# Investigating a Class of Generalized Caputo-Type Fractional Integro-Differential Equations

Authors: Saeed M. Ali; Wasfi Shatanawi; Mohammed D. Kassim; Mohammed S. Abdo; S. Saleh
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we prove some new uniqueness and Ulam-Hyers stability results of a nonlinear generalized fractional integro-differential equation in the frame of Caputo derivative involving a new kernel in terms of another functionψ. Our approach is based on Babenko’s technique, Banach’s fixed point theorem, and Banach’s space of absolutely continuous functions. The obtained results are demonstrated by constructing numerical examples.


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## Body

## 1. Introduction

It is notable that fractional calculus was and still is a new tool that uses fractional differential and integral equations to construct more modern mathematical models that can precisely describe complex frameworks. There are many definitions of fractional integrals (FIs) and fractional derivatives (FDs) accessible in the literature, for instance, the Riemann-Liouville and Caputo definitions that assumed a significant part in the advancement of the theory of fractional analysis. Referring to all books and papers in this field will be extremely many. In this regard, here, we refer to the most important of main references, e.g., Samko et al. [1] gave a broad comprehensive mathematical handling of fractional derivatives and integrals. Podlubny [2] and Kilbas et al. [3] have been introduced many useful results related to fractional differential equations (FDEs). Several applications have been implemented recently by a wide range of works on this subject, see [4–8].However, the currently common operator is the generalized FD regarding another function, see [1, 3]. Agrawal [7] studied further various properties for generalized fractional derivatives and integrals. More recently, Almeida [9] inspired an idea of that generalization by projecting this generalization onto the definition of the Caputo fractional derivative with respect to another function, so-called ψ-Caputo, and introduced many interesting properties, which are more general than the classical Caputo FD. Jarad and Abdeljawad [10] provided interesting properties for generalized FDs and Laplace transform. Specifically, ψ-Caputo type FDEs with initial, boundary, and nonlocal conditions have been investigated by many researchers using fixed-point theories, see Almeida et al. [11, 12], Abdo et al. [13], and Wahash et al. [14]. A recent survey on ψ-Caputo type FDEs can be found in [15–18]. For more results in this direction, we refer to interesting works provided by Zhang et al. [19], Zhao et al. [20], Baitiche et al. [21], Benchohra et al. [22], Ravichandran et al. [23], Trujillo et al. [24], and Furati et al. [25].Li in [17, 18] investigated some interesting results of the integral equations and the integro-differential equations involving Hadamard-type. In this work, our goal is to intend to address a general extension of these studies. Precisely, we consider the following ψ-Caputo type fractional integro-differential equation (FIDE)
(1)CDaψ,ϱφx+a1CDaψ,ϱ1φx+a2Iaψ,ϱ2φx=∫axGζ,φ′ζdζ,φa=0,where
(i)
0<ϱ1<ϱ<1,ϱ2>0, and a1,a2∈ℂ(ii)
The symbolDCaψ,σ denotes the generalized Caputo FD of order σ∈ϱ,ϱ1(iii)
The notationIaψ,ϱ2 means the generalized Riemann-Liouville FI of order ϱ2(iv)
G:a,b×ℝ⟶ℝ is a continuous function, and 0≤a<b<∞(v)
φ∈AC0a,b such that Iaψ,ϱ2 and Daψ,σ exist and are both continuous in a,bObserve that the considered system (1) covers the previous standard cases of nonlinear FIDEs by defining the kernel, i.e., if ψx=x, ψx=logx, and ψx=xρ, then, the problem (1) reduces to the Caputo type FIDE, Caputo-Hadamard type FIDE, and Caputo-Katugampola type FIDE, respectively.The aim of this work is to develop the nonlinear FIDEs. In particular, we investigate the uniqueness and Ulam-Hyers stability of solution for the problem (1) by Banach’s fixed point theorem and Babenko’s technique [26]. Note that the presentation and structuring of the arguments for our problem are new, and our results generalize and cover some of the known results in the literature. In addition, the obtained results here are valid when the left hand side of the considered problem (1) involves many FDs and FIs. For more details, see Remark 22.The remainder of this paper is organized as follows: in Section2, we present some important tools related the fractional calculus and the functional spaces, in which we aim to determine our analysis strategies. Section 3 gives the main results and their illustrative examples. Finally, our brief conclusion is included in Section 4.


## 2. Preliminaries

In this section, we present some properties, lemmas, definitions, and important estimations needed in the proof of our result.Defining the Banach space as(2)AC0a,b=φ:φ∈ACa,bwithφa=0andφ0=∫abφ′ζdζ<∞.Next, we present some important definitions and properties of advanced fractional calculus.Definition 1 [3, 9].
Theψ-Riemann-Liouville FI and ψ-Caputo FD are defined by
(3)Iaψ,ϱωx=1Γϱ∫axψx−ψζϱ−1ψ′ζωζdζ,ϱ>0,DCaψ,ϱωx=Iaψ,n−ϱ1ψ′xddxωx=1Γn−ϱ∫axψx−ψζn−ϱ−1ψ′ζωψnζdζ,ϱ>0,respectively, where
(4)n=−−ϱ,ωψnx=1ψ′xddxnωx.Definition 2 [27].
The incomplete gamma function is represented by(5)γϱ,ζ=∫0ζuϱ−1e−udu=ζϱΓϱe−ζ∑i=0∞ζiΓϱ+i+1,ϱ>0,ζ≥0.Property 3 [3, 9].
Letϱ≥0, and κ>0. Then
(6)Iaψ,ϱψx−ψaκ=Γκ+1Γκ+ϱ+1ψx−ψaκ+ϱ,x>a,(7)DCa,ψ,ϱψx−ψaκ=Γκ+1Γκ−ϱ+1ψx−ψaκ−ϱ,x>a.Property 4 [3, 9].
Letϱ,κ>0, and ω∈AC0a,b. Then
(8)DCaψ,ϱIaψ,ϱωζ=ωζ,ζ>a,(9)Iaψ,ϱIaψ,κωζ=Iaψ,ϱ+κωζ,ζ>a,(10)DCaψ,κIaψ,ϱωζ=Iaψ,ϱ−κωζ,ϱ>κ,ζ>a,(11)Iaψ,ϱωa=0.
In the following, some very significant lemmas will be given.Lemma 5.
Letϱ,κ∈0,1.If ω∈AC0a,b, then
(12)Iaψ,ϱDCaψ,ϱωζ=ωζ,ζ>a,(13)Iaψ,ϱDCaψ,κωζ=Iaψ,ϱ−κωζ,ϱ>κ,ζ>a.Proof.
Letω∈AC0a,b. Then by Definition 1 and Property 4, we get
(14)Iaψ,ϱDCaψ,ϱωζ=Iaψ,ϱIaψ,1−ϱωψ1ζ=Iaψ,1ωψ1ζ=ωζ−ωa=ωζ,ζ>a,(15)Iaψ,ϱDCaψ,κωζ=Iaψ,ϱIaψ,1−κωψ1ζ=Iaψ,ϱ−κIaψ,1ωψ1ζ=Iaψ,ϱ−κωζ−ωa=Iaψ,ϱ−κωζ,ϱ>κ,ζ>a.Lemma 7.
Letϱ>0. Then, Iaψ,ϱ is bounded from AC0a,b into itself, and
(16)Iaψ,ϱω0≤1Γϱ+1ψb−ψaϱω0.Proof.
Letω∈AC0a,b. Then
(17)ωζ=∫aζω′sds=∫aζzsds,wherezζ=ω′ζandωa=0.
By virtue of Definition1, we get
(18)Iaψ,ϱωx=Iaψ,ϱ∫aζzsdsx=1Γϱ∫axψx−ψζϱ−1ψ′ζ∫aζzsdsdζ.
Taking advantage of the Dirichlet’s formula, we have(19)Iaψ,ϱωx=1Γϱ∫axzs∫sxψx−ψζϱ−1ψ′ζdζds=1Γϱ∫axzs−ψx−ψζϱϱζ=sxds=1Γϱ∫axzsψx−ψsϱϱds≤1Γϱ+1ψb−ψaϱ∫axzsds=1Γϱ+1ψb−ψaϱ∫axω′sds=1Γϱ+1ψb−ψaϱω0.
Now, we will provide and prove the next lemma:Lemma 9.
Ifϱ≥0, then
(20)Iaψ,ϱeψx=eψaψx−ψaϱ∑i=0∞ψx−ψaiΓϱ+i+1.Proof.
Using Definition1, we have
(21)Iaψ,ϱeψx=1Γϱ∫axψx−ψζϱ−1ψ′ζeψζdζ.
Performing the substitutions=ψx−ψζ, we get
(22)Iaψ,ϱeψx=1Γϱ∫0ψx−ψasϱ−1eψx−sds=eψxΓϱ∫0ψx−ψasϱ−1e−sds.
From Definition2, we obtain
(23)Iaψ,ϱeψx=γϱ,ψx−ψaeψxΓϱ=eψaψx−ψaϱ∑i=0∞ψx−ψaiΓϱ+i+1.


## 3. Main Results

Theorem 11.
Letai∈ℂi=1,2, 0<ϱ1<ϱ<1, and ϱ2>0. If h∈AC0a,b, then, the following linear problem
(24)CDaψ,ϱφx+a1CDaψ,ϱ1φx+a2Iaψ,ϱ2φx=hx,φa=0,has a solution in the space AC0a,b, that is
(25)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx.Proof.
Applying the operatorIaψ,ϱ to both sides of Eq. (24), we obtain
(26)Iaψ,ϱDCaψ,ϱφx+a1Iaψ,ϱDCaψ,ϱ1φx+a2Iaψ,ϱIaψ,ϱ2φx=Iaψ,ϱhx.
According to Lemma5, we find that
(27)φx+a1Iaψ,ϱ−ϱ1φx+a2Iaψ,ϱ+ϱ2φx=Iaψ,ϱhx.
Observe thatφa=0 and 0<ϱ1<ϱ<1. It follows that
(28)1+a1Iaψ,ϱ−ϱ1+a2Iaψ,ϱ+ϱ2φx=Iaψ,ϱhx.
In view of Babenko approach, we have(29)φx=1+a1Iaψ,ϱ−ϱ1+a2Iaψ,ϱ+ϱ2−1Iaψ,ϱhx.
By using the multinomial theorem and Property4, we obtain
(30)φx=∑ℓ=0∞−1ℓa1Iaψ,ϱ−ϱ1+a2Iaψ,ϱ+ϱ2ℓIaψ,ϱhx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1Iaψ,ϱ−ϱ1ℓ1a2Iaψ,ϱ+ϱ2ℓ2Iaψ,ϱhx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx.
Asx⟶a, we get φa=0. Now, we need to prove the series is absolutely continuous on a,b and converges in the space AC0a,b. Indeed, by Lemma 7, we have
(31)Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx0≤ηh0,where
(32)η=ψb−ψaℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱΓℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ+1.
It follows that(33)φ0≤η∑ℓ=0∞∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2ψb−ψaℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱΓℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ+1h0=η∑ℓ=0∞∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ψb−ψaϱ−ϱ1ℓ1a2ψb−ψaϱ+ϱ2ℓ2Γℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ+1h0=ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2h0,where
(34)Eϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2<∞,which is the value at v1=a1ψb−ψaϱ−ϱ1,v2=a2ψb−ψaϱ+ϱ2 of the multivariate Mittag-Leffler function Eϱ−ϱ1,ϱ+ϱ2,ϱ+1v1,v2 given in [3]. So, we conclude that the series to the right of Eq. (25) is convergent. Obviously, φx∈ACa,b due to h∈ACa,b. To affirm that the obtained series could be a solution, we must see that it fulfills Eq. (24), i.e.,
(35)DCaψ,ϱ∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx+a1DCaψ,ϱ1∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx+a2Iaψ,ϱ2∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱhx=DCaψ,ϱIaψ,ρhx+∑ℓ=1∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ρ−ρ1+ℓ2ρ+ρ2+ρhx+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1+1a2ℓ2Iaψ,ℓ1+1ϱ−ϱ1+ℓ2ϱ+ϱ2hx+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a0ℓ2+1Iaψ,ℓ1ϱ−ϱ1+ℓ2+1ϱ+ϱ2hx=hx+∑ℓ=1∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ρ−ρ1+ℓ2ρ+ρ2hx+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1+1a2ℓ2Iaψ,ℓ1+1ϱ−ϱ1+ℓ2ϱ+ϱ2hx+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2+1Iaψ,ℓ1ϱ−ϱ1+ℓ2+1ϱ+ϱ2hx=hx,by the cancellation. Notice that each series is absolutely convergent and also the term arrangements are possibly cancellated. In fact,
(36)−∑ℓ1+ℓ2=1ℓℓ1,ℓ2a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2hx+∑ℓ1+ℓ2=0ℓℓ1,ℓ2a1ℓ1+1a2ℓnIaψ,ℓ1+1ϱ−ϱ1+ℓ2ϱ+ϱ2hx+∑ℓ1+ℓ2=0ℓℓ1,ℓ2a1ℓ1a2ℓ2+1Iaψ,ℓ1ϱ−ϱ1+ℓ2+1ϱ+ϱ2hx=0.
The remainder terms cancel each other similarly. Plainly, the uniqueness follows promptly from the fact that the FIDE(37)DCaψ,ϱφx+a1DCaψ,ϱ1φx+a2Iaψ,ϱ2φx=0,only has solution zero due to the Babenko approach.Remark 13.
Notice that the solution of Eq. (24) in AC0a,b is stable, if for all ε>0, there exists δ>0 such that φ0<ε with h0<δ. Taking advantage of the following inequality
(38)φ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2h0,we conclude that φ is stable.Example 1.
The followingψ-Caputo type FIDE
(39)DCaψ,0.9φx+2DCaψ,0.7φx−Iaψ,0.4φx=ψx−ψaκ,has the solution in AC0a,b, that is
(40)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ22ℓ1−1ℓ2×Γκ+1Γκ+0.2ℓ1+1.3ℓ2+1.9ψx−ψaκ+0.2ℓ1+1.3ℓ2+0.9.
So, according to Theorem11, we have
(41)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ22ℓ1−1ℓ2×Iaψ,0.2ℓ1+1.3ℓ2+0.9ψx−ψaκ.
From Property3, we get
(42)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ22ℓ1−1ℓ2×Γκ+1Γκ+0.2ℓ1+1.3ℓ2+1.9ψx−ψaκ+0.2ℓ1+1.3ℓ2+0.9.Example 2.
The followingψ-Caputo type FIDE
(43)DCaψ,0.8φx+DCaψ,0.7φx−3Iaψ,0.2φx=eψx,has the solution in AC0a,b described as
(44)φx=eψa∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2−3ℓ2ψx−ψa0.1ℓ1+ℓ2+0.8×∑i=0∞ψx−ψaiΓ0.1ℓ1+ℓ2+1.8+i.
So, as stated by Theorem11, we obtain
(45)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ21ℓ1−3ℓ2Iaψ,0.1ℓ1+ℓ2+0.8eψx.
By virtue of Lemma9, we obtain
(46)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2−3ℓ2eψaψx−ψa0.1ℓ1+ℓ2+0.8×∑i=0∞ψx−ψaiΓ0.1ℓ1+ℓ2+1.8+i.
The uniqueness result of Eq. (1) will be proved through the following theorem.Theorem 14.
LetG:a,b×ℝ⟶ℝ be a continuous function, and assume that there exists a constant C such that
(47)Gx,φ1−Gx,φ2≤Cφ1−φ2,x∈a,b,φ1,φ2∈ℝ.
If(48)CηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2<1,then, the problem (1) has a unique solution in AC0a,b.Proof.
Consider the operatorI on AC0a,b defined by
(49)Iφ=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2×Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ρ∫axGζ,φ′ζdζ.
Forφ∈AC0a,b, we have ∫axGζ,φ′ζdζ∈AC0a,b, since φ′ζ∈La,b and Gζ,φ′ζ∈La,b.
Hence,(50)∫axGζ,φ′ζdζ0=∫abGx,φ′xdx≤∫abGx,φ′x−Gx,0dx+∫abGx,0dx≤C∫abφ′xdx+∫abGx,0dx<∞.
Using the inequality (38), we obtain
(51)Iφ0<∞andIφa=0.
Besides,Iφ is absolutely continuous on a,b via Theorem 11. So, I:AC0a,b⟶AC0a,b. Now, we just have to show that I is a contraction mapping. Let φ,φ∗∈AC0a,b. Then
(52)Iφ−Iφ∗0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×∫axGζ,φ′ζdζ−∫axGζ,φ∗′ζdζ0.
Since(53)∫axGζ,φ′ζdζ−∫axGζ,φ∗′ζdζ0=∫abGx,φ′x−Gx,φ∗′xdx≤C∫abφ′−φ∗′dx=Cφ−φ∗0,we obtain
(54)Iφ−Iφ∗0≤CηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×φ−φ∗0.
Inequality (48) leads us to that I is contraction mapping.


### 3.1. Ulam-Hyers Stability (UHS)

The first results about this type of stability emerged in 1940 by Ulam [28, 29]. From that point forward, the UHS is studied via several researchers. With the vast development of fractional calculus, the studying of stability for FDEs also attracted the numerous authors, see [30–32].In this regard, we investigate some recent results on the UHS and generalized UHS of (1). For ε>0,x∈a,b, and φ1∈AC0a,b, the following inequality
(55)DCaψ,ϱφ1x+a1DCaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x−∫axGζ,φ′ζdζ≤ε,is satisfied.Remark 16.
Letε>0. Then, φ1∈AC0a,b satisfies (55) iff there exists ξx∈AC0a,b with ξ0=0 such that
(i)
ξ0=∫axξ′ζdζ≤ε, for x∈a,b(ii)
forx∈a,b(56)DCaψ,ϱφ1x+a1DCaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x=∫axGζ,φ1′ζdζ+∫axξ′ζdζ.Lemma 17.
The solution of the problem (56) with φ10=0 satisfies the following inequality
(57)φ1−ZG0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε,where
(58)ZGx≔∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2×Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ρ∫axGζ,φ1′ζdζ,and η is defined by (32).Proof.
By virtue of Lemma 8, the solution of Eq. (56) is described as
(59)φ1x=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2×Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ∫axGζ,φ1′ζdζ+∫axξ′ζdζ.
It follows from Eq. (59), Remark 16, and Eq. (38) that
(60)φ1−ZG0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×∫axGζ,φ1′ζdζ+∫axξ′ζdζ−∫axGζ,φ1′ζdζ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ξ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε.Theorem 19 (UHS).
Suppose that the hypotheses of Theorem14 with Eq. (55) are satisfied. Then, Eq. (1) is UH stable.Proof.
Assume thatε>0 and φ1∈AC0a,b satisfy Eq. (55), and let φ∈AC0a,b be a unique solution of
(61)CDaψ,ϱφ1x+a1CDaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x=∫axGζ,φ′ζdζ,φa=φ1a=0,
that is(62)φx=φa+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ρ∫axGζ,φ′ζdζ.
Sinceφa=φ1a=0, we obtain
(63)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ∫axGζ,φ′ζdζ.
According to Lemma17 and (38), we get
(64)φ1−φ0≤φ1−ZG0+ZG−φ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε+ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×∫axGζ,φ1′ζdζ−∫axGζ,φ′ζdζ0.
Using the assumption of Theorem14, we have
(65)∫axGζ,φ1′ζdζ−∫axGζ,φ′ζdζ0≤Cφ1−φ0.
So,(66)φ1−φ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε+ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×Cφ1−φ0.
From the inequality (48), we find that
(67)φ1−φ0≤CGε,where CG≔R/1−RC and
(68)R≔ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2.Corollary 21.
Under assumptions of Theorem19, if we put Φε=CGε along with Φ0=0, then Eq. (1) is a generalized UH stable.Example 3.
Leta=1 and b=ψ−11+ψ1. Then, there exists a unique solution for the following nonlinear ψ-Caputo-type FIDE
(69)DCaψ,0.9φx+Iaψ,0.6φx=∫axeζ2C3+eζ2sinφ′ζ+ecosζ+ln1+ζdζ,where the constant C is to be determined. It is clear that
(70)Gx,z=ex2C3+ex2sinz+ecosx+ln1+x,is continuous from 1,ψ−11+ψ1×ℝ to ℝ and satisfies
(71)Gx,z1−Gx,z2=ex2C3+ex2sinz1−ex2C3+ex2sinz1≤ex2C3+ex2sinz1−sinz1≤ex2C3+ex2z1−z1≤1Cz1−z1.
Obviously,ψb−ψa=1 and
(72)∑ℓ=0∞∑ℓ2=ℓℓℓ21ψb−ψa1.5ℓ21Γ1.5ℓ2+1.9=∑ℓ=0∞1Γ1.5ℓ+1.9.
Forℓ≥1, we have
(73)ℓ+1≤1.5ℓ+1.9,(74)1Γ1.5ℓ+1.9≤1Γℓ+1=1ℓ!.
Therefore,(75)∑ℓ=0∞∑ℓ2=ℓℓℓ21Γ1.5ℓ2+1.9≤1Γ1.9+∑ℓ=1∞1k!=1Γ1.9−1+∑ℓ=0∞1k!≤0.04+e.
Let us choose a positiveC such that
(76)C<10.04+e.
It follows from Theorem14 that Eq. (69) has a unique solution.
Moreover, by Theorem19, and for any solution φ1x∈AC0a,b of the inequality
(77)DCaψ,0.9φx+Iaψ,0.6φx−∫axeζ2C3+eζ2sinφ1′ζ+ecosζ+ln1+ζdζ≤ε,forx∈a,b,there exists a unique solution φx∈AC0a,b of Eq. (69) such that
(78)φ1−φ0≤CGε,where CG≔R/1−RC>0, R=η/0.04+e and η=1/Γℓ10.9+1.5ℓ2+1.9. Consequently, Eq. (69) is UH stable.Remark 22.
All previous results can be generalized in which the left-hand side of (1) may contain several FDs and FIs. For example, Theorem 11 can be generalized as follows:Theorem 23.
Supposeai∈ℂi=1,⋯,n,bi+n∈ℂi=1,⋯,m−n with 0<α1<⋯<αn<α<1 and 0≤κn+1<κn+2<⋯<κm∈ℝ. If h∈AC0a,b, then the following linear problem
(79)CDaψ,αφx+a1CDaψ,α1φx+…+anCDaψ,αnφx+bn+1Iaψ,kn+1φx+bn+2Iaψ,kn+2φx+…+bmIaψ,kmφx=hx,φa=0,has a solution
(80)φx=∑ℓ=0∞−1ℓ∑ℓ1+⋯+ℓm=ℓℓℓ1,ℓ2,⋯,ℓm×a1ℓ1⋯bmℓmIaψ,ℓ1α−α1+⋯+ℓmα+κm+αhx.where DCaψ,δ is the ψ-Caputo FD of order δ>0∈α,αi;i=1,⋯,n and Iaψ,σ is generalized FI of order σ>0∈κj;j=n+1,⋯,m.


## 3.1. Ulam-Hyers Stability (UHS)

The first results about this type of stability emerged in 1940 by Ulam [28, 29]. From that point forward, the UHS is studied via several researchers. With the vast development of fractional calculus, the studying of stability for FDEs also attracted the numerous authors, see [30–32].In this regard, we investigate some recent results on the UHS and generalized UHS of (1). For ε>0,x∈a,b, and φ1∈AC0a,b, the following inequality
(55)DCaψ,ϱφ1x+a1DCaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x−∫axGζ,φ′ζdζ≤ε,is satisfied.Remark 16.
Letε>0. Then, φ1∈AC0a,b satisfies (55) iff there exists ξx∈AC0a,b with ξ0=0 such that
(i)
ξ0=∫axξ′ζdζ≤ε, for x∈a,b(ii)
forx∈a,b(56)DCaψ,ϱφ1x+a1DCaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x=∫axGζ,φ1′ζdζ+∫axξ′ζdζ.Lemma 17.
The solution of the problem (56) with φ10=0 satisfies the following inequality
(57)φ1−ZG0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε,where
(58)ZGx≔∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2×Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ρ∫axGζ,φ1′ζdζ,and η is defined by (32).Proof.
By virtue of Lemma 8, the solution of Eq. (56) is described as
(59)φ1x=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2a1ℓ1a2ℓ2×Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ∫axGζ,φ1′ζdζ+∫axξ′ζdζ.
It follows from Eq. (59), Remark 16, and Eq. (38) that
(60)φ1−ZG0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×∫axGζ,φ1′ζdζ+∫axξ′ζdζ−∫axGζ,φ1′ζdζ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ξ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε.Theorem 19 (UHS).
Suppose that the hypotheses of Theorem14 with Eq. (55) are satisfied. Then, Eq. (1) is UH stable.Proof.
Assume thatε>0 and φ1∈AC0a,b satisfy Eq. (55), and let φ∈AC0a,b be a unique solution of
(61)CDaψ,ϱφ1x+a1CDaψ,ϱ1φ1x+a2Iaψ,ϱ2φ1x=∫axGζ,φ′ζdζ,φa=φ1a=0,
that is(62)φx=φa+∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ρ∫axGζ,φ′ζdζ.
Sinceφa=φ1a=0, we obtain
(63)φx=∑ℓ=0∞−1ℓ∑ℓ1+ℓ2=ℓℓℓ1,ℓ2×a1ℓ1a2ℓ2Iaψ,ℓ1ϱ−ϱ1+ℓ2ϱ+ϱ2+ϱ∫axGζ,φ′ζdζ.
According to Lemma17 and (38), we get
(64)φ1−φ0≤φ1−ZG0+ZG−φ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε+ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×∫axGζ,φ1′ζdζ−∫axGζ,φ′ζdζ0.
Using the assumption of Theorem14, we have
(65)∫axGζ,φ1′ζdζ−∫axGζ,φ′ζdζ0≤Cφ1−φ0.
So,(66)φ1−φ0≤ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2ε+ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2×Cφ1−φ0.
From the inequality (48), we find that
(67)φ1−φ0≤CGε,where CG≔R/1−RC and
(68)R≔ηEϱ−ϱ1,ϱ+ϱ2,ϱ+1a1ψb−ψaϱ−ϱ1,a2ψb−ψaϱ+ϱ2.Corollary 21.
Under assumptions of Theorem19, if we put Φε=CGε along with Φ0=0, then Eq. (1) is a generalized UH stable.Example 3.
Leta=1 and b=ψ−11+ψ1. Then, there exists a unique solution for the following nonlinear ψ-Caputo-type FIDE
(69)DCaψ,0.9φx+Iaψ,0.6φx=∫axeζ2C3+eζ2sinφ′ζ+ecosζ+ln1+ζdζ,where the constant C is to be determined. It is clear that
(70)Gx,z=ex2C3+ex2sinz+ecosx+ln1+x,is continuous from 1,ψ−11+ψ1×ℝ to ℝ and satisfies
(71)Gx,z1−Gx,z2=ex2C3+ex2sinz1−ex2C3+ex2sinz1≤ex2C3+ex2sinz1−sinz1≤ex2C3+ex2z1−z1≤1Cz1−z1.
Obviously,ψb−ψa=1 and
(72)∑ℓ=0∞∑ℓ2=ℓℓℓ21ψb−ψa1.5ℓ21Γ1.5ℓ2+1.9=∑ℓ=0∞1Γ1.5ℓ+1.9.
Forℓ≥1, we have
(73)ℓ+1≤1.5ℓ+1.9,(74)1Γ1.5ℓ+1.9≤1Γℓ+1=1ℓ!.
Therefore,(75)∑ℓ=0∞∑ℓ2=ℓℓℓ21Γ1.5ℓ2+1.9≤1Γ1.9+∑ℓ=1∞1k!=1Γ1.9−1+∑ℓ=0∞1k!≤0.04+e.
Let us choose a positiveC such that
(76)C<10.04+e.
It follows from Theorem14 that Eq. (69) has a unique solution.
Moreover, by Theorem19, and for any solution φ1x∈AC0a,b of the inequality
(77)DCaψ,0.9φx+Iaψ,0.6φx−∫axeζ2C3+eζ2sinφ1′ζ+ecosζ+ln1+ζdζ≤ε,forx∈a,b,there exists a unique solution φx∈AC0a,b of Eq. (69) such that
(78)φ1−φ0≤CGε,where CG≔R/1−RC>0, R=η/0.04+e and η=1/Γℓ10.9+1.5ℓ2+1.9. Consequently, Eq. (69) is UH stable.Remark 22.
All previous results can be generalized in which the left-hand side of (1) may contain several FDs and FIs. For example, Theorem 11 can be generalized as follows:Theorem 23.
Supposeai∈ℂi=1,⋯,n,bi+n∈ℂi=1,⋯,m−n with 0<α1<⋯<αn<α<1 and 0≤κn+1<κn+2<⋯<κm∈ℝ. If h∈AC0a,b, then the following linear problem
(79)CDaψ,αφx+a1CDaψ,α1φx+…+anCDaψ,αnφx+bn+1Iaψ,kn+1φx+bn+2Iaψ,kn+2φx+…+bmIaψ,kmφx=hx,φa=0,has a solution
(80)φx=∑ℓ=0∞−1ℓ∑ℓ1+⋯+ℓm=ℓℓℓ1,ℓ2,⋯,ℓm×a1ℓ1⋯bmℓmIaψ,ℓ1α−α1+⋯+ℓmα+κm+αhx.where DCaψ,δ is the ψ-Caputo FD of order δ>0∈α,αi;i=1,⋯,n and Iaψ,σ is generalized FI of order σ>0∈κj;j=n+1,⋯,m.


## 4. Conclusions

ψ-Caputo FD, a general fractional operator, is of great use because of its wide freedom to cover many classical fractional operators. In this work, we have studied the uniqueness of solution for the nonlinear ψ-Caputo type FIDE (1) by using the Banach space AC0a,b, Banach’s fixed point theorem, and Babenko’s method. Moreover, the UH stability results to the proposed problem have been discussed. Also, some pertinent examples have been provided to justify the main results. The obtained results in this study extended and developed the current results introduced by [17, 18]. We have already concluded that our results are valid when the left-hand side of the considered problem (1) involves many FDs and IDs as shown in Remark 22. Furthermore, problem (1) covers previous standard cases of nonlinear FDEs and FIDEs by selecting the suitable standard kernel in the studied problem. More specifically, our results generalize some known results in literature like those that include Hadamard and Katugampola FDs.For future research, we will consider a class of nonlinear FIDEs with the fuzzy initial conditions in a fractional case. It would also be interesting to study the same results for our current problem under theψ-Hilfer operator [33] or Atangana-Baleanu operator [8].



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