# Hyers–Ulam–Mittag-Leffler Stability for a System of Fractional Neutral Differential Equations

Authors: Manzoor Ahmad; Jiqiang Jiang; Akbar Zada; Zeeshan Ali; Zhengqing Fu; Jiafa Xu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This article concerns with the existence and uniqueness for a new model of implicit coupled system of neutral fractional differential equations involving Caputo fractional derivatives with respect to the Chebyshev norm. In addition, we prove the Hyers–Ulam–Mittag-Leffler stability for the considered system through the Picard operator. For application of the theory, we add an example at the end. The obtained results can be extended for the Bielecki norm.


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## Body

## 1. Introduction

Models of fractional differential equationsFDEs have many applications in various fields of engineering and science such as mechanics, electricity, biology, chemistry, physics, and control and signal processing[1, 2]. For the different materials and phenomena, the heredity characteristics are well explained by FDEs, and as a result, many research papers and books have been published in this field [3–7]. FDEs in which the highest fractional derivatives FDs of unknown term appear both with and without delays are known as neutral FDEs. In the last few years, the study of neutral FDEs have developed dramatically. This is due the fact that the qualitative behavior of the aforesaid equations are quite different from those of nonneutral FDEs. Neutral FDEs also play a key role and has many advantages. For instance, they give more better description over population fluctuations. Also, neutral FDEs with delay appear in models of electrical networks containing lossless transmission lines, for more details see [8]. Different attempts have been made for the investigation of solutions of fractional and neutral FDEs [9–17].Another imperative and more remarkable area of research is committed to the stability analysis of the solutions for theFDEs and ordinary orders. Many targets are achieved in this regard, for some recent work we refer the reader to see [18–35]. Niazi et al. [36] investigated the existence, uniqueness (EU), and Hyers–Ulam–Mittag-Leffler (HUML) stability for the following neutral FDEs:(1)D0κcuω=gω,uω,D0σcuω,ω∈0,1,uω=φ1ω,ω∈−ξ,0,where 1<κ≤2,0<σ<1, D0κcandD0σc are the Caputo derivatives, and g:0,1×Cξ×Cξ⟶ℝ is continuous. Here, Cξ denotes the Banach space of all continuous functions φ1:−ξ,1⟶ℝ with norm(2)φ1=maxφ1ω:−ξ≤ω≤1.Ifuω∈Cξ, then is defined by(3)uωω=uω+τ,τ∈−ξ,1,ω∈0,1.As far as we know, results onEU and HUML stability for a coupled system of neutral FDEs have not been investigated by the researchers. Many real-world problems required to be modeled into a coupled system of FDEs as they cannot be described into a single fractional differential equation, see [37] and references cited therein.Motivated by the abovementioned work, in this article, we study theEU and HUML stability of the following implicit coupled neutral FDEs system involving Caputo FDs. The proposed coupled system (CS) is given by(4)D0κcuω=gω,uω,D0σcvω,ω∈J,D0σcvω=hω,D0κcuω,vω,ω∈J,uω=φ1ω,ω∈−ξ,0,vω=φ2ω,ω∈−ξ,0,where J=0,1,1<κ,σ≤2 and D0κcandD0σc represent the Caputo FDs of orders κandσ, respectively. The set of all continuous functions φ1,φ2:−ξ,1⟶ℝ with norms φ1=maxφ1ω:−ξ≤ω≤1 and φ2=maxφ2ω:−ξ≤ω≤1 is a Banach space denoted by Cξ, and hence their product is also a Banach space. The functions g,h:J×Cξ×Cξ⟶ℝ are continuous. If u,v:−ξ,1⟶ℝ are continuous, then ∀ω∈J, and the functions uω,vω∈Cξ are presented by uωτ=uω+τ,vωτ=vω+τ,τ∈−ξ,1,−ξ<0.


## 2. Preliminaries

This section is concerned with some notions, definitions, and preliminary results used throughout the article.Definition 1.
(see [38]). The κ>0 order Riemann–Liouville integral for θ∗ is(5)Iaκθ∗ω=1Γκ∫aωω−sκ−1θ∗sds,where the integral is point-wise convergent.Definition 2.
(see [38]). Suppose θ∗ be a given function on a,b, the κ∈n−1,n order Caputo derivative of θ∗ is stated by(6)Daκcθ∗ω=∫aωω−sn−κ−1Γn−κdnθ∗sdsnds,with κ=n−1. Furthermore, if domain of θ∗ is a,b and κ∈0,1, then(7)Daκcθ∗ω=1Γ1−κ∫aωθ∗′sω−sκds,where θ∗′s=dθ∗s/ds.Theorem 1 (see [39]).
Letκ∈n−1,n, then θ∗∈Ca,b,ℝ, which implies dκ/dtκθ∗ω=0 has the formula(8)θ∗ω=∑k=0κckωk,where ck∈ℝ, k=1,2,…,κ, κ+1=n.Definition 3.
(see [40]). For a metric space Yκ,σ,d, an operator Aκ,σ:Yκ,σ⟶Yκ,σ is said to be Picard operator PO if there is x∗∈Yκ,σ so that(a)
FAκ,σ=x∗ where FAκ,σ=x∈Yκ,σ:Aκ,σx=x.(b)
The sequenceAκ,σnx0n∈N has the limit x∗∀x0∈Yκ,σ.Definition 4.
(see [40]). For an ordered metric space Yκ,σ,d,≼, if Aκ,σ:Yκ,σ⟶Yκ,σ is increasing PO with Aκ,σx∗=x∗, then ∀x∈Yκ,σ, x≤Aκ,σx implies x≤x∗ and x≥Aκ,σx implies x≥x∗.Lemma 1 (see [41]).
Letθ1,θ2∈CJ,ℝ+. If κ>0, θ2 is increasing and there is m≥0 so that(9)θ1ω≤θ2ω+m∫0ωω−sκ−1θ1sds,ω∈J.
Then,(10)θ1ω≤θ2ω+m∫0ω∑k=1∞mΓκnΓnκω−snκ−1θ1sds,ω∈J.Lemma 2 (see [42]).
Forλ>0 and κ>−1, we have(11)∫0ωω−sκ−1sλds=ΓλΓκ+1Γλ+κ+1ωκ+λ.


## 3. Main Results

In this section, we provide results regarding theEU and HUML stability for the solution of the considered system on the compact interval J=0,1, using the PO and Henry–Gronwall lemma [41]. Suppose g,h:J×Cξ×Cξ⟶ℝ be continuous and Cξ be a Banach space of all the functions φ1,φ2:−ξ,1⟶ℝ which are continuous with norms φ1=maxφ1ω:−ξ≤ω≤1 and φ2=maxφ2ω:−ξ≤ω≤1. Consider the system(12)D0κcuω=gω,uω,D0σcvω,ω∈J,D0σcvω=hω,D0κcuω,vω,ω∈J,uω=φ1ω,ω∈−ξ,0,vω=φ2ω,ω∈−ξ,0,where 1<κ,σ≤2 and φ1,φ2∈C−ξ,1,ℝ+. For ϵ1,ϵ2, we have the following inequalities:(13)D0κcuω−gω,uω,D0σcvω≤ϵ1Eκωκ,ω∈J,D0σcvω−hω,D0κcuω,vω≤ϵ2Eσωσ,ω∈J,where Eκ⋅ and Eσ⋅ represent the Mittag-Leffler ML function defined by(14)Eκ⋅=∑k=0∞⋅kΓkκ+1,Eσ⋅=∑k=0∞⋅kΓkσ+1,⋅∈ℂ,Reκ,Reσ>0.Definition 5.
Extending the definition of Hyers–UlamHU stability for a CS [43], we say that system (12) is HUML stable with respect to Eλωλ=maxEκωκ,Eσωσ if there is cEλ=maxcEκ,cEσ>0 so that for every ε=maxϵ1,ϵ2>0 and each solution u¯,v¯∈Cξ×Cξ of the inequalities (13) there is a unique solution u,v∈Cξ×Cξ so that(15)u,v−u¯,v¯≤cEλϵEλωλ,λ=maxκ,σ.Remark 1.
Functionsu¯,v¯∈C1J,ℝ are, respectively, the solutions of the above inequalities ⟺ there is θ1,θ2∈C1J,ℝ so that(1)
θ1ω≤ϵ1Eκωκ, θ2ω≤ϵ2Eσωσ,ω∈J;(2)
D0κcuω=gω,uω,Dσcvω+θ1ω,ω∈J,D0σcvω=hω,Dκcuω,vω+θ2ω,ω∈J.Theorem 2.
For1<κ,σ≤2, if u¯,v¯∈C1J,ℝ+ are, respectively, the solutions of(17)D0κcuω=gω,uω,D0σcvω,ω∈J,uω=φ1ω,ω∈−ξ,0,D0σcvω=hω,D0κcuω,vω,ω∈J,vω=φ2ω,ω∈−ξ,0.
Then,u¯,v¯ satisfies the following integral inequalities:(18)u¯ω−u¯0−u¯′0ω−12u¯″0ω2−1Γκ∫0ωω−sκ−1gs,u¯s,D0σcv¯sds≤ϵ1Eκωκ,v¯ω−v¯0−v¯′ωω−12v¯″0ω2−1Γσ∫0ωω−sσ−1hs,D0κcu¯s,v¯sds≤ϵ2Eσωσ.Proof.
From Remark1, we have(19)D0κcu¯ω=gω,u¯ω,D0σcv¯ω+θ1ω,1<κ,σ≤2,ω∈J.
Then,(20)u¯ω−u¯0−u¯′0ω−12u¯′′0ω2=1Γκ∫0ωω−sκ−1gs,u¯s,D0σcv¯sds+1Γκ∫0ωω−sκ−1θ1sds.
So, we have(21)u¯ω−u¯ɛ−u¯′0ɛω−12u¯″0ω2−1Γκ∫0ωω−sκ−1gs,u¯s,D0σcv¯sds≤1Γκ∫0ωω−sκ−1θ1s≤ϵ1Γκ∫0ωω−sκ−1Eκsκ=ϵ1Γκ∑k=0∞1Γkκ+1∫0ωω−sκ−1skκds=ϵ1Γκ∑k=0∞ωk+1κΓκΓkκ+1Γkκ+1Γk+1κ+1≤ϵ1Γκ∑m=0∞ωmκΓmκ+1≤ϵ1Eκωκ.
Using the same technique, we can get(22)v¯ω−v¯0−v¯′0ω−12ω2v¯″0−1Γσ∫0ωω−sσ−1hs,cD0κu¯s,v¯sds≤ϵ2Eσωσ.Theorem 3.
Let the following assumptions hold:M1g,h∈CJ×ℝ×ℝ,ℝ, θ1,θ2∈C1J,ℝ, D0κcuω≤1/Γ2−κuω,D0σcvω≤1/Γ2−σvω;M2 There are Q1,Q2>0, so that ∀ω∈J and uj,vj∈ℝ,j=1,2(23)gω,u1,u2−gω,v1,v2≤Q1∑j=12uj−vj,hω,u1,u2−hω,v1,v2≤Q2∑j=12uj−vj;M3(24)maxω∈JQ1Γκ−11κκ−1+1Γ3−σ,Q2Γσ−11σσ−1+1Γ3−κ=K<1.Then,(i)
System (12) has a unique solution in C2−ξ,1,ℝ∩C2J,ℝ;(ii)
roblem (4) is HUML stable.Proof.
(i)
Consider(25)D0κcuω=gω,uω,D0σcvω,1<κ,σ≤2,ω∈J.The solution is equivalent to(26)uω−u0−u′0ω−12u″0ω2=1Γκ∫0ωω−sκ−1gs,us,D0σcvsds,uω=u0+u′0ω+12u″0ω2+1Γκ∫0ωω−sκ−1gs,us,D0σcvsds.Thus,(27)uω=u0+u′0ω+12u″0ω2+1Γκ∫0ωω−sκ−1gs,us,D0σcvsds,ω∈J,φ1ω,ω∈−ξ,1.Similarly, on considering(28)D0σcvω=hω,D0κcuω,vω,1<κ,σ≤2,ω∈J,we can obtain(29)vω=v0+v′0ω+12v″0ω2+1Γσ∫0ωω−sσ−1hs,D0κcus,vsds,ω∈J,φ2ω,ω∈−ξ,1.LetYξ=u∈C−ξ,1,ℝ;D0κcu∈C2−ξ,1,ℝ and define norm on Yξ by u=maxω∈Juω+maxω∈JD0κcuω and Yξ=v∈C−ξ,1,ℝ;D0σv∈C2−ξ,1,ℝ define norm on Yξ by v=maxω∈Jvω+maxω∈JD0σcvω, where C−ξ,1,ℝ and C2−ξ,1,ℝ, respectively, represent the class of continuous and continuously differentiable functions from −ξ,1 to ℝ. The norm is defined in such a way that the norm of each term depends on the derivatives of the fractional order of the other terms of Yξ. The product Yξ×Yξ is also a Banach space with norm u,v=u+v.Define operatorsK=H,G:Yξ×Yξ⟶Yξ×Yξ by(30)Hu,vω=u0u′0ω+12u″0ω2+1Γκ∫0ωω−sκ−1gs,us,D0σcvsds,ω∈J,φ1ω,ω∈−ξ,0,Gu,vω=v0+v′0ω+12v″0ω2+1Γσ∫0ωω−sσ−1hs,D0κcus,vsds,ω∈J,φ2ω,ω∈−ξ,0.First, we show thatK is a contraction mapping. It is clear that(31)Ku,vω−Ku¯,v¯ω=Hu,vω,Gu,vω−Hu¯,v¯ω,Gu¯,v¯ω≤Hu,vω−Hu¯,v¯ω+Gu,vω−Gu¯,v¯ω=φ1ω−φ1ω+φ2ω−φ2ω=0,∀ω∈−ξ,0.Forω∈J, we have(32)max0≤s≤ωus,vs−u¯s,v¯s≤max0≤s≤ωus−u¯s+max0≤s≤ωvs−v¯s≤max0≤s≤ωus+τ−u¯s+τ+max0≤s≤ωvs+τ−v¯s+τ=maxτ≤s+τ≤ω+τus+τ−u¯s+τ+maxτ≤s+τ≤ω+τvs+τ−v¯s+τ≤max−ξ≤s∗≤ωus∗−u¯s∗+max−ξ≤s∗≤ωvs∗−v¯s∗≤max−ξ≤s∗≤1us∗−u¯s∗+max−ξ≤s∗≤1vs∗−v¯s∗=us∗−u¯s∗+vs∗−v¯s∗=u,v−u¯,v¯,wheres∗=s+τ and −ξ≤τ<0. Thus,(33)Ku,v−Ku¯,v¯=H,Gu,v−H,Gu¯,v¯≤Hu,v−Hu¯,v¯+Gu,v−Gu¯,v¯≤1Γκ∫0ωω−sκ−1gs,us,D0σcvs−gs,u¯s,D0σcv¯sds+1Γσ∫0ωω−sσ−1hs,D0κcus,vs−hs,D0κcu¯s,v¯sds≤Q1Γκ∫0ωω−sκ−1max0≤s≤ωus+τ−u¯s+τ+max0≤s≤ωD0σcvs+τ−D0σcv¯s+τds+Q2Γσ∫0ωω−sσ−1max0≤s≤ωD0κcus+τ−D0κcu¯s+τ+max0≤s≤ωvs+τ−v¯s+τds≤Q1Γκ∫0ωω−sκ−1max−ξ≤s∗≤1us∗−u¯s∗+max−ξ≤s∗≤1D0σcvs∗−D0σcv¯s∗ds+Q2Γσ∫0ωω−sσ−1max−ξ≤s∗≤1D0κcus∗−cD0κu¯s∗+max−ξ≤s∗≤1vs∗−v¯s∗ds≤Q1max−ξ≤s∗≤1us∗−u¯s∗+max−ξ≤s∗≤1D0σcvs∗−v¯s∗Γκ∫0ωω−sκ−1ds+Q2max−ξ≤s∗≤1vs∗−v¯s∗+max−ξ≤s∗≤1D0κcus∗−u¯s∗Γσ∫0ωω−sσ−1ds≤Q1u−u¯Γκ+1+Q2v−v¯Γσ+1.Also,(34)D0σc,D0κcKu,v−D0σc,D0κcKu¯,v¯=maxω∈JD0σcHu,v,D0κcGu,v−D0σcHu¯,v¯,D0κcGu¯,v¯≤maxω∈JD0σcHu,v−D0σcHu,v+maxω∈JD0κcGu,v−D0κcGu¯,v¯1Γ2−σmaxω∈J∫0ωω−s1−σκ−1κ−2Γκ∫0ωω−sκ−3gs,us,cD0σvsds−κ−1κ−2Γκ∫0ωω−sκ−3gs,u¯s,D0σcv¯sds+1Γ2−κmaxω∈J∫0ωω−s1−κσ−1σ−2Γσ∫0ωω−sσ−3hs,D0κcus,vsds−σ−1σ−2Γσ∫0ωω−sσ−3hs,D0κcu¯s,v¯sds≤Q1Γ2−σΓκ−1u−u¯∫0ωω−s1−σds+Q2Γ2−κΓσ−1v−v¯∫0ωω−s1−κds≤Q1Γ3−σΓκ−1u−u¯+Q2Γ3−κΓσ−1v−v¯.So, we obtain(35)Ku,v−Ku¯,v¯≤Q1u−u¯Γκ+1+Q2v−v¯Γσ+1+Q1Γ3−σΓκ−1u−u¯+Q2Γ3−κΓσ−1v−v¯≤Q1Γκ−11κκ−1+1Γ3−σu−u¯+Q2Γσ−11σσ−1+1Γ3−κv−v¯≤Ku,v−u¯,v¯.This shows thatK is a contraction operator.(ii)
Next, letu¯,v¯∈C2J,ℝ be the approximate solution of (13) and u,v∈C2J,ℝ be the unique solutions of system (12), that is,(36)D0κcuω=gω,uω,Dσcvω,ω∈J,1<κ,σ≤2,D0σcvω=hω,D0+κcuω,vω,ω∈J,1<κ,σ≤2,uω=u¯ω,vω=v¯ω,ω∈−ξ,0,then(37)uω=u0+u′0ω+12u″0ω2+1Γκ∫0ωω−sκ−1gs,us,D0σcvsdsvω=v0+v′0ω+12v″0ω2+1Γσ∫0ωω−sσ−1hs,D0κcus,vsds.
Forω∈−ξ,0, we have(38)u¯,v¯−u,v≤u¯−u+v¯−v=φ1−φ1+φ2−φ2=0.
Also, forω∈J, we have(39)u,vω−u¯,v¯ω≤uω−u¯ω+vω−v¯ω.
Now,(40)uω−u¯ω=uω−u0−u′0ω−12u″ω2−1Γκ∫0ωω−sκ−1gs,u¯s,D0σcv¯sds≤uω−u0−u′0ω−12u″ω2−1Γκ∫0ωω−sκ−1gs,us,D0σcvsds+1Γκ∫0ωω−sκ−1gs,us,D0σcvsds−1Γκ∫0ωω−sκ−1gs,u¯s,D0σcv¯sds≤ϵ1Eκωκ+1Γκ∫0ωω−sκ−1gs,us,D0κcvs−gs,u¯s,D0σcv¯sds≤ϵ1Eκωκ+Q1Γκ∫0ωω−sκ−1us−u¯s+D0σcvs−D0σcv¯sds≤ϵ1Eκωκ+Q1Γκ∫0ωω−sκ−1us−u¯sds+∫0ωω−sκ−1D0σcvs−D0σcv¯sds.
Similarly, we can get(41)vω−v¯ω≤ϵ2Eσωσ+Q2Γσ∫0ωω−sσ−1D0κcus−cD0κu¯sds+∫0ωω−sσ−1vs−v¯sds.
Letχ,ϑ∈C−ξ,1,ℝ+ and in view of inequalities (40) and (41) take Aκ,σ=Aκ,Aσ, where Aκ,Aσ:C−ξ,1×−ξ,1,ℝ+⟶C−ξ,1,ℝ+ are defined by(42)Aκχ,ϑω=ϵ1Eκωκ+Q1Γκ∫0ωω−sκ−1χsds+∫0ωω−sκ−1D0σcϑsds,ω∈J,0,ω∈−ξ,0,Aσχ,ϑω=ϵ2Eσωσ+Q2Γσ∫0ωω−σ−1ϑsds+∫0ωω−sσ−1D0κcχsds,ω∈J,0,ω∈−ξ,0.
We will show thatAκ,σ is PO and consequently need to show that Aκ,σ is a contraction. Consider(43)Aκ,σχ,ϑω−Aκ,σχ∗,ϑ∗ω≤Aκχ,ϑω−Aκχ∗,ϑ∗ω+Aσχ,ϑω−Aσχ∗,ϑ∗ω≤Q1Γκ∫0ωω−sκ−1χs−χs∗ds+∫0ωω−sκ−1D0σcϑs−D0σcϑs∗ds+Q2Γσ∫0ωω−sσ−1ϑs−ϑs∗ds+∫0ωω−sσ−1D0κcχs−D0κcχs∗ds≤Q1Γκmax0≤s≤ωχs−χs∗+max0≤s≤ωD0σcϑs−D0σcϑs∗∫0ωω−sκ−1ds+Q2Γσmax0≤s≤ωϑs−ϑs∗+max0≤s≤ωD0κcχs−cD0σχs∗∫0ωω−sσ−1ds≤Q1Γκ+1χ−χ∗+Q2Γσ+1ϑ−ϑ∗.
Moreover,(44)D0σc,D0κcAκ,σχ,ϑ−D0σc,D0κcAκ,σχ∗,ϑ∗=D0σcAκχ,ϑ,D0κcAσχ,ϑ−D0σcAκχ∗,ϑ∗,D0κcAσχ∗,ϑ∗≤D0σcAκχ,ϑ−D0σcAκχ∗,ϑ∗+D0κcAσχ,ϑ−D0κcAσχ∗,ϑ∗≤1Γ2−σ∫0ωω−s1−σκ−1κ−2Γκ∫0ωω−sκ−3maxω∈Jχs−χs∗ds+κ−1κ−2Γκ∫0ωω−sκ−3maxω∈JD0σcϑs−D0σcϑs∗ds+1Γ2−κ∫0ωω−s1−κσ−1σ−2Γσ∫0ωω−sσ−3maxω∈Jϑs−ϑs∗ds+σ−1σ−2Γσ∫0ωω−sσ−3maxω∈JD0κcχs−D0κcχs∗ds≤Q1Γ3−σΓκ−1χ−χ∗+Q2Γ3−κΓσ−1ϑ−ϑ∗.
Thus, we get(45)Aκ,σχ,ϑω−Aκ,σχ∗,ϑ∗ω≤Q1Γκ−11κκ−1+1Γ3−σχ−χ∗+Q2Γσ−11Γ3−κ+1σσ−1ϑ−ϑ∗.
Therefore,Aκ,σχ,ϑω−Aκ,σχ∗,ϑ∗ω≤Kχ,ϑ−χ∗,ϑ∗∀χ,ϑ,χ∗,ϑ∗∈C−ξ,1,ℝ+, showing that Aκ,σ is a contraction operator. By Definition 3, Aκ,σ is a PO and FAκ,σ=x∗,y∗, then(46)x∗ω≤ϵ1Eκωκ+Q1Γκ∫0ωω−sκ−1xs∗ds+Q1Γκ∫0ωω−sκ−1D0σcys∗ds,y∗ω≤ϵ2Eσωσ+Q2Γσ∫0ωω−sσ−1ys∗ds+Q2Γσ∫0ωω−sσ−1D0σcxs∗ds.
We try to show thatx∗ωandy∗ω are increasing, and for this, take ℏ1,ℏ2∈0,1,ℏ1<ℏ2 and q1∗=mins∈Jxs∗+D0σcys∗,q2∗=mins∈Jys∗+D0κcxs∗. Thus,(47)x∗ℏ2−x∗ℏ2≤ϵ1Eκℏ2κ−Eκℏ2κ+Q1Γκ∫0ℏ1ℏ2−sκ−1−ℏ1−sκ−1xs∗+D0σcys∗ds+Q1Γκ∫0ℏ2ℏ2−sκ−1xs∗+D0σcys∗ds≥ϵ1Eκℏ2κ−Eκℏ2κ+q1∗Q1Γκ∫0ℏ1ℏ2−sκ−1−ℏ1−sκ−1ds+q1∗Q1Γκ∫0ℏ2ℏ2−sκ−1ds≥ϵ1Eκℏ2κ−Eκℏ2κ+q1∗Q1Γκ+1ℏ2κ−ℏ1κ>0.
Similarly, we can get(48)y∗ω≥ϵ2Eσℏ2σ−Eσℏ2σ+q2∗Q2Γσ+1ℏ2σ−ℏ1σ>0.
So that the solutionx∗ω,y∗ω is increasing and D0κcx∗,D0σcy∗>0. Also(49)x∗ω≤ϵ1Eκωκ+Q1Γκ∫0ωω−sκ−1x∗sds+Q1ΓκΓ2−σ∫0ωω−sκ−1x∗sds≤ϵ1Eκωκ+Q1Γκ1+1Γ2−σ∫0ωω−sκ−1x∗sds.
Thus, by using Gronwall’s lemma we have(50)x∗ω≤cEκϵ1Eκωκ,where cEκ=EκQ1/Γκ1+1/Γ2−σ. In the same way(51)y∗ω≤cEσϵ2Eσωσ,where cEσ=EσQ2/Γσ1+1/Γ2−κ,ω∈J. Particularly, if x=u¯−u and y=v¯−v, then by (40) and (41), we have xω≤Aκ and yω≤Aσ; therefore, by Gronwall’s lemma we obtain(52)uω−u¯ω≤cEκϵ1Eκωκ,vω−v¯ω≤cEσϵ2Eσωσ.
Thus, from the above we have(53)u,v−u¯,v¯≤u−u¯+v−v¯≤cEκϵ1Eκωκ+cEσϵ2Eσωσ≤2cEλϵEλωλ.
Hence, it follows that (4) is ML stable.


## 4. Illustrative Example

As an application of our results, we consider the following example.Example 1.
Consider the problem(54)D03/2cuω=14eωuω−1+D05/3cvω−1,ω∈J=0,1,D05/3cvω=110e2ω+1+D03/2cuω−1+vω−1,ω∈J=0,1,uω=0,vω=0,ω∈−1,0.
Set the functions as(55)g3/2ω,uω,vω=14eω1+u+v,h5/3ω,uω,vω=110e2t+1+u+v,and then for every u,v,u¯,v¯ from ℝ, ω∈0,1 we have(56)g3/2ω,uω,vω−g3/2ω,u¯ω,v¯ω≤14u−u¯+14v−v¯,h5/3ω,uω,vω−h5/3ω,u¯ω,v¯ω≤110u−u¯+110v−v¯.
Therefore,M2 is satisfied with κ=3/2andσ=5/3, and from the inequalities in (56), Q1=1/4andQ2=1/10. Thus, we have K=0.34601<1. Therefore, by Theorem 2, (54) has a unique solution. After calculations, we obtain(57)u,v−u¯,v¯≤c3/2ϵE3/2ω3/2,ω∈0,1.
Using Theorem3, the solution of (54) is Hyers–Ulam–ML stable.Remark 2.
If we consider another spaceBu,v=u,v:u,v∈C−ξ,1,D0κcu,D0σcv∈C1−ξ,1 with modified Bielecki’s norms defined by(58)u=maxω∈Juωe−λω+D0σcvωe−ρω,v=maxω∈Jvωe−λω+D0κcuωe−ρω,where 0<ρ<λ,J=0,1, then the results similar to Theorem 3 can be obtained for the solution of (4).


## 5. Conclusion

We gave sufficient conditions for theEU of the solutions to the nonlinear implicit CS of neutral FDEs. Our main tool was the Banach contraction principle. Likewise under specific conditions, we have found the HUML stability results for the solution of the CS given in (4).



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*Source: 2786041-2020-05-21.xml*