϰθ−∑i=1mIηi;ψGiθHθ=Ip−1;ψFθ+c0,c0∈ℝ.
By multiplyingψ′θeλΨ0θ to both sides of (16), we find that(20)eλΨ0θddθDq;ψcϰθ−∑i=1mIηi;ψGiθHθ+λψ′θeλΨ0θDq;ψcϰθ−∑i=1mIηi;ψGiθHθ=ψ′θeλΨ0θIp−1;ψFθ+c0ψ′θeλΨ0θ,c0∈ℝ.
On the contrary, we have(21)ddθDq;ψcϰθ−∑i=1mIηi;ψGiθHθ.eλΨ0θ=eλΨ0θddθDq;ψcϰθ−∑i=1mIηi;ψGiθHθ+λψ′θeλΨ0θDq;ψcϰθ−∑i=1mIηi;ψGiθHθ.
From (20) and (21), we find that(22)ddθDq;ψcϰθ−∑i=1mIηi;ψGiθHθ.eλΨ0θ=Ip−1;ψFθψ′θeλΨ0θ+c0ψ′θeλΨ0θ.
Integrating from 0 toθ and using the fact that Gi0=0, i=1,…,m, and from the condition Dq;ψcϰ0=0 in (17), we have(23)Dq;ψcϰθ−∑i=1mIηi;ψGiθHθ.eλΨ0θ=∫0θIp−1;ψFτψ′τeλΨ0τdτ+c0λe−λΨ0θ−1+c1,where c1∈ℝ; by multiplying e−λΨ0θ to both sides, we obtain(24)Dq;ψcϰθ−∑i=1mIηi;ψGiθHθ=∫0θIp−1;ψFτψ′τe−λΨ0θ−Ψ0τdτ+c0λ1−e−λΨ0θ+c1e−λΨ0θ.
Next, applyingψ-RL fractional integral of order q to both sides of (24) and using Lemma 1, we obtain(25)ϰθ=Iq;ψHs∫0sIp−1;ψFτψ′τe−λΨ0s−Ψ0τdτ+c0λ1−e−λΨ0s+c1e−λΨ0sθ+∑i=1mIηi+q;ψGisθ+c2,where c2∈ℝ with the help of conditions Dq;ψcϰ0=0, ϰ0=0, and Gi0=0, i=1,…,m; we find c1=c2=0. Then, we apply the third condition of (17) in (25), and we obtain(26)∫0℧Ip−1;ψFτψ′τe−λΨ0℧−Ψ0τdτ+c0λ1−e−λΨ0℧=ρ∫0ξIp−1;ψFτψ′τe−λΨ0ξ−Ψ0τdτ+c0λ1−e−λΨ0ξ.
Some computations give us(27)c0=λΔ∫0℧Ip−1;ψFτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψFτψ′τe−λΨ0ξ−Ψ0τdτ.
Insertingc0,c1, and c2 in (25), leads to solution (18).
Conversely, by Lemma1 and by taking Dq;ψc on both sides of (25), we obtain(28)Dq;ψcϰθ−∑i=1mIηi;ψGiθHθ=∫0θIp−1;ψFτψ′τe−λΨ0θ−Ψ0τdτ+∫0℧Ip−1;ψFτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψFτψ′τe−λΨ0ξ−Ψ0τdτ1−e−λΨ0θΔ.
Next, operatingDp;ψc+λDp−1;ψc on both sides of the above equation, with the help of Lemma 1, we obtain(29)Dp;ψc+λDp−1;ψcDq;ψcϰθ−∑i=1mIηi;ψGiθHθ=Dp;ψc+λDp−1;ψc∫0θIp−1;ψFτψ′τe−λΨ0θ−Ψ0τdτ+∫0℧Ip−1;ψFτψ′τ×e−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψFτψ′τe−λΨ0ξ−Ψ0τdτ1−e−λΨ0θΔ=Dp;ψc+λDp−1;ψc∫0θIp−1;ψFτψ′τe−λΨ0θ−Ψ0τdτ−c0λ=Dp;ψce−λΨ0θ−c0Dp−1;ψce−λΨ0θ=Dp−1;ψc1ψ′θddθ+λ∫0θIp−1;ψFτψ′τe−λΨ0θ−Ψ0τdτ=Dp−1;ψcIp−1;ψFθ=Fθ.
Now, it remains to review the boundary conditions (17) of our problem (16). Substituting θ=0 in (18) with the fact that gi0=0,i=1,…,m, leads to u0=0. Next, we apply Dq;ψc on (18); then, we substitute θ=0; it follows that Dq;ψcϰ0=0. Substituting θ=℧ and θ=ξ, we find that the two resulting equations are equal, and from it, we get that(30)Dq;ψcϰθ−∑i=1mIηi;ψGiθHθθ=℧=ρDq;ψcϰθ−∑i=1mIηi;ψGiθHθθ=ξ.
This means thatϰθ satisfies (16) and (17). Therefore, ϰθ is solution of problem (16) and (17).Lemma 3.
ForF,H∈E,we have(1)
∫0θΨq−1θ,s/ΓqHs∫0s∫0τΨp−2τ,σ/Γp−1Fσdσψ′τe−λΨ0s−Ψ0τdτds≤Ψ0p+q−1℧/λΓp+qΓq1−e−λΨ0θHF(2)
∫0℧∫0τΨp−2θ,σ/Γp−1Fσdσψ′τe−λΨ0℧−Ψ0τdτ≤Ψ0p−1℧/λΓp1−e−λΨ0℧F(3)
∫0ξ∫0τψ′p−2θ,σ/Γp−1Fσdσψ′τe−λΨ0ξ−Ψ0τdτ≤Ψ0p−1ξ/λΓp1−e−λΨ0ξFproof.
To prove property(1), we have(31)∫0τΨp−2θ,σΓp−1dσ=Ψ0p−1τΓp,∫0sΨ0p−1τΓpψ′τe−λΨ0s−Ψ0τdτ≤Ψ0p−1sλΓp∫0sψ′τe−λΨ0s−Ψ0τdτ=Ψ0p−1sλΓp1−e−λΨ0s.
From the above integrals and left side of(1), we obtain(32)∫0θΨq−1θ,sΓqHs∫0s∫0τΨp−2θ,σΓp−1Fσdσψ′τe−λΨ0s−Ψ0τdτds≤HF∫0θΨq−1θ,sΓqΨ0p−1sλΓp1−e−λΨ0sds≤HF1−e−λΨ0℧λΓp∫0θΨq−1θ,sΓqΨ0p−1sds≤Ψ0p+q−1℧λΓp+q1−e−λΨ0℧HF.
The proofs of properties(2) and (3) are similar to proof of (1).Now, we consider the following assumptions:(H1)h∈Cℑ×ℝ,ℝ∖0 and f∈Cℑ×ℝ2,ℝ, and there exist positive functions Lθ and Mθ, such that(33)hθ,ϰ−hθ,ϰ∗≤Lθϰ−ϰ∗,fθ,ϰ1,ϰ2−fθ,ϰ1∗,ϰ2∗≤Mθϰ1−ϰ1∗+ϰ2−ϰ2∗,forθ∈ℑ and ϰ1,ϰ2,ϰ1∗,ϰ2∗∈ℝ.(H2) There exist functionsφi,χ,ϑ∈E such that(34)giθ,ϰ≤φiθ,for eachθ,ϰ∈ℑ×ℝ,hθ,ϰ≤χθ,for eachθ,ϰ∈ℑ×ℝ,fθ,ϰ,ϰ¯≤ϑθ,for eachθ,ϰ,ϰ¯∈ℑ×ℝ×ℝ.(H3) There exist0<Λ and ϒ<1, such that(35)1−e−λΨ0℧λΛχM+ϒLϑ<1,where(36)Λ=Ψ0p+q−1℧Γp+q+δΓγ−1Ψ0γ+p+q−2℧Γγ+p+q−1+Ψ0q℧ΔΓq+1×1−e−λΨ0℧Ψ0p−1℧Γp+δΓγ−1Ψ0γ+p−2℧Γγ+p−1+ρ1−e−λΨ0ξΨ0p−1ξΓp+δΓγ−1Ψ0γ+p−2ξΓγ+p−1,ϒ=Ψ0p+q−1℧Γp+q+Ψ0q℧ΔΓq+1Γp−11−e−λΨ0℧Ψ0p−2℧+ρ1−e−λΨ0ξΨ0p−2ξ.


### 3.1. Existence of Solutions

In this section, we prove the existence of a solution for problems (6) and (7) by applying Dhage FPT [27].Theorem 1.
Suppose (H1)–(H3) hold; then, problems (6) and (7) have at least one solution inE.proof.
First, we setsupθ∈ℑLθ=L,supθ∈ℑMθ=M,supθ∈ℑχθ=χ,supθ∈ℑϑθ=ϑ, and supθ∈ℑφiθ=φ,i=1,2,…,m.
Now, we defineBr⊂E as(37)Br=ϰ∈E:ϰ≤r.
Define two operatorsC:E⟶E and D:E⟶E as(38)Cϰθ=1Γp−1∫0θψ′sψθ−ψsp−2fs,ϰs,δIγ;ψϰsds,θ∈ℑ,Dϰθ=hθ,ϰθ,θ∈ℑ.
Then, using assumptions (H1)-(H2), we have, forϰ,ϰ¯∈Br and each θ∈ℑ,(39)Cϰθ−Cϰ¯θ≤1ΓSp−1∫0θΨp−2θ,sfs,ϰs,δIγ;ψϰs−fs,ϰ¯s,δIγ;ψϰ¯sds≤1Γp−1∫0θΨp−2θ,sMsϰs−ϰ¯s+δΓγ∫0sΨγ−1τ,sϰτ−ϰ¯τdτds≤Mϰ⋅−ϰ¯⋅1Γp−1∫0θΨp−2θ,s1+δΓγ+1Ψ0γsds≤1ΓpΨ0p−1θ+δΓγ−1Γγ+p−1Ψ0γ+p−2θMϰ⋅−ϰ¯⋅,Cϰθ≤Ψ0p−2θΓp−1ϑ.
Also,(40)Dϰθ−Dϰ¯θ≤Lϰ⋅−ϰ¯⋅,Dϰθ≤χ.
Now, we also consider two operatorsA:E⟶E and B:Br⟶E defined by(41)Aϰθ=Iq;ψDϰs∫0sCϰτψ′τe−λΨ0s−Ψ0τdτθ+1ΔIq;ψDϰs1−e−λΨ0sθ×∫0℧Cϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ,Bϰθ=∑i=1mIηi+q;ψgis,ϰsθ.
We need to prove thatA and B satisfy all assumptions of Dhage’s theorem [27]. This can be proven in the forthcoming steps:Step 1:
A is a contraction map. Indeed, let ϰθ,ϰ¯θ∈Br. Then,(42)Aϰθ−Aϰ¯θ≤Iq;ψDϰs∫0sCϰτ−Cϰ¯τψ′τe−λΨ0s−Ψ0τdτθ+Iq;ψDϰs−Dϰ¯s%∫0sCϰ¯τψ′τe−λΨ0s−Ψ0τdτθ+1ΔIq;ψDϰs%1−e−λΨ0sθ∫0℧%Cϰτ−Cϰ¯τψ′τ×e−λΨ0℧−Ψ0τdτ+ρ∫0ξCϰτ−Cϰ¯τ%ψ′τe−λΨ0ξ−Ψ0τdτ%+1ΔIq;ψDϰs−D%ϰ¯s1−e−λΨ0sθ∫0℧Cϰτψ′τ×e−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ.
Using Lemma3 and hypotheses (H1)-(H2), we obtain(43)Aϰθ−Aϰ¯θ≤1−e−λΨ0℧λχMΨ0p+q−1℧Γp+q+δΓγ−1Ψ0γ+p+q−2℧Γγ+p+q−1+Ψ0q℧ΔΓq+11−e−λΨ0℧Ψ0p−1℧Γp+δΓγ−1Ψ0γ+p−2℧Γγ+p−1+ρ1−e−λΨ0ξΨ0p−1ξΓp+δΓγ−1Ψ0γ+p−2ξΓγ+p−1+LϑΨ0p+q−1℧Γp+q+Ψ0q℧ΔΓq+1Γp−1×1−e−λΨ0℧Ψ0p−2℧+ρ1−e−λΨ0ξ×Ψ0p−2ξϰ⋅−ϰ¯⋅.
Moreover,(44)Aϰθ−Aϰ¯θ≤1−e−λΨ0℧λΛχM+ϒLϑϰ−ϰ¯.
Hence, by (50), A is a contraction map.Step 2
:B is compact and continuous on Br. Firstly, we prove that B is continuous on Br.
Letϰnθ be a sequence such that ϰnθ⟶Br in Br. It follows from Lebesgue dominant convergence theorem that, for all θ∈ℑ,(45)Bϰnθ=limn⟶∞∑i=1m1%Γηi+q∫aθψ′sψθ−ψsηi+q−1gis,ϰnsds=∑i=1m1Γηi+q∫aθΨηi+q−1θ,slimn⟶∞g%is,ϰnsds=∑i=1m1Γηi+q∫aθΨηi+q−1θ,sgis,ϰsds.
Hence,limn⟶∞Bϰnθ=Bϰθ. Thus, B is a continuous on Br. Besides, we prove that B is uniformly bounded on Br. Indeed, for any ϰ∈Br, we have(46)Bϰθ≤supθ∈ℑ∑i=1m1Γηi+q∫aθΨηi+q−1θ,sgis,ϰsds≤∑i=1mΨ0ηi+q℧Γηi+q+1φi≔Φ.
Therefore,Bϰθ≤Φ, for all θ∈ℑ, which implies that B is uniformly bounded on Br. Now, we show that B is equicontinuous. Let θ1,θ2∈ℑ with θ1>θ2. Then, for any ϰθ∈Br, we have(47)Bϰθ1−Bϰθ2≤supθ∈ℑ∑i=1m1Γηi+q∫0θ2Ψηi+q−1θ1,s−Ψηi+q−1θ2,sgis,ϰsds+∑i=1m1Γηi+q∫θ2θ1Ψηi+q−1θ1,sgis,ϰsds≤∑i=1mφiΓηi+q+12ψθ1−ψθ2ηi+q+Ψ0ηi+qθ2−Ψ0ηi+qθ1≤∑i=1m2φiΓηi+q+1ψθ1−ψθ2ηi+q.
Asθ2⟶θ1, Bϰθ1−Bϰθ2⟶0. This means that B is equicontinuous. Thus, Arzelá–Ascoli theorem shows that B is a compact operator on Br.Step 3
: we prove thatϰ=Aϰ+Bϰ¯, for all ϰ¯∈Br⇒ϰ∈Br. For any ϰ¯∈Br, we have(48)ϰθ=Aϰθ+Bϰ¯θ≤Aϰθ+Bϰ¯θ≤supθ∈ℑ|Iq;ψDϰs∫0sCϰτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Cϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ+1ΔIq;ψDϰs1−e−λΨ0sθ|+∑i=1mIηi+q;ψgis,ϰ¯sθ≤1−e−λΨ0℧λΨ0p+q−1℧ΓqΓp+q+Ψ0q℧Γq+1Δ×Ψ0p−1℧Γp1−e−λΨ0℧−ρΨ0p−1ξΓp1−e−λΨ0ξ×ϑχ+∑i=1mΨ0ηi+q℧Γηi+q+1φi≤r,which implies ϰ≤r, so ϰ∈Br. Hence, all assumptions of Dhage’s theorem [27] are satisfied. So, the equation ϰθ=Aϰθ+Bϰθ has at least one solution in Br. So, there exists a solution of problems (6) and (7) in ℑ=0,℧.


### 3.2. Uniqueness of Solutions

Here, we prove the uniqueness theorem of (6) and (7) relying on Banach’s FPT [28].Theorem 2.
Suppose that (H1)-(H2) and the following hypothesis hold:(H4)gi∈Cℑ×ℝ,ℝ, and there exist positive functionKiθ, such that(49)giθ,ϰ−giθ,ϰ¯≤Kiθϰ−ϰ¯.
If(50)Ξ≔1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧<1,then problems (6) and (7) have a unique solution.proof.
We set the operatorG:E⟶E as(51)Gϰθ=Aϰθ+Bϰθ.
ConsiderBR=ϰ∈E:ϰ≤R, and we set supθ∈ℑKiθ=K,i=1,2,…,m. First, we show that GBR∈BR. As in the previous proof (Step 3) of Theorem 1, we can obtain the following.
Forϰ∈BR and θ∈ℑ,(52)Gϰ≤1−e−λΨ0℧λΨ0p+q−1℧ΓqΓp+q+Ψ0q℧Γq+1ΔΨ0p−1℧Γp1−e−λΨ0℧−ρΨ0p−1ξΓp1−e−λΨ0ξϑχ+∑i=1mΨ0ηi+q℧Γηi+q+1φi≤R.
This shows thatGBR⊆BR. Next, we prove that G is a contraction. For ϰ,ϰ¯∈BR,(53)Gϰθ−Gϰ¯θ≤Aϰθ−Aϰ¯θ+Bϰθ−Bϰ¯θ,(54)Bϰθ−Bϰ¯θ≤supθ∈ℑ∑i=1m1Γηi+q∫0θΨηi+q−1θ,sgis,ϰs−gis,ϰ¯sds≤∑i=1mKiΓηi+q+1Ψ0ηi+q℧ϰ−ϰ¯.
From (44), (50), and (54), we obtain(55)Gϰθ−Gϰ¯θ≤1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧ϰ−ϰ¯.
AsΞ<1, G is contractive map. Consequently, by Banach’s FPT [28], we conclude that G has a unique fixed point, which is a solution of (6) and (7).


### 3.3. Stability Analysis

In this portion, we discuss the U-H and generalized U-H stabilities of the solution of the proposed problem. We adopt the following definitions from [29].Letε>0. Consider the subsequent inequality:(56)LψpDq;ψcϰθ−∑i=1mIηi;ψgiθ,ϰθhθ,ϰθ−fθ,ϰθ,δIγ;ψϰθ≤ε,θ∈ℑ.Definition 3.
Problems (6) and (7) is said to be U-H stable if there existsCf>0s.t for eachε>0, and for each solutionw∈Eof (56), there exists a solutionϰ∈Eof (6) and (7) with(57)wθ−ϰθ≤Cfε,θ∈ℑ.Definition 4.
Problems (6) and (7) are said to be generalized U-H stable if there existsΘ∈Cℝ+,ℝ+withΘ0=0s.t; for each solutionw∈Eof (56), there exists a solutionϰ∈Eof (6), (7) with(58)wθ−ϰθ≤Θε,θ∈ℑ.Remark 1.
A functionw∈Esatisfies problems (6) and (7) if there exists aϕ∈E(which depends onw) such that(i)
ϕθ≤ε,θ∈ℑ.(ii)
Forθ∈ℑ,(59)LψpDq;ψcwθ−∑i=1mIηi;ψgiθ,wθhθ,wθ=fθ,wθ,δIγ;ψwθ+ϕθ.Theorem 3.
Let (H1) and (H3) are fulfilled. Then, problems (6) and (7) are U-H and generalized U-H stableproof.
Letw∈E be a solution of inequality (56), for each ε>0. Then, from Remark 1 and Lemma 2, we have(60)wθ=Iq;ψhs,ws∫0sIp−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ws1−e−λΨ0sθ+∑i=1mIηi+q;ψgis,wsθ.
Then, by Remark1, Lemma 2, and (H1)-(H3), we obtain(61)wθ−Gwθ=|Iq;ψhs,ws∫0sIp−1;ψϕτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψϕτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψϕτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ws1−e−λΨ0sθ|≤εχΩ,where(62)Ω=1−e−λΨ0ξΨ0p+q−1℧λΓp+qΓq+Ψ0p−1℧λΓp1−e−λΨ0℧+ρΨ0p−1ξλΓp1−e−λΨ0ξΨ0q℧ΔΓq+1,which is satisfied inequality (56); then, for each θ∈ℑ, we have(63)wθ−ϰθ=|wθ−Iq;ψhs,ϰs∫0sIp−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ϰs1−e−λΨ0sθ+∑i=1mIηi+q;ψgis,ϰsθ|≤wθ−Gwθ+Gwθ−Gϰθ≤Ωχε+1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧wθ−ϰθ.
Then,(64)wθ−ϰθ=Ωχ1−Ξ1ε,where(65)Ξ1=ΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧.
By settingCf=Ωχ/1−Ξ1, we obtain(66)wθ−ϰθ≤Cfε.
Therefore, BVP (6) and (7) is U-H stable.
Similarly, forΘ∈Cℝ+,ℝ+ such that Θε=Cfε along with Θ0=0, the solution of problems (6) and (7) is generalized U-H stable.


## 3.1. Existence of Solutions

In this section, we prove the existence of a solution for problems (6) and (7) by applying Dhage FPT [27].Theorem 1.
Suppose (H1)–(H3) hold; then, problems (6) and (7) have at least one solution inE.proof.
First, we setsupθ∈ℑLθ=L,supθ∈ℑMθ=M,supθ∈ℑχθ=χ,supθ∈ℑϑθ=ϑ, and supθ∈ℑφiθ=φ,i=1,2,…,m.
Now, we defineBr⊂E as(37)Br=ϰ∈E:ϰ≤r.
Define two operatorsC:E⟶E and D:E⟶E as(38)Cϰθ=1Γp−1∫0θψ′sψθ−ψsp−2fs,ϰs,δIγ;ψϰsds,θ∈ℑ,Dϰθ=hθ,ϰθ,θ∈ℑ.
Then, using assumptions (H1)-(H2), we have, forϰ,ϰ¯∈Br and each θ∈ℑ,(39)Cϰθ−Cϰ¯θ≤1ΓSp−1∫0θΨp−2θ,sfs,ϰs,δIγ;ψϰs−fs,ϰ¯s,δIγ;ψϰ¯sds≤1Γp−1∫0θΨp−2θ,sMsϰs−ϰ¯s+δΓγ∫0sΨγ−1τ,sϰτ−ϰ¯τdτds≤Mϰ⋅−ϰ¯⋅1Γp−1∫0θΨp−2θ,s1+δΓγ+1Ψ0γsds≤1ΓpΨ0p−1θ+δΓγ−1Γγ+p−1Ψ0γ+p−2θMϰ⋅−ϰ¯⋅,Cϰθ≤Ψ0p−2θΓp−1ϑ.
Also,(40)Dϰθ−Dϰ¯θ≤Lϰ⋅−ϰ¯⋅,Dϰθ≤χ.
Now, we also consider two operatorsA:E⟶E and B:Br⟶E defined by(41)Aϰθ=Iq;ψDϰs∫0sCϰτψ′τe−λΨ0s−Ψ0τdτθ+1ΔIq;ψDϰs1−e−λΨ0sθ×∫0℧Cϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ,Bϰθ=∑i=1mIηi+q;ψgis,ϰsθ.
We need to prove thatA and B satisfy all assumptions of Dhage’s theorem [27]. This can be proven in the forthcoming steps:Step 1:
A is a contraction map. Indeed, let ϰθ,ϰ¯θ∈Br. Then,(42)Aϰθ−Aϰ¯θ≤Iq;ψDϰs∫0sCϰτ−Cϰ¯τψ′τe−λΨ0s−Ψ0τdτθ+Iq;ψDϰs−Dϰ¯s%∫0sCϰ¯τψ′τe−λΨ0s−Ψ0τdτθ+1ΔIq;ψDϰs%1−e−λΨ0sθ∫0℧%Cϰτ−Cϰ¯τψ′τ×e−λΨ0℧−Ψ0τdτ+ρ∫0ξCϰτ−Cϰ¯τ%ψ′τe−λΨ0ξ−Ψ0τdτ%+1ΔIq;ψDϰs−D%ϰ¯s1−e−λΨ0sθ∫0℧Cϰτψ′τ×e−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ.
Using Lemma3 and hypotheses (H1)-(H2), we obtain(43)Aϰθ−Aϰ¯θ≤1−e−λΨ0℧λχMΨ0p+q−1℧Γp+q+δΓγ−1Ψ0γ+p+q−2℧Γγ+p+q−1+Ψ0q℧ΔΓq+11−e−λΨ0℧Ψ0p−1℧Γp+δΓγ−1Ψ0γ+p−2℧Γγ+p−1+ρ1−e−λΨ0ξΨ0p−1ξΓp+δΓγ−1Ψ0γ+p−2ξΓγ+p−1+LϑΨ0p+q−1℧Γp+q+Ψ0q℧ΔΓq+1Γp−1×1−e−λΨ0℧Ψ0p−2℧+ρ1−e−λΨ0ξ×Ψ0p−2ξϰ⋅−ϰ¯⋅.
Moreover,(44)Aϰθ−Aϰ¯θ≤1−e−λΨ0℧λΛχM+ϒLϑϰ−ϰ¯.
Hence, by (50), A is a contraction map.Step 2
:B is compact and continuous on Br. Firstly, we prove that B is continuous on Br.
Letϰnθ be a sequence such that ϰnθ⟶Br in Br. It follows from Lebesgue dominant convergence theorem that, for all θ∈ℑ,(45)Bϰnθ=limn⟶∞∑i=1m1%Γηi+q∫aθψ′sψθ−ψsηi+q−1gis,ϰnsds=∑i=1m1Γηi+q∫aθΨηi+q−1θ,slimn⟶∞g%is,ϰnsds=∑i=1m1Γηi+q∫aθΨηi+q−1θ,sgis,ϰsds.
Hence,limn⟶∞Bϰnθ=Bϰθ. Thus, B is a continuous on Br. Besides, we prove that B is uniformly bounded on Br. Indeed, for any ϰ∈Br, we have(46)Bϰθ≤supθ∈ℑ∑i=1m1Γηi+q∫aθΨηi+q−1θ,sgis,ϰsds≤∑i=1mΨ0ηi+q℧Γηi+q+1φi≔Φ.
Therefore,Bϰθ≤Φ, for all θ∈ℑ, which implies that B is uniformly bounded on Br. Now, we show that B is equicontinuous. Let θ1,θ2∈ℑ with θ1>θ2. Then, for any ϰθ∈Br, we have(47)Bϰθ1−Bϰθ2≤supθ∈ℑ∑i=1m1Γηi+q∫0θ2Ψηi+q−1θ1,s−Ψηi+q−1θ2,sgis,ϰsds+∑i=1m1Γηi+q∫θ2θ1Ψηi+q−1θ1,sgis,ϰsds≤∑i=1mφiΓηi+q+12ψθ1−ψθ2ηi+q+Ψ0ηi+qθ2−Ψ0ηi+qθ1≤∑i=1m2φiΓηi+q+1ψθ1−ψθ2ηi+q.
Asθ2⟶θ1, Bϰθ1−Bϰθ2⟶0. This means that B is equicontinuous. Thus, Arzelá–Ascoli theorem shows that B is a compact operator on Br.Step 3
: we prove thatϰ=Aϰ+Bϰ¯, for all ϰ¯∈Br⇒ϰ∈Br. For any ϰ¯∈Br, we have(48)ϰθ=Aϰθ+Bϰ¯θ≤Aϰθ+Bϰ¯θ≤supθ∈ℑ|Iq;ψDϰs∫0sCϰτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Cϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξCϰτψ′τe−λΨ0ξ−Ψ0τdτ+1ΔIq;ψDϰs1−e−λΨ0sθ|+∑i=1mIηi+q;ψgis,ϰ¯sθ≤1−e−λΨ0℧λΨ0p+q−1℧ΓqΓp+q+Ψ0q℧Γq+1Δ×Ψ0p−1℧Γp1−e−λΨ0℧−ρΨ0p−1ξΓp1−e−λΨ0ξ×ϑχ+∑i=1mΨ0ηi+q℧Γηi+q+1φi≤r,which implies ϰ≤r, so ϰ∈Br. Hence, all assumptions of Dhage’s theorem [27] are satisfied. So, the equation ϰθ=Aϰθ+Bϰθ has at least one solution in Br. So, there exists a solution of problems (6) and (7) in ℑ=0,℧.


## 3.2. Uniqueness of Solutions

Here, we prove the uniqueness theorem of (6) and (7) relying on Banach’s FPT [28].Theorem 2.
Suppose that (H1)-(H2) and the following hypothesis hold:(H4)gi∈Cℑ×ℝ,ℝ, and there exist positive functionKiθ, such that(49)giθ,ϰ−giθ,ϰ¯≤Kiθϰ−ϰ¯.
If(50)Ξ≔1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧<1,then problems (6) and (7) have a unique solution.proof.
We set the operatorG:E⟶E as(51)Gϰθ=Aϰθ+Bϰθ.
ConsiderBR=ϰ∈E:ϰ≤R, and we set supθ∈ℑKiθ=K,i=1,2,…,m. First, we show that GBR∈BR. As in the previous proof (Step 3) of Theorem 1, we can obtain the following.
Forϰ∈BR and θ∈ℑ,(52)Gϰ≤1−e−λΨ0℧λΨ0p+q−1℧ΓqΓp+q+Ψ0q℧Γq+1ΔΨ0p−1℧Γp1−e−λΨ0℧−ρΨ0p−1ξΓp1−e−λΨ0ξϑχ+∑i=1mΨ0ηi+q℧Γηi+q+1φi≤R.
This shows thatGBR⊆BR. Next, we prove that G is a contraction. For ϰ,ϰ¯∈BR,(53)Gϰθ−Gϰ¯θ≤Aϰθ−Aϰ¯θ+Bϰθ−Bϰ¯θ,(54)Bϰθ−Bϰ¯θ≤supθ∈ℑ∑i=1m1Γηi+q∫0θΨηi+q−1θ,sgis,ϰs−gis,ϰ¯sds≤∑i=1mKiΓηi+q+1Ψ0ηi+q℧ϰ−ϰ¯.
From (44), (50), and (54), we obtain(55)Gϰθ−Gϰ¯θ≤1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧ϰ−ϰ¯.
AsΞ<1, G is contractive map. Consequently, by Banach’s FPT [28], we conclude that G has a unique fixed point, which is a solution of (6) and (7).


## 3.3. Stability Analysis

In this portion, we discuss the U-H and generalized U-H stabilities of the solution of the proposed problem. We adopt the following definitions from [29].Letε>0. Consider the subsequent inequality:(56)LψpDq;ψcϰθ−∑i=1mIηi;ψgiθ,ϰθhθ,ϰθ−fθ,ϰθ,δIγ;ψϰθ≤ε,θ∈ℑ.Definition 3.
Problems (6) and (7) is said to be U-H stable if there existsCf>0s.t for eachε>0, and for each solutionw∈Eof (56), there exists a solutionϰ∈Eof (6) and (7) with(57)wθ−ϰθ≤Cfε,θ∈ℑ.Definition 4.
Problems (6) and (7) are said to be generalized U-H stable if there existsΘ∈Cℝ+,ℝ+withΘ0=0s.t; for each solutionw∈Eof (56), there exists a solutionϰ∈Eof (6), (7) with(58)wθ−ϰθ≤Θε,θ∈ℑ.Remark 1.
A functionw∈Esatisfies problems (6) and (7) if there exists aϕ∈E(which depends onw) such that(i)
ϕθ≤ε,θ∈ℑ.(ii)
Forθ∈ℑ,(59)LψpDq;ψcwθ−∑i=1mIηi;ψgiθ,wθhθ,wθ=fθ,wθ,δIγ;ψwθ+ϕθ.Theorem 3.
Let (H1) and (H3) are fulfilled. Then, problems (6) and (7) are U-H and generalized U-H stableproof.
Letw∈E be a solution of inequality (56), for each ε>0. Then, from Remark 1 and Lemma 2, we have(60)wθ=Iq;ψhs,ws∫0sIp−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψfθ,wθ,δIγ;ψwθ+ϕθτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ws1−e−λΨ0sθ+∑i=1mIηi+q;ψgis,wsθ.
Then, by Remark1, Lemma 2, and (H1)-(H3), we obtain(61)wθ−Gwθ=|Iq;ψhs,ws∫0sIp−1;ψϕτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψϕτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψϕτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ws1−e−λΨ0sθ|≤εχΩ,where(62)Ω=1−e−λΨ0ξΨ0p+q−1℧λΓp+qΓq+Ψ0p−1℧λΓp1−e−λΨ0℧+ρΨ0p−1ξλΓp1−e−λΨ0ξΨ0q℧ΔΓq+1,which is satisfied inequality (56); then, for each θ∈ℑ, we have(63)wθ−ϰθ=|wθ−Iq;ψhs,ϰs∫0sIp−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0s−Ψ0τdτθ+∫0℧Ip−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0℧−Ψ0τdτ−ρ∫0ξIp−1;ψfτ,ϰτ,δIγ;ψϰτψ′τe−λΨ0ξ−Ψ0τdτ×1ΔIq;ψhs,ϰs1−e−λΨ0sθ+∑i=1mIηi+q;ψgis,ϰsθ|≤wθ−Gwθ+Gwθ−Gϰθ≤Ωχε+1−e−λΨ0℧λΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧wθ−ϰθ.
Then,(64)wθ−ϰθ=Ωχ1−Ξ1ε,where(65)Ξ1=ΛχM+ϒLϑ+∑i=1mKiΓηi+q+1Ψ0ηi+q℧.
By settingCf=Ωχ/1−Ξ1, we obtain(66)wθ−ϰθ≤Cfε.
Therefore, BVP (6) and (7) is U-H stable.
Similarly, forΘ∈Cℝ+,ℝ+ such that Θε=Cfε along with Θ0=0, the solution of problems (6) and (7) is generalized U-H stable.


## 4. Example

Consider the followingψ-Caputo HFSIDE:(67)D3/2;θc+3D1/2;θcD4/5;θcϰθ−∑i=14Iηi;ψgiθ,ϰθhθ,ϰθ=fθ,ϰθ,12I5/2;θϰθ,θ∈0,1,ϰ0=0,CD4/5;θϰ0=0,D4/5;θcϰθ−∑i=14Iηi;ψgiθ,ϰθhθ,ϰθθ=1=713D4/5;θcϰθ−∑i=14Iηi;ψgiθ,ϰθhθ,ϰθθ=5/6,where(68)∑i=14Iηi;θgiθ,ϰθs=I1/4;θ118ss2+1+sins+cosϰsθ+I9/5;θsinϰs4π36+s2θ+I4/3;θsin2ϰs12s+12θ+I5/2;θ13π64+s3ϰs2+ϰsθ,hθ,ϰθ=e−θsinϰθθ+30+160θ2+1,fθ,ϰθ,12I5/2;θϰθ=1θ+81ϰθ1+ϰθ+arctan12I5/2;θϰθ.Herep=3/2,q=4/5,m=4,η1=7/4,η2=4/3,η3=2/3,η4=5/6,δ=1/2,γ=5/2,ρ=7/13,ξ=5/6, and g1=1/18θθ2+1+sinθ+cosϰθ, g2=sinxθ/4π36+θ2, g3=sin2xθ/12θ+12, and g4=1/3π64+θ3xθ/2+xθ.The hypothesis (H1), (H2), and (H4) are satisfied with the following positives functions:Lθ=e−θ/θ+3,χθ=e−θ/θ+3+θ2+1/6,Mθ=ϑθ=3/2θ+81,K1θ=1/18,φ1θ=1/18θθ2+1+2,K2θ=φ2θ=1/4π36+θ2, K3θ=φ3θ=1/12θ+12, and K4θ=φ4θ=1/3π64+θ3, which gives the norms, L=1/3,χ=1/2,M=ϑ=1/6,K1=1/18,φ1=2+2/18,K2=φ2=1/24π, K3=φ3=1/12,andK4=φ4=1/24π.With the given data, we find that(69)Δ≃0.45595101,Λ≃5.3500159,ϒ≃2.8388423,and hypothesis (H3) is satisfied by(70)1−e−λλΛχM+ϒLϑ+∑i=1mKiΓηi+q+1≃0.3153266651<1.In the view of Theorem2, problem (67) has a unique solution. In addition, Theorem 3 ensures that (6) and (7) are U-H and generalized U-H stable. As shown in Theorem 3, for every ϵ>0, if w∈ℝ satisfies(71)LψpDq;ψcwθ−∑i=1mIηi;ψgiθ,wθhθ,wθ−fθ,wθ,δIγ;ψwθ≤ε,θ∈0,1,then there exists a unique solution ϰ∈ℝ such that(72)wθ−ϰθ≤Cfε,θ∈0,1,where(73)Cf≃0.81<1.Hence, problem (67) is U-H and generalized U-H stable.


## 5. Conclusion

In this study, we have successfully investigated the existence, uniqueness, and two kinds of stability in the sense of Ulam of the solutions for a new classψ-Caputo-type hybrid FDEs with hybrid conditions. The existence of solutions is provided by using a generalization of Krasnoselskii’s FPT due to Dhage [27], whereas the uniqueness result is achieved by Banach’s FPT. After that, we have studied the concept of U-L and generalized U-L stabilities of (6) and (7). Also, we have presented an illustrative example to support our main results from a numerical point of view.In future works, many results can be established when one takes a more generalized operator. Precisely, it will be of interest to study the current problem in this work for the fractional operator with variable order [30] and ψ-Hilfer fractional operator [31].



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