# New Existence and Stability Results forψ,w-Caputo−Fabrizio Fractional Nonlocal Implicit Problems

Authors: Mohammed S. Abdo; Wafa Shammakh; Hadeel Z. Alzumi
Journal: Journal of Mathematics
 (2023)

---

## Abstract

In the context ofψ-weighted Caputo−Fabrizio fractional derivatives, we develop and extend the existence and Ulam−Hyers stability results for nonlocal implicit differential equations. The fixed-point theorems due to Banach and Krasnoselskii are the foundation for the proof of existence and uniqueness results. Additionally, the Ulam−Hyers stability demonstrates the assurance of the existence of solutions via Gronwal inequality. Also, we offer an example as an application to explain and validate the acquired results. Finally, in terms of our outcome, we designate a more general problem for the ψ,w-Caputo−Fabrizio fractional system that includes analogous problems to the problem at hand.


---

## Body

## 1. Introduction

It is noteworthy that fractional calculus (FC) has received considerable attention from researchers due to its wide variety of applications in several scientific domains. The significant concepts and definitions of FC have been introduced by Osler [1] and Kilbas et al. [2]. Samko et al. [3] and Diethelm and Ford [4] provided some basic history of FC and its applications in engineering and various fields of science.Many classes of fractional differential equations (FDEs) have been extensively investigated and analyzed in the past decades; for instance, theories involving the existence of unique solutions have been notarized in [5–7] and references therein. Numerical and analytical methods are developed with the aim of solving such equations then tracked as useful in modeling some real-world problems, as shown in [8–10].The qualitative properties of solutions address an indispensable piece of the theory of FDEs. The beforehand previously mentioned district has been investigated well for classical DEs. Regardless, for FDEs, there are various aspects and viewpoints that require further investigation and analysis. The existence and uniqueness have been intensively studied by using Riemann−Liouville (R-L), Caputo, Hilfer, and other FDs, as shown in [11–16] and references therein.Generalized FDs and integrals and their applications were discussed by several authors. For example, Kilbas et al. [2] introduced some nice properties of ψ-Reiman−Liouville FD. The ψ-Caputo FD has been defined by Almedia [17]. Then, Sousa and Oliveira proposed another generalization in the Hilfer sense [18].In the previously mentioned derivatives, there exists a singular kernel. In this manner, as of late, some authors presented some new kinds of FDs in which they have supplanted a singular kernel with a nonsingular kernel, as shown in [19–21]. The nonlocal FDs with nonsingular kernels have been demonstrated as a decent tool to model real-world problems in various areas of engineering and science, as shown in [22–25].On the other hand, Jarad et al. [26] introduced the concept of weighted FDs. Some recent papers dealt with the theory of existence and Ulam−Hyers (UH) (and Generalized Ulam−Hyers (GUH)) stability of different types of FDEs due to their importance in many areas of exploration; for instance, Shaikh et al. [27] established the existence and uniqueness results of the following CF-type Cauchy problem:(1)D0+ϑCFυϰ=fϰ,υϰ,ϰ∈0,1,0<ϑ<1,υ0=υ0.The existence and UH stability results in the following CF fractional implicit equation:(2)D0+ϑCFυϰ=fϰ,υϰ,D0+ϑCFυϰ,ϰ∈0,T,0<ϑ<1.υ0=υ0 and aυ0+aυT=c were investigated by Abbas et al. [28] in b-metric spaces and by Salim et al. [29] in Banach spaces, respectively. The equation (2) with 0<ϑ≤2 was considered by Gul et al. [30].On the other hand, Abdo et al. [31], considered the following weighted Caputo problem:(3)D0+ϑ;ψ,wCρϰ=φϰ,ρϰ,0<ϑ≤1,ρ0=ρ0.In this regard, Al-Refai and Jarrah [32], introduced the concept of the ψ,w-Caputo−Fabrizio FD, where ψϰ is a monotone function and wϰ is a weight function. They also obtained the uniqueness result of the Cauchy problem. Motivated by studies [31, 32], we consider the following weighted implicit nonlocal FDE:(4)Da;wμCFρϰ=φϰ,ρϰ,Da;wμCFρϰ,ρa=c+gρ,and the following ψ,w type implicit nonlocal FDE:(5)Da;wμ;ψCFρϰ=φϰ,ρϰ,Da;wμ;ψCFρϰ,ρa=c+gρ,where 0<μ<1, ϰ∈℧:=a,b, c∈ℝ, g∈C℧, φ:℧×ℝ×ℝ⟶ℝ is a given function, and Da;wμ;ψCF is a ψ,w- Caputo−Fabrizio FD, and w,ψ∈C1℧ with w,w′,ψ>0 on ℧.We pay attention to the topic of the novel weighted operators with another function. As far as we are aware, no studies using theψ,w-Caputo−Fabrizio FDs have been published that address the qualitative aspects of the aforementioned problems. Consequently, to enhance and enrich the literature on this new trend, which is extremely restricted right now, we develop and extend the existence, and Ulam–Hyers stability results for problems (4) and (5) based on Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, and Gronwal inequality. Besides, we also give a more general problem as a system, that covers the problems at hand.Remark 1.
(i)
Ifψϰ=ϰ, then problem (5) reduces to problem (4)(ii)
Ifψϰ=ϰ, w=1, and g≡0, then problem (5) reduces to the implicit problem (2), as shown in [28, 29](iii)
Ifψϰ=ϰ, w=1, and g≡0, then problem (5) reduces to the Cauchy problem (1) without implicit term, as shown in [27](iv)
Our current results for the problem (5) stay available on problem (4)(v)
With different values ofψ, our current problems cover many problems associated with less general operators; for instance, the operator presented by Caputo and Fabrizio [19]
The accessories of this paper are arranged as follows: Section2 gives some fundamental results about advanced FC. Our key findings for the problems (4) and (5) are obtained in Section 3. A comprehensive example that verifies the validity of the theories is provided in Section 4. Section 5 includes the conclusions of the work.


## 2. Primitive Results

In this section, we begin by giving some notations and basic nomenclature. Let℧:=a,b, a<b<∞, and ℝ be the set of real numbers. Ca,b and ACa,b denote the set of continuous and absolutely continuous functions, respectively, on ℧, endowed with the usual supremum norm. Let ψϰ and wϰ be the monotone and weight function, respectively, with w,ψ∈C1℧ and w,w′,ψ′>0 on ℧.Definition 1 (see [32]).
Let0<ϑ<1, and ρ∈AC℧,ℝ. The left ψ,w-Caputo−Fabrizio FD is defined as(6)Da;wμ;ψCFρϰ=ℵμ1−μ1wϰ∫aϰe−λμψϰ−ψζddζwρζdζ,where λμ=μ/1−μ, and ℵμ is a normalization function satisfying ℵ0=ℵ1=1.
The previous operator can be written as(7)Da;wμ;ψCFρϰ=ℵμ1−μe−λμψϰwϰ∫aϰeλμψζddζwρζdζ.Definition 2 (see [32]).
Let0<ϑ<1, and ρ∈AC℧,ℝ. The left ψ,w-Caputo−Fabrizio FI is defined as(8)Ia;wμ;ψCFρϰ=1−μℵμρϰ+μℵμ1wϰ∫aϰψ′ζwζρζdζ.Lemma 1 (see [32]).
Letρ∈AC℧,ℝ. Then,(9)Da;wμ;ψCFIa;wμ;ψCFρϰ=ρϰ,Da;wμ;ψCFIa;wμ;ψCFρϰ=ρϰ−waρawϰ.In particular, ifρa=0, we have Da;wμ;ψCFIa;wμ;ψCFρϰ=ρϰ.Lemma 2 (see [32]).
Let0<ϑ<1, and ρ∈AC℧,ℝ with fa=0. Then, the following FDE:(10)Da;wμ;ψCFIa;wμ;ψCFρϰ=fϰ,ρa=c,has the unique solution(11)ρϰ=wawϰc+1−μℵμfϰ+μℵμ1wϰ∫aϰψ′ζwζfζdζ,ϰ∈℧.For our forthcoming analysis, we need Banach’s contraction map [33] and Krasnoselskii’s fixed point theorem [34].Lemma 3 (Gronwall’s Lemma [35]).
Leta,b>0, and d≥0. Let’s assume that functions z,y:℧⟶ℝ+ are continuous if(12)zϰ≤d+∫aϰyζzζdζ,∀ϰ∈℧.Then,(13)zϰ≤dexp∫aϰyζdζ,∀ϰ∈℧.


## 3. Main Results

In this section, we give some qualitative analyses ofψ,w -Caputo−Fabrizio type problems (4) and (5).Lemma 4.
Let0<ϑ<1, ρ∈AC℧,ℝ and g∈C℧,ℝ be continuous with fa=0. Then, the following ψ-weighted FDE:(14)Da;wμ;ψCFρϰ=fϰ,ϰ∈℧,ρa=c+gρ,has the unique solution(15)ρϰ=wawϰc+gρ+aμfϰ+bμwϰ∫aϰψ′ζwζfζdζ,ϰ∈℧,where aμ=1−μ/ℵμ, and bμ=μ/ℵμ.Proof.
Let’s assume thatρ satisfies the first equation of (14). From Corollary 2.1 in [32], the equationDa;wμ;ψCFρϰ=fϰimplies that(16)ρϰ=waρawϰ+aμfϰ+bμwϰ∫aϰψ′ζwζfζdζ.
So, by the nonlocal conditionρa=c+gρ, we obtain(17)ρϰ=wawac+gρ+aμfϰ+bμwϰ∫aϰψ′ζwζfζdζ,which is (15).
Conversely, ifρ satisfies (15), then by Lemma 1, we have(18)Da;wμ;ψCFρϰ=Da;wμ;ψCFwawϰc+gρ+Da;wμ;ψCFaμfϰ+bμwϰ∫aϰψ′ζwζfζdζ=Ia;wμ;ψCFDa;wμ;ψCFfϰ=fϰ,where Da;wμ;ψCFwa/wϰc+gρϰ=0. Moreover, ρa=c+gρ.Hence, we can deduce the next corollary:Corollary 1.
Let0<ϑ<1, φ∈AC℧,ℝ and g∈C℧,ℝ with φρ0=0. Then, the problem (5) is equivalent to(19)ρϰ=wawϰc+gρ+aμφρϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧,where φρ∈C℧,ℝ with φρϰ:=φϰ,ρϰ,φρϰ.We define the operatorK:AC℧,ℝ⟶AC℧,ℝ by(20)Kρϰ=wawϰc+gρ+aμφρϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧.Then, the fixed point of operatorK is equivalent to the solution of the ψ,w-type problem (5).The first result is based on Banach FPT [33].Theorem 1.
assume that:(Hy1): There exists Lφ,Mφ>0 such that(21)φϰ,υ,ω−φϰ,υ¯,ω¯≤Lφυ−υ¯+Mφω−ω¯,for eachϰ∈℧, υ,ω,υ¯,ω¯∈ℝ.(Hy2): There exists Lg>0 such that 0<Lg<1 and gυ−gω≤Lgυ−ω, for υ,ω∈C℧,ℝ.If(22)Lg+aμ+bμwbwaψb−ψaLφ1−Mφ<1.Then, theψ,w-type problem (5) has a unique solution on AC℧,ℝ.Proof.
Letφρϰ=φϰ,ρϰ,φρϰ and φρ¯ϰ=φϰ,ρ¯ϰ,φρ¯ϰ. Then, by (Hy1), we have (23)φρϰ−φρ¯ϰ=φϰ,ρϰ,φρϰ−φϰ,ρ¯ϰ,φρ¯ϰ≤Lφρϰ−ρ¯ϰ+Mφφρϰ−φρ¯ϰ,which implies(24)φρϰ−φρ¯ϰ≤Lφ1−Mφρϰ−ρ¯ϰ.
Sinceψ′,w>0, and applying the mean value theorem for integrals, we obtain(25)∫aϰψ′ζwζdζ=wκ∫aϰψ′ζdζ=wκψϰ−ψa,for some a<κ<b, then(26)∫aϰψ′ζwζdζ≤wbψb−ψa.
We supposeρϰ,ρ¯ϰ∈AC℧,ℝ, we have from (Hy2), (24), and (26) that(27)Kρϰ−Kρ¯ϰ≤supϰ∈℧wawϰgρ−gρ¯+aμφρϰ−φρ¯ϰ+bμwϰ∫aϰψ′ζwζφρζ−φρ¯ζdζ≤Lgρ−ρ¯+aμLφ1−Mφρ−ρ¯+Lφ1−Mφρ−ρ¯wbwabμψb−ψa=Lg+aμ+wbwabμψb−ψaLφ1−Mφρ−ρ¯.
As the condition (22), K is a contraction and by the Banach’s fixed point theorem, K has a unique fixed point which is unique solution of (5).Next, we give existence results based on the Krasnoselskii fixed point theorem [34].Theorem 2.
Letφ:℧×ℝ×ℝ⟶ℝ, and g:C℧,ℝ⟶ℝ be continuous satisfying (Hy1) and (Hy2). In addition, we assume that(Hy3): There exist constants nφ,mφ,sφ>0 with 0<sφ<1 such that(28)φϰ,ρ1,ρ2≤nφ+mφρ1+sφρ1,for eachϰ∈℧, ρ1,ρ2∈ℝ.(Hy4): gρ≤μgρ, for ρ∈C℧,ℝ, and μg>0.Then, theψ,w-type problem (5) has a least one solution if(29)μg+aμ+wbwabμψb−ψamφ1−sφ<1.Proof.
From (20), we define the operators K1,K2:AC℧,ℝ⟶AC℧,ℝ by(30)K1ρϰ=wawϰc+gρ+aμφρϰ,ϰ∈℧,and(31)K2ρϰ=bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧,where K1ρ+K2ρϰ=Kρϰ. Let us define ℱr=ρ∈AC℧,ℝ:ρ≤r. We choose(32)r≥c+aμ+wb/wabμψb−ψanφ/1−sφ1−μg+aμ+wb/wabμψb−ψamφ/1−sφ.
Forφρζ=φζ,ρζ,φρζ, we use assumption (Hy3) to get(33)φρζ=φζ,ρζ,φρζ≤nφ+mφρζ+sφφρζ,which implies(34)φρζ≤nφ+mφρζ1−sφ.
For anyρ,ρ⋆∈ℱr, we have(35)K1ρ+K2ρ⋆ϰ≤K1ρϰ+K2ρ⋆ϰ≤supϰ∈℧wawϰc+gρ+aμφρϰ+supϰ∈℧bμwϰ∫aϰψ′ζwζφρ⋆ζdζ≤c+μgρ+aμnφ+mφρ1−sφ+nφ+mφρ⋆1−sφwbwabμψb−ψa≤μg+aμ+wbwabμψb−ψamφ1−sφr+c+aμ+wbwabμψb−ψanφ1−sφ≤r.
Due to (29), we deduce that K1ρ+K2ρ⋆≤r.
Sinceφ and g are continuous, we show that K1 is a contraction operator. For each ρϰ,σϰ∈AC℧,ℝ, and for any ϰ∈℧, we have(36)K1ρϰ−K1σϰ≤supϰ∈℧wawϰgρ−gσ+aμφρϰ−φσϰ≤Lgρ−σ+aμLφ1−Mφρ−σ.
Hence,(37)K1ρ−K1σ≤Lg+aμLφ1−Mφρ−σ.
From (22), Lg+aμLφ/1−Mφ<1. So, K1 is a contraction.
Next, to prove thatK2 is a compact and continuous operator, we provide the following steps:Step 1:K2 is continuous. Let ρnn≥1 be in ℬr such that ρn⟶ρ in ℬr. Then, K2ρnϰ−K2ρϰ⟶0, as n⟶∞, due to continuity of φ,ψ and w.Step 2:K2 is uniformly bounded. By (31) we have(38)K2ρϰ≤supϰ∈℧K2ρϰ≤supϰ∈℧bμwϰ∫aϰψ′ζwζφρζdζ.From (26) and (34), and for any ρ∈ℱr, (38) becomes(39)K2ρϰ≤supϰ∈℧bμwϰ∫aϰψ′ζwζnφ+mφρζ1−sφdζ≤nφ+mφρ1−sφbμwbwaψb−ψa≤nφ+mφrbμ1−sφwbwaψb−ψa≤r⋆.Thus,K2 is uniformly bounded on ℱr.Step 3:K2 is compact. Let ρ∈ℱr, and ϰ∈℧ with ϰϵ<ϰδ∈℧. Then,(40)K2ρϰδ−K2ρϰϵ≤bμwϰδ∫aϰδψ′ζwζφρζdζ−bμwϰϵ∫aϰϵψ′ζwζφρζdζ≤bμwϰδ−bμwϰϵ∫aϰδψ′ζwζφρζdζ+bμwϰϵ∫ϰϵϰδψ′ζwζφρζdζ≤bμwϰδ−wϰϵ∫aϰδψ′ζwζφρζdζ+bμwϰϵ∫ϰϵϰδψ′ζwζφρζdζ≤wϰδ−wϰϵnφ+mφrbμ1−sφwϰϵψϰϵ−ψa+nφ+mφrbμ1−sφwϰδwϰϵψϰδ−ψϰϵ.
Since(41)∫ϰϵϰδψ′ζwζdζ≤wϰδψϰδ−ψϰϵ,∫aϰϵψ′ζwζdζ≤wϰϵψϰϵ−ψa,due to mean value theorem for integrals. Consequently, (40) gives(42)K2ρϰδ−K2ρϰϵ⟶0,asϰδ−ϰϵ⟶0.
Thus,K2 is equicontinuous on ℱr. As per preceding steps, K2 is relatively compact on ℱr , and by the Arzela−Ascoli theorem, K2 has at least one fixed point. By virtue of Krasnoselskii’s theorem [34], the ψ -weighted problem (5) has a least one solution.


### 3.1. UH Stability Analysis

In this section, we give the UH stability and generalized UH stability results for theψ-weighted problem (5).Definition 3.
(5) is UH stable if there exists a χφ>0 such that for each ε>0 and each solution σ∈AC℧,ℝ of the inequality(43)Da;wμ;ψCFσϰ−φϰ,σϰ,Da;wμ;ψCFσϰ≤ε,ϰ∈℧,there exists a solution ρ∈AC℧,ℝ of (5), satisfying(44)σϰ−ρϰ≤χφε.Remark 2.
σ∈AC℧,ℝ satisfies (44) if and only if there exists ϖ∈AC℧,ℝ with(i)
ϖϰ≤ε, ϰ∈℧,(ii)
For allϰ∈℧,(45)Da;wμ;ψCFσϰ=φϰ,σϰ,Da;wμ;ψCFσϰ+ϖϰ.Lemma 5.
Let0<μ<1, and σ∈AC℧,ℝ be a solution of (43). Then, σ satisfies(46)σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ≤εbμwbwaψb−ψa,ϰ∈℧,where(47)Aσϰ≔=wawϰc+gσ+aμφσϰ,and φσϰ=φt,ωt,φσϰ.Proof.
Letσ be a solution of (43). It follows from (ii) of Remark 2 that(48)Da;wμ;ψCFσϰ=φϰ,σϰ,Da;wμ;ψCFσϰ+ϖϰσ0=c+gσ.
Then, the solution of problem (48) is(49)σϰ=Aσϰ+bμwϰ∫aϰψ′ζwζφσζ+ϖζdζ,ϰ∈℧.
Once more by (i) of Remark2, we get(50)σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ≤bμwϰ∫aϰψ′ζwζϖζdζ≤bμwϰε∫aϰψ′ζwζdζ≤εwbwabμψb−ψa.Theorem 3.
Assume that the conditions of Theorem1 are fulfilled. Then, the solution of the ψ-weighted problem (5) is UH and generalized UH stable.Proof.
Letσ∈AC℧,ℝ be a solution of (43), and ρ∈AC℧,ℝ be a unique solution of ψ-weighted problem (5)(51)Da;wμ;ψCFρϰ=φϰ,ρϰ,Da;wμ;ψCFρϰρa=σa.
From Corollary1, we get(52)ρϰ=Aρϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧,where Aρϰ:=wa/wϰc+gρ+aμφρϰ. Clearly, if ρa=σa, then gρ=gσ. Also, by (Hy1) and (Hy2) along with (24), we have(53)Aρϰ−Aσϰ≤wawϰgρ−gσ+aμφρϰ−φσϰ≤aμLφ1−Mφρ−σ.
From (22), we get Aρϰ=Aσϰ. Hence, (52) becomes(54)ρϰ=Aσϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧.
Using Lemma5, (Hy1), and (Hy2), we have(55)σϰ−ρϰ=σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφρζdζ≤σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ+bμwϰ∫aϰψ′ζwζφσζ−φρζdζ≤εwbwabμψb−ψa+aμLφ1−Mφ∫aϰψ′ζwζσζ−ρζdζ.
Using Gronwall’s Lemma [35], we obtain(56)σϰ−ρϰ≤εwbwabμψb−ψaexpaμLφ1−Mφ∫aϰψ′ζwζdζ≤εwbwabμψb−ψaexpaμLφ1−Mφwbwabμψb−ψa=εχφ,where(57)χφ=wbwabμψb−ψaexpaμLφ1−Mφwbwabμψb−ψa.
Hence, (5) is UH stable. Moreover, if there exists a nondecreasing function ϕ:ℝ+⟶ℝ+ such that ϕε=ε. Then, from (56), we have(58)σϰ−ρϰ≤χφϕϵ,ϰ∈℧,with ϕ0=0, which proves (5) is GUH stable.


## 3.1. UH Stability Analysis

In this section, we give the UH stability and generalized UH stability results for theψ-weighted problem (5).Definition 3.
(5) is UH stable if there exists a χφ>0 such that for each ε>0 and each solution σ∈AC℧,ℝ of the inequality(43)Da;wμ;ψCFσϰ−φϰ,σϰ,Da;wμ;ψCFσϰ≤ε,ϰ∈℧,there exists a solution ρ∈AC℧,ℝ of (5), satisfying(44)σϰ−ρϰ≤χφε.Remark 2.
σ∈AC℧,ℝ satisfies (44) if and only if there exists ϖ∈AC℧,ℝ with(i)
ϖϰ≤ε, ϰ∈℧,(ii)
For allϰ∈℧,(45)Da;wμ;ψCFσϰ=φϰ,σϰ,Da;wμ;ψCFσϰ+ϖϰ.Lemma 5.
Let0<μ<1, and σ∈AC℧,ℝ be a solution of (43). Then, σ satisfies(46)σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ≤εbμwbwaψb−ψa,ϰ∈℧,where(47)Aσϰ≔=wawϰc+gσ+aμφσϰ,and φσϰ=φt,ωt,φσϰ.Proof.
Letσ be a solution of (43). It follows from (ii) of Remark 2 that(48)Da;wμ;ψCFσϰ=φϰ,σϰ,Da;wμ;ψCFσϰ+ϖϰσ0=c+gσ.
Then, the solution of problem (48) is(49)σϰ=Aσϰ+bμwϰ∫aϰψ′ζwζφσζ+ϖζdζ,ϰ∈℧.
Once more by (i) of Remark2, we get(50)σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ≤bμwϰ∫aϰψ′ζwζϖζdζ≤bμwϰε∫aϰψ′ζwζdζ≤εwbwabμψb−ψa.Theorem 3.
Assume that the conditions of Theorem1 are fulfilled. Then, the solution of the ψ-weighted problem (5) is UH and generalized UH stable.Proof.
Letσ∈AC℧,ℝ be a solution of (43), and ρ∈AC℧,ℝ be a unique solution of ψ-weighted problem (5)(51)Da;wμ;ψCFρϰ=φϰ,ρϰ,Da;wμ;ψCFρϰρa=σa.
From Corollary1, we get(52)ρϰ=Aρϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧,where Aρϰ:=wa/wϰc+gρ+aμφρϰ. Clearly, if ρa=σa, then gρ=gσ. Also, by (Hy1) and (Hy2) along with (24), we have(53)Aρϰ−Aσϰ≤wawϰgρ−gσ+aμφρϰ−φσϰ≤aμLφ1−Mφρ−σ.
From (22), we get Aρϰ=Aσϰ. Hence, (52) becomes(54)ρϰ=Aσϰ+bμwϰ∫aϰψ′ζwζφρζdζ,ϰ∈℧.
Using Lemma5, (Hy1), and (Hy2), we have(55)σϰ−ρϰ=σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφρζdζ≤σϰ−Aσϰ−bμwϰ∫aϰψ′ζwζφσζdζ+bμwϰ∫aϰψ′ζwζφσζ−φρζdζ≤εwbwabμψb−ψa+aμLφ1−Mφ∫aϰψ′ζwζσζ−ρζdζ.
Using Gronwall’s Lemma [35], we obtain(56)σϰ−ρϰ≤εwbwabμψb−ψaexpaμLφ1−Mφ∫aϰψ′ζwζdζ≤εwbwabμψb−ψaexpaμLφ1−Mφwbwabμψb−ψa=εχφ,where(57)χφ=wbwabμψb−ψaexpaμLφ1−Mφwbwabμψb−ψa.
Hence, (5) is UH stable. Moreover, if there exists a nondecreasing function ϕ:ℝ+⟶ℝ+ such that ϕε=ε. Then, from (56), we have(58)σϰ−ρϰ≤χφϕϵ,ϰ∈℧,with ϕ0=0, which proves (5) is GUH stable.


## 4. Example

Here, we provide an example to illustrate the obtained results.Example 1.
consider the followingψ,w-Caputo−Fabrizio type problem:(59)D0+,ϰ/31/4,ϰ/2CFρϰ=ϰ−1310sinρϰ+sinD0+,ϰ/31/4,ϰ/2CFρϰ5+ϰ+ϰ−1320,ϰ∈1,2,υ1=13+∑i=1ndiρϰi.
Clearly,μ=1/4,ψϰ=ϰ/2, wϰ=ϰ/3, a=1, b=2, c=1/3, 1<ϰ1=1.5<⋯<ϰn<2=b, and di are positive constants with ∑i=1ndi<2/5. Set(60)φϰ,ρ,σ=ϰ−1310sinρ+sinσ5+ϰ+12,for ϰ∈1,2, ρ,σ∈0,∞, and(61)gρ=∑i=1ndiρϰi,ρ∈0,∞.(I)
Application of Theorem1: Note that φ1,ρ1,σ1=0. Now, let ϰ∈1,2, and ρ,σ,ρ⋆,σ⋆∈0,∞. Then,(62)φϰ,ρ,σ−φϰ,ρ⋆,σ⋆=ϰ−1310sinρ+sinσ5+ϰ−sinρ⋆+sinσ⋆5+ϰ≤ϰ−1310sinρ−sinρ⋆5+ϰ+sinσ−sinσ⋆5+ϰ≤170ρ−ρ⋆+σ−σ⋆,gρ−gσ=∑i=1ndiρϰi−∑i=1ndiσϰi≤∑i=1ndiρ−σ≤25ρ−σ.Thus, (Hy1) and (Hy2) hold with Lφ=Mφ=1/70, and Lg=2/5. In addition, the condition (22) holds. Indeed, we have ℵμ=1, aμ=3/4, bμ=1/4 and(63)Lg+aμ+bμwbwaψb−ψaLφ1−Mφ=143345<1.Thus, Theorem1 shows that (59) has a unique solution on 1,2.(II)
Application of Theorem3: For ε>0 with χφ=1/8e736>0. It follows from Theorem 3 that (59) is HU and GUH stable.(III)
Application of Theorem2: For ϰ∈1,2, and ρ,σ∈0,∞, we have(64)φϰ,ρ,σ=ϰ−1310sinρ5+ϰ+ϰ−1310sinσ5+ϰ+ϰ−1320≤ϰ−1320+ϰ−13105+ϰρ+ϰ−13105+ϰσ≤120+170ρ+170σ,gυ=∑i=1ndiρϰi≤∑i=1ndiρ≤25υ.
Consequently, (Hy3) and ((Hy4) hold with nφ=1/20,mφ=sφ=1/70 and μg=2/5. Also, we have(65)μg+aμ+wbwabμψb−ψamφ1−sφ=143345<1.
Thus, all the assumptions of Theorem2 are satisfied. Hence, (59) has a solution on [1, 2].Remark 3.
consider a more general problem as a system that contains a number of problems similar to our current problem (5) as follows:(66)Da;wμi;ψCFρiϰ=φiϰ,ρiϰ,Da;wμi;ψCFρiϰ,ϰ∈℧ρia=ci+gρi,i=1,…,n∈ℕ,where 0<μi<1, Da;wμi;ψCF is the generalized Caputo−Fabrizio FD of order μi.The system (66) can be written as(67)Da;wY;ψCFPϰ=Φϰ,Pϰ,Da;wμi;ψCFPϰ,ϰ∈℧,Pa=C+gP,where(68)Pϰ=ρ1ϰρ2ϰ⋮ρnϰ,Φϰ,Pϰ,Da;wμi;ψCFPϰ=φ1ϰ,ρ1ϰ,Da;wμi;ψCFρ1ϰφ2ϰ,ρ1ϰ,Da;wμi;ψCFρ1ϰ⋮φnϰ,ρ1ϰ,Da;wμi;ψCFρ1ϰ,Pa=ρ1aρ2a⋮ρna,C=c1c2⋮cn,Y=μ1μ2⋮μn.By using Corollary1, the system (67) has the following solution:(69)Pϰ=wawϰC+gP+aYΦPϰ+bYwϰ∫aϰψ′ζwζΦPζdζ,ϰ∈℧,where ΦPϰ=Φϰ,Pϰ,ΦPϰ,aY=1−Y/ℵY, and bY=Y/ℵY.Remark 4.
Following the methodology of proof used in the preceding parts, we can obtain the same results (Theorems1–3) for the nonlinear system (67) in view of the formula (69).


## 5. Conclusions

In this regard, Al-Refai and Jarrah [32] defined the weighted Caputo−Fabrizio FD of the Caputo sense. As an extra contribution to this topic, we developed and extended the existence, uniqueness, and UH stability results for nonlocal implicit equations involving ψ,w -Caputo−Fabrizio FDs. Our approach has been based on Banach’s and Krasnoselskii’s fixed point theorem. As an application, we have given a convenient example that validates the theoretical results. Finally, in light of our present results, a more general problem for the nonlocal implicit system has been presented that contains similar problems to the problem considered.In the future direction, it will be interesting to study the current systems underψ,w-Atangana−Baleanu of the Caputo sense, introduced recently in [21, 36].



---

*Source: 6123608-2023-03-01.xml*