# Analysis of Stochastic Weighted Impulsive Neutralψ-Hilfer Integro-Fractional Differential System with Delay

Authors: Manzoor Ahmad; Akbar Zada; Jihad Ahmad; Mohamed A. Abd El Salam
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper is devoted to the study of stochastic fractional differential equations. Particularly, we study the existence and uniqueness of solutions of nonlinear stochastic weighted impulsiveψ-Hilfer neutral integro-fractional differential system with infinite delay. We obtain the results with the help of fixed-point theorems and Banach contraction principle. Additionally, we investigate the Ulam–Hyers stability of the solutions using basic schemes of fractional calculus. For application of the theory, we add an example.


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## Body

## 1. Introduction

Fractional calculus (FC) is the generalization of integer orders differentiations and integrations to fractional orders [1]. It is observed that many interdisciplinary applications can be modeled elegantly with the help of fractional differential operators. The fractional parameters allow a class of characterization of several eventual uncertainties presented in the dynamical models [2]. Fractional differential models are considered to be more accurate as compared to integer order differential models [3]. Many researchers have elaborated applications of fractional derivatives [4] in the time-dependent damping behavior of several viscoelastic behavior science. Furthermore, FC has also great applications in nonlinear oscillation of earthquake [5], fluid dynamics [6–12], traffic models [13], seepage flow in porous media [14] etc. Repudiating fractional differential equations (FDEs) is like saying that irrational numbers do not exist.Researchers are investigating and analyzing FDEs with different approaches. Very recently Sousa and Olivera [15] extended the concept of Hilfer fractional derivative to ψ-Hilfer fractional derivative and discussed its important properties and applications. The ψ-Hilfer fractional derivative is quite different from other classical fractional derivatives because the kernel is in the form of function. Moreover, it has been proved that ψ-Hilfer fractional derivative generalizes many existing fractional derivatives. FDEs in which the highest fractional derivatives of unknown term appear both with and without delays are known as neutral FDEs. In the last few years, the study of neutral FDEs has developed dramatically. This is due to the fact that the qualitative behavior of neutral FDEs is quite different from those of nonneutral FDEs. Neutral FDEs also play an important role and have many applications. For instance, it gives more better sketch of population fluctuations. Also, neutral FDEs with delay appear in models of electrical networks containing lossless transmission lines [16].In the qualitative behavior of dynamical systems, the concept of stability is very important because every workable control system is designed to be stable. In physical applications, stability of solutions is very meaningful because deviations in mathematical models inevitably result from errors in measurement. A stable solution can become unstable due to such deviations. If the water flow in a sea is not stable, then jumping of an ant into the sea will probable cause flood. The investigations of stability properties of solutions have attracted many researchers through its potential applications. Especially, the Ulam–Hyers (UH) stability analysis and its applications have been studied by many researchers [17]. The definition of UH stability has applicable significance since it means that if one is investigating the UH stability, then one does not require to reach to the exact solution. It guarantees that there is always exact solution to every approximate solution [18]. This is quite helpful in many fields such as numerical analysis [19], economics [20], optimizations theory [21], and biology [22], where finding the exact solution is very difficult or time consuming, see [23].Due to lack of investigations on the existence and UH stability to the solutions of weighted impulsive implicit neutralψ-Hilfer stochastic integro FDEs in the literature, it is of great interest to perform some investigations in this area. Y. Guo et al. [24] investigated the impulsive neutral functional stochastic differential equation for existence and UH stability given by(1)D0+αyu−gu,yu=fu,yu+ϑu,yudduωu,u∈0,b,b>0,u≠uk,ΔI0+2−αyuk=Ikyuk−,ΔI0+1−αyuk=Jkyuk−,k=1,2,…,m,I0+2−αy0−g0,y0=ϕ1∈By,I0+1−αy0−g(0,y0=ϕ2∈By,where D0+. represent the Riemann–Liouville fractional derivative of order α∈(1,2], 0<u1<u2<,…,<um+1=b. The functions f,g,ϑ:0,T×By are continuous and ωu(u∈(0,T]) is Wiener process, and By is a phase space. The notations Ik,Jk represent appropriate functions, I0+1−α,I0+2−α are Riemann–Liouville fractional integrals having orders 1−α and 2−α, respectively. ΔI0+1−α and ΔI0+2−α are defined by(2)ΔI0+1−αyuk=I0+1−αyuk+−I0+1−αyuk−,ΔI0+2−αyuk=I0+2−αyuk+−I0+2−αyuk−.The histories(yu:((−∞,b]⟶ℝ)∈By is defined by yuθ=yu+θ, θ≤0.Sathiyaraj et al. [25] studied Ulam–Hyers stability results to the solutions of the following fractional stochastic differential system involving Hilfer fractional derivative:(3)D0+α,βHyu=Ayu+ϑu,yu,D0+α,βHyu+∫0uηs,ys,D0+α,βHysdωs,u∈0,T,T>0,I0+1−γyu=y0,γ=α+β−αβ,where D0+.H represent the Hilfer fractional derivative of order α∈0,1 and type β∈0,1, y∈ℝn, and A is a matrix of dimension n×n. The nonlinear functions ϑ,η:(0,T]×ℝn×ℝn⟶ℝn and η:(0,T]×ℝn×ℝn⟶ℝn×ℝn are continuous.There exists a vast literature over existence and uniqueness (EU) of solutions related to FDEs involving Caputo or Riemann–Liouville fractional derivatives, etc., while in setting of nonlinear weighted impulsive implicit neutralψ-Hilfer stochastic integro FDEs, as far as we know, existence, uniqueness, and stability have not been discussed.Motivated by the work discussed above, in this study, we study the EU and Ulam–Hyers stability of the following nonlinear weighted impulsive neutralψ-Hilfer stochastic integro-FDEs’ system:(4)D0+p,q;ψℌyu−gu,yu=μζu,yu+Fu,yu,yuI0+p;ψyu,dωudu,u∈J=0,T−u1,u2,…um,T>0,μ∈ℝ+,ΔI0+1−ρ;ψyuk=Ikyuk−,ρ=p+q−pq,k=1,2,…m,ΔI0+1−ρ;ψg(uk,y(uk))=0,k=1,2,…m,I0+1−ρ;ψ(yu)=φ,u∈(−∞,0],where D0+p,q;ψℌ. represent the ψ-Hilfer fractional derivative of order p0<p<1 and type q0<q≤1, I0+1−ρ;ψ is the Riemann–Liouville fractional left-sided integral of order 1−ρ, and ψ∈C1J,ℝ is the increasing function with (dψu/du)≠0, ∀u∈J, 0<u1<u2<…<um<um+1=T,(5)ΔI0+1−ρ;ψyuk=I0+1−ρ;ψyuk+−I0+1−ρ;ψyuk−,I0+1−ρ;ψyuk+=limϵ⟶0I0+1−ρ;ψyuk+ϵ,I0+1−ρ;ψyuk−=limϵ⟶0I0+1−ρ;ψyuk−ϵ.The functionsζ,g:J×By⟶H, and F:J×By×By×By⟶H are appropriate functions, By is a phase space of mappings from −∞,0 into Hilbert space H, φ∈By, with φu=φu, for u∈(−∞,0] and norm φBy=supu∈(−∞,T]φu. Furthermore, ωuu∈J is a Wiener process, g0,φu=0. The histories (yu:(−∞,T]⟶H)∈By is defined by yuθ=yu+θ, θ≤0.The rest of this study is arranged into the following way. In Section2, we recall some helpful definitions and results relating to both fractional derivatives and fractional integrals. The EU of solutions to the considered system (4) is discussed in Section 3. Ulam–Hyers stability result is established in Section 4. A particular example is given in Section 5.


## 2. Preliminaries

This portion is dedicated to introduce some notions, definitions, and preliminary results.Throughout this study, letΩ,ℱ,P denote the complete filtered probability space such that ℱ0 contains all P−null sets of ℱ. Also, (ωu)u≥0 is supposed to be Q-Wiener process on the probability space Ω,ℱ,P with the covariance operator Q satisfying TraQ<∞. Consider K1 and H1 as two separable Hilbert spaces and LK1,H1 as the space of all bounded linear operators from K1 into H1. The notation . denotes the same norms in K1, H1, and LK1,H1; we also denote the inner product of K1 and H1 by (.,.). Furthermore, we assume that there exists a complete orthonormal system ekk=1,2,… in H1 and a bounded sequence of nonnegative real numbers λkk=1,2,… such that Qek=λkek and a sequence of independent Brownian motion βkk≥1 satisfying(6)(ωu,e)H1=∑k=0∞λkek,eH1βku.LetJ0=(0,u1] and Jk=(uk,uk+1], for k=1,2,…,m.Consider the spacePC1−ρ;ψJ,H1=y:J⟶H1 is continuous everywhere except some points uk at which yuk+ and yuk− do exist and yuk−=yuk, k=1,2,…,m}. We also introduce the spaces:(7)C1−ρ;ψJk,H1=y:Jk⟶H1,(ψu−ψ0)1−ρ∈CJk,H1,and(8)PC1−ρ;ψJ,H1=y:J⟶H1,for eachu∈Jk,(ψu−ψ0)1−ρ∈CJk,H1,with norms(9)yC1−ρ;ψ=supu∈JkE(ψu−ψ0)1−ρy,k=1,2,…,m,(10)yPC1−ρ;ψ=supu∈JkE(ψu−ψ0)1−ρy,k=1,2,…,m,where E. is mathematical expectation.Before coming to fractional order neutral differential equations, we define the abstract phase spaceBy. Let y:(−∞,0]⟶(0,∞+) be a continuous function satisfying l=∫−∞0yudu<∞+. The Hilbert space By,.By induced by y is given by(11)By=φ:(−∞,0]⟶H1,such that for anyr>0,φ.is measurable and bounded on [−r,0]such that∫−∞0yssups≤θ≤0Eφθ212dθ<∞+,endowed with norm,(12)φBy=∫−∞0yssups≤θ≤0Eφθ212dθ.Define the space:(13)By′=φ:(−∞,T]⟶H1,φ∈C1(Jk,H1),k=1,2,…,mthere existφuk−andφuk+along withφuk−=φuk,φ0=ϕ∈By,where φk is the restriction of φ into Jk and J0.We use the notation.By′ for the seminorm in the space By′ defined by(14)φBy′=ϕBy+maxφkJk(ψu−ψ0)1−ρ;ψ,k=1,2,…,m,where(15)ϕ=φ0,φkJk(ψu−ψ0)1−ρ;ψ=supθ∈JkEψθ−ψ01−ρ;ψφθ21/2.Definition 1.
(see [15]). Let ρ>0 and ψ be increasing, positive monotone function on J such that ψ′ is continuous on a,b; then, the ψ-Riemann–Liouville fractional integral of a function ℏ:a,b⟶H1 with respect to another function ψ is defined as(16)Ia+ρ;ψℏu=1Γρ∫auψ′s(ψu−ψs)ρ−1ℏsds,u≥0.Definition 2.
(see [15]). The ψ-Hilfer fractional derivative Da+p,q;ψℌ of a function ℏ:a,b⟶H1 of order p∈(n−1,n] and type q∈(0,1] is given by(17)Da+p,q;ψℌℏu=Ia+qn−p;ψ1ψ′uddunIa+1−qn−p;ψℏu,u∈J.Lemma 1 (see [15]).
Letp,q,ρ>0. Then,(18)Ia+p;ψIa+q;ψℏu=Ia+p+q;ψℏu.
If(19)ℏu=(ψu−ψa)ρ−1,then(20)Ia+p;ψℏu=Γρ(ψu−ψa)p+ρ−1Γp+ρ.Lemma 2 (see [15]).
Ifℏ∈C1−ρJk,H1, p∈0,1 and q∈(0,1], then(21)Ia+p;ψℌDa+p,q;ψℏu=ℏu−(ψu−ψa)ρ−1ΓρIa+1−q1−p;ψℏa.Lemma 3 (see [15]).
Letp∈0,1 and ℏ:J⟶H1 be continuous; then, for any c∈J and a function y∈C1−ρJk,H1 defined by(22)yu=(ψu−ψ0)1−ρΓρI0+1−ρ;ψyc−I0+1−ρ+p;ψℏu|t=t1+I0+p;ψℏu,which is the solution of ψ-Hilfer fractional differential equation, D0+p,q;ψℌyu=ℏu.Lemma 4 (see [26].
Banach fixed-point theorem). LetC≠∅ be a closed subset of a Banach space Y; then, any contraction mapping T:C⟶C has a unique fixed point.


## 3. Main Results

In this portion, we demonstrate and exhibit the existence and uniqueness for the solution of the considered system on the interval−∞,T under Banach contraction principle and Mönch’s fixed-point theorem. We also discuss the UH stability for the solution of considered problem (4). Before coming to the main results, we assume some hypothesis as follows.A0: let the function F:J×By×By×By⟶H1 satisfy the following assumptions:(i)
The functionFu,x,y,x, for all x,y,z∈By, is measurable and continuous, for a.e. u∈J.(ii)
There exists constantα1∈0,α, m1,m2,m3∈L1/α1J,0,∞+, and positive integrable function Ω1:0,∞+⟶0,∞+ such that(23)EFu,x,y,z2≤m1uΩ1xBy+m2uΩ1yBy+m3uΩ1D0+p,q;ψℌI0+p;ψzBy,for all u,x,y,z∈J×By×By×By while Ω1 satisfies(24)limη⟶∞infΩ1ηη=0.(iii)
There exists constantα2∈0,α and functions η1,η2,η3∈L1/α2J,0,∞+ such that, for any bounded subsets F1,F2,F3⊂By,(25)β(F(u,F1,F2,F3))≤η1u[supθ(−∞,0]β(F1θ)]+η2u[supθ(−∞,0]β(F2θ)]+η3u[supθ(−∞,0]β(F3θ)],for a.eu∈J, where F1θ=uθ,u∈F1, F2θ=vθ,v∈F2, and F3θ=wθ,w∈F3, and β is Hausdorff measure of noncompactness.(iv)
We can find constantsℒu,ℒv,ℒw all greater than zero, such that, for all x,y,x,x′,y′,z′∈By×By×By and u∈J,(26)E(ψu−ψ0)1−ρFu,x,y,z−Fu,x′,y′,z′2≤ℒux−x′2+ℒvy−y′2+ℒwD0+p,q;ψℌz−D0+p,q;ψℌz′2.A1: let the function g:J×By⟶H1 satisfy the following conditions.(v)
The functiong is continuous, and there exists a positive constant Qg such that(27)Eψu−ψ01−ρgu,y2≤Qg1+yBy2,u∈J.(vi)
There exists constantα3∈0,α, and a function η4∈L1/α3J,0,∞+ such that, for any bounded subset F4⊂By,(28)β(g(u,F4))≤η4u[supθ(−∞,0]β(F4θ)],η∗=supu∈Jη4u.(vii)
There exists a constantℒg>0, such that, for all y∈By and u∈J,(29)E(ψu−ψ0)1−ρgu,y−gu,y′2≤ℒgy−y′2.A2: let the function ζu,ϕ satisfy the following conditions.(viii)
ζu,ϕ is continuous for a.e. u∈J and is measurable for all ϕ∈By.(ix)
The functionζ is continuous, and we can find a positive constants Qζ and ℒζ such that(30)Eψu−ψ01−ρζu,y2≤QζyBy2,Eψu−ψ01−ρζu,y−ζu,y′2≤ℒζy−y′By2,u∈J.(x)
There exists constantα4∈0,α, and a function η5∈L1/α3J,0,∞+ such that, for any bounded subset F5⊂By,(31)β(ζ(u,F5))≤η5u[supθ(−∞,0]β(F5θ)].A3: Ik:H1⟶H1 is continuous and satisfies the conditions as follows.(xi)
Let there exist constantsck,Kk>0, k=1,2,…,m, and F5⊂By′ such that(32)Iky≤ckyBy′,β(ψu−ψ0)1−ρIk(F5)≤Kk[supθ(−∞,T]β(F5(θ)].(xii)
There exists a constantℒ1=maxℒk>0,k=1,2,…,m such that, for all y,y′∈By′,(33)Iky−Iky′≤ℒky−y′By′.A4:(34)ℳ=Qgl2+(ψT−ψ0)2+2p−2ρΓ2p+1μ2Qζ+(ψT−ψ0)2−2ρΓ2ρ∑k=1mck2<1η∗+2μ(ψT−ψ0)pΓp+1η1L1α2J,ℝ++(ψT−ψ0)ρ−1Γρ∑k=1mKk+(ψT−ψ0)p+1/2Γp+1η2L1α4J,ℝ+<1.Before studying the qualitative behavior of solutions to (4), we consider a simplified form of (4) given by(35)D0+p,q;ψℌyu−gu,yu=σu,u∈J=]0,T]−u1,u2,…um,T>0,ΔI0+1−ρ;ψyuk=Ikyuk−,ρ=p+q−pq,k=1,2,…m,ΔI0+1−ρ;ψguk,yuk=0,k=1,2,…m,I0+1−ρ;ψyu=φ,u∈−∞,0,where D0+p,q;ψℌ. represent the ψ-Hilfer fractional derivative of order p0<p<1 and type q0<q≤1, I0+1−ρ;ψ is the ψ-Riemann–Liouville fractional left-sided integral of order 1−ρ.Theorem 1.
Let0<p<1, 0<q≤1, and σ:J⟶H1 be continuous, then any solution y∈PC1−ρ;ψJ,H1 of(36)D0+p,q;ψℌyu−gu,yu=σu,u∈J,ΔI0+1−ρ;ψyuk=Ikyuk−,ρ=p+q−pq,k=1,2,…m,ΔI0+1−ρ;ψg(uk,yuk=0,k=1,2,…m,I0+1−ρ;ψyu=φ,u∈−∞,0,.has the form(37)yu=gu,yu+(ψu−ψ0)ρ−1Γρϕu+1Γp∫0uψ′s(ψu−ψs)p−1σsds,u∈J0,and(38)yu=g(u,yu)+ψu−ψ0ρ−1Γρ(ϕu+∑j=1kIk(y(uk)))+1Γp∫0uψ′sψu−ψsp−1σsds,u∈(uk,uk+1],k=1,2,…,m..Proof.
For anyu∈J0=(0,u1], we consider(39)D0+p,q;ψℌ[yu−g(u,yu)]=σu,0<p<1,0<q≤1,u∈(0,u1].
Applying the integralI0+p;ψ on both sides of (39) and using Lemma 2, we have(40)I0+p;ψℌD0+p,q;ψyu−gu,yu=yu−gu,yu−(ψu−ψ0)ρ−1ΓρI0+1−ρ;ψy0=I0+p;ψσu,u∈J0,which implies that(41)yu=gu,yu+(ψu−ψ0)ρ−1Γρϕu+1Γp∫0uψ′s(ψu−ψs)p−1σsds,u∈J0.
Now, foru∈(u1,u2] and(42)D0+p,q;ψℌ[yu−g(u,yu)]=σu,0<p<1,0<q≤1,u∈(0,u1].[yu−g(u,yu)]=σu,u∈(u1,u2],under the condition(43)I0+1−ρ;ψyu1+−I0+1−ρ;ψyu1−=I1yu1−.
From (41) and Lemma 3, one has(44)yu=gu,yu+(ψu−ψ0)ρ−1ΓρI0+1−ρ;ψyu1+−I0+1−ρ+p;ψσu1++1Γp∫0uψ′s(ψu−ψs)p−1σsds=gu,yu+(ψu−ψ0)ρ−1ΓρI0+1−ρ;ψyu1−+I1yu1−−I0+1−ρ+p;ψσu1++1Γp∫0uψ′sψu−ψsp−1σsds,u∈(u1,u2].
From (41), we have(45)I0+1−ρ;ψyu1−−I0+1−ρ+p;ψσu1+=y0u=ϕu.
Using (45) in (44), we obtain(46)yu=gu,yu+(ψu−ψ0)ρ−1ΓρI0+1−ρ;ψyu1−−I0+1−ρ+p;ψσu1++I1yu1−+1Γp∫0uψ′s(ψu−ψs)p−1σsds=gu,yu+(ψu−ψ0)ρ−1Γρϕu+I1yu1−+1Γp∫0uψ′sψu−ψsp−1σsds,u∈(u1,u2].
Continuing in the same manner, foru∈(uk,uk+1], we have(47)yu=gu,yu+ψu−ψ0ρ−1Γρϕu+∑j=1kIkyuk+1Γp∫0uψ′sψu−ψsp−1σsds,u∈uk,uk+1,k=1,2,…,m.
Next on the basis of Theorem1, we suppose the solution of equation (4).Definition 3.
Lety:(−∞,T]⟶H1 be the solution of equation (4); then, y satisfies the following ψ-Hilfer fractional integral equation:(48)yu=ϕu,u∈(−∞,0],gu,yu+(ψu−ψ0)ρ−1Γρϕu+μΓp∫0uψ′s(ψu−ψs)p−1ζs,ysds+1Γp∫0uψ′sψu−ψsp−1Fs,ys,ys,I0+p;ψysdωs,u∈J0,gu,yu+(ψu−ψ0)ρ−1Γρϕu+∑k=1mIkyuk+μΓp∫0uψ′sψu−ψsp−1ζs,ysds+1Γp∫0uψ′s(ψu−ψs)p−1F(s,ys,ys,I0+p;ψys)dωs,u∈(uk,uk+1],k=1,2,…,m,.which is a mild solution of equation (4).Lemma 1 (see [27]).
Lety∈By′; then, yu∈By, satisfying(49)lEy212≤yuBy≤ϕBy+lsups∈0,u]Eψs−ψ01−ρys212,where l=∫−∞0fudu<∞+ and y0u=ϕ∈By.
Next, we consider some properties and definitions of noncompactness.
The Hausdorff measure of noncompactnessβ. defined on each bounded set Ω⊂B, where B is a Banach space which is given by(50)βΩ=infε>0;Ωhas finiteεnet inB.Lemma 2 (see [28]).
In the real Banach spaceB, if Ω,Θ⊂B are bounded, then the following properties are satisfied. (1) Monotone: if, for all bounded subsets Ω,Θ of B, Ω⊆Θ, then βΩ≤βΘ; (2) nonsingular: βx∪Ω=βΩ, for each x∈H1 and every nonempty subset Ω⊂H1; (3) regular: the subset Ω⊂H1 is precompact if and only if βΩ=0; (4) for Ω+Θ=u+v;u∈Ω,v∈Θ, βΩ+Θ≤βΩ+βΘ; (5) βΩ∪Θ≤maxβΩ,βΘ; (6) βμΩ≤μβΩ; (7) if the subset W⊂CJ,B is both bounded and equicontinuous, then t⟶βWu is continuous on J and(51)βW≤maxu∈JβWu,β∫0uWsds≤∫0uβWsds,u∈J,where∫0uWsds=∫0uvsds,v∈W,u∈J,.
(8) Ifxnn=1∞ is a sequence of Bochner integrable functions from the interval J into the Banach space B satisfying xn≤m˜u, for almost u∈J and each n≥1, then the function Φu=βxnn=1∞∈LJ,ℝ+ and(52)β∫0uxnsds,n≥1=2∫0uΦsds.
(9) IfW is bounded, then, for every ε>0, there is a sequence unn=1∞⊂W such that(53)βW≤2βxnn=1∞+ε.
Establishing our main results, the above lemma about Hausdorff measurement of noncompactness will be used.Lemma 3 (see [29]).
LetD be a closed and convex subset of a Banach space B, 0∈D. If a map F:D⟶B is continuous, which satisfies Mönchs condition, i.e., M⊂D, M⊂co¯0∪FM implies M¯ is compact. Then, F has a fixed point in D.Lemma 4 (see [30]).
IfW⊂C0,T,L20V,X and ω is the standard Wiener process, then(54)β∫0uWsdωs≤T.TraQβWs,where(55)∫0uWsdωs=∫0uvsdωs,for allv∈W,u∈0,T.Theorem 2.
Let the conditionsA0i,ii to A4 be true; then, system (4) possess at least one solution on the interval J.Proof.
Transform the considered system (4) into equivalent fixed-point problem. Define an operator T:By′⟶By′ by(56)Tyu=gu,yu+ψu−ψ0ρ−1Γρϕu+μΓp∫0uψ′sψu−ψsp−1ζs,ysds+1Γp∫0uψ′sψu−ψsp−1Fs,ys,ys,I0+p;ψy(s)dωs,u∈J0,gu,yu+ψu−ψ0ρ−1Γρϕu+∑k=1mIkyuk+1Γp∫0uψ′sψu−ψsp−1Fs,ys,ys,I0+p;ψy(s)dωs,+μΓ(p)∫0uψ′sψu−ψsp−1ζs,ysdsu∈uk,uk+1,k=1,2,…,m,ϕu,u∈−∞,0..
The operatorT has fixed point in the interval −∞,T if and only if (4) has a solution. For ϕ∈By, we define(57)ϕ˜u=0,u∈J,ϕu,u∈−∞,0.
Then,ϕ˜∈By′. Let yu=xu+ϕ˜u, u∈(−∞,0]; then, it can be easily observed that x0=0, for u∈(−∞,0], where xu is given by(58)xu=gu,xu+ϕ˜u+ψu−ψ0ρ−1Γρϕu+μΓp∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γp∫0uψ′sψu−ψsp−1F(s,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜sdωs,u∈J0,gu,xu+ϕ˜u+ψu−ψ(0)ρ−1Γ(ρ)ϕ(u)+∑k=1m(Ik(x(uk))+ϕ˜(uk)+μΓp∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γp∫0uψ′sψu−ψsp−1F(s,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜sdωs,u∈uk,uk+1,k=1,2,…,m,if and only if yu=ϕu for u∈(−∞,0] and(59)yu=gu,yu+(ψu−ψ0)ρ−1Γρϕu+μΓp∫0uψ′sψu−ψsp−1ζs,ysds+1Γp∫0uψ′sψu−ψsp−1Fs,ys,ys,I0+p;ψysdωs,u∈J0,gu,yu+(ψu−ψ0)ρ−1Γρϕu+∑k=1mIkyuk+μΓp∫0uψ′sψu−ψsp−1ζs,ysds+1Γp∫0uψ′s(ψu−ψs)p−1F(s,ys,ys,I0+p;ψys)dωs,u∈(uk,uk+1],k=1,2,…,m..
Setting the spaceBy′′,.By′′, induced by By′, as(60)B′y′=x:x∈By′,x0=0,with the following norm:(61)xB′y′=supθ∈0,TEψθ−ψ01−ρx212,x0=0.
Consider a close ballℬr=x∈By′′:xBy′≤r. Then, for every r, ℬr is a closed, bounded, and convex subset of By′′. For any x∈By′′, we have(62)xu+ϕ˜uBy≤xuBy+ϕ˜uBy≤lsups∈0,u(E|ψs−ψ(01−ρx2|12)+ϕ˜uBy≤lxBy′+ϕBy=lr+ϕBy=r1say.
Define an operatorT:By′′⟶By′′ by(63)Txu=gu,xu+ϕ˜u+ψu−ψ0ρ−1Γρϕu+μΓp∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γp∫0uψ′sψu−ψsp−1F(s,ys+ϕ˜s,ys+ϕ˜s,I0+p;ψys+ϕ˜sdωs,u∈J0,gu,xu+ϕ˜u+ψu−ψ0ρ−1Γρϕu+∑k=1m(Ikxuk+ϕ˜uk+μΓp∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γp∫0uψ′sψu−ψsp−1F(s,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜s)dωs,u∈uk,uk+1,k=1,2,…,m..
We will complete the proof in several steps.Step 1
: we prove that there exists some positive real numberr for which Tℬr⊂ℬr. If it is not true, then, for each r>0, there is some xr∈ℬrur∈(−∞,b] such that TxrurBr′′2>r2.
Now, it follows from the assumptions that(64)Eψur−ψ0)1−ρTxrur2≤5Eψur−ψ01−ρgur,xrur+ϕ˜ur2+5Eψur−ψ01−ρΓp∫0urψ′sψur−ψ0p−1μζ(s,xrs+ϕ˜rsds2+5Eψur−ψ01−ρΓp∫0uψ′sψur−ψsp−1Fs,xrs+ϕ˜rs,xrs+ϕ˜rs,I0+p;ψxrs+ϕ˜rsdωs2+5E1Γρφ12+5E1Γρ∑k=1mIkxruk−+ϕ˜uk−2=5∑j=15Inj.
Now, we determineIn1 to In5 from the conditions A0i−ii and A2i−ii:(65)In1=Eψur−ψ01−ρgur,xrur+ϕ˜ur2≤Qg1+xrur+ϕ˜ur)By≤Qg+Qgr12,In2=Eψur−ψ01−ρΓ(p)∫0urψ′sψur−ψ0p−1μζs,xrs+ϕ˜sds2≤μ2ψur−ψ02−2ρΓ2(p)∫0urψ′sψur−ψ0p−1ds×∫0urψ′sψur−ψ0p−1Eζt,xrs+ϕ˜s2ds≤μ2ψu−ψ02+2p−2ρΓ2p+1Qζxr+ϕ˜By2≤μ2ψu−ψ02+2p−2ρΓ2p+1Qζr32,r3=r+ϕBysay,In3=Eψur−ψ01−ρΓp∫0urψ′sψu−ψsp−1F(s,xrs+ϕ˜s,xrs+ϕ˜s,ℌD0+p,q;ψxrs+ϕ˜sdωs2≤ψtr−ψ02−2ρ.TraQΓ2p∫0urψ′sψur−ψ0p−1ds∫0urψ′sψur−ψ0p−1EF(s,xrs+ϕ˜s,xrs+ϕ˜s,I0+p;ψxrs+ϕ˜s2ds≤ψur−ψ02+2p−2ρ.TraQΓ2p+1[m1uΩ1xrs+ϕ˜sBy+m2uΩ1xrs+ϕ˜sBy+m3uΩ1D0+p,q;ψℌI0+p;ψxrs+ϕ˜sBy],xr0+ϕ0=0,≤ψT−ψ02+2p−2ρ.TraQΓ2p+1m1uΩ1r2+m2u+m3uΩ1r1,In4≤1Γ2ρϕ2≤ψT−ψ02−2ρΓ2(ρ)ϕ2,In5=E1Γρ∑k=1mIkxruk−+ϕ˜uk−2≤ψT−ψ02−2ρΓ2ρ∑k=1mck2r2.
Hence, we have(66)r2<TxrurBy″≤Qg+Qgr12+(ψT−ψ0)2+2p−2ρΓ2p+1m1uΩ1r2+m2u+m3uΩ1r1+μ2Qζr32+ψT−ψ02−2ρΓ2ρϕ2+∑k=1mck2r2.
Dividing each term byr2 and taking r⟶∞, then, from the above,(67)limr⟶∞r1r=limr⟶∞lr+ϕByr=l,limr⟶∞r3r=limr⟶∞r+ϕByr=1,andlimη⟶∞infΩ1ηη=0,yields(68)1<Qgl2+(ψT−ψ0)2+2p−2ρΓ2p+1μ2Qζ+(ψT−ψ0)2−2ρΓ2ρ∑k=1mck2.
This contradictsA4. Thus, there exists some r for which Tℬr⊂ℬr.Step 2
: letxnn=1∞+⊂ℬr such that xn⟶x in ℬr as n⟶∞.
Now, for anyu∈(0,u1], we have(69)Eψu−ψ01−ρ(Txnu−Txu)2≤3Eψu−ψ01−ρgu,xnu+ϕ˜u−gu,xu+ϕ˜u2+3Eψu−ψ01−ρΓp∫0uψ′sψu−ψ0p−1μζs,xns+ϕ˜s−ζs,xs+ϕ˜sds+3Eψu−ψ01−ρΓp∫0uψ′sψu−ψsp−1Fs,xns+ϕ˜s,xns+ϕ˜s,I0+p;ψxns+ϕ˜s−Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜sdωs2≤3ψu−ψ02−2ρℒglxn−xBy2+μ2ψu−ψ02−2ρ+2pΓ2p+1ℒζxn−xBy+3ψu−ψ02−2ρ+2pTraQΓ2p+1ℒuxn−xBy2+ℒvlxn−xBy2+ℒwxn−xBy2⟶0,asn⟶∞,∀u∈0,u1.
Also, for allu∈uk,uk+1, where k=1,2,…m, we have(70)Eψu−ψ01−ρ(Txnu−Txu)2≤4Eψu−ψ01−ρgu,xnu+ϕ˜u−gu,xu+ϕ˜u2+4Eψu−ψ01−ρΓp∫0uψ′sψu−ψ0p−1μζs,xns+ϕ˜s−ζs,xs+ϕ˜sds+4Eψu−ψ01−ρΓp∫0uψ′sψu−ψsp−1Fs,xns+ϕ˜s,xns+ϕ˜s,ℌD0+p,q;ψxns+ϕ˜s−Fs,xs+ϕ˜s,xs+ϕ˜s,ℌD0+p,q;ψxs+ϕ˜sdωs2+4E1Γp∑k=1mIkxnuk−+ϕ˜uk−−Ikxuk−+ϕ˜uk−2≤4ψu−ψ02−2ρℒglxn−xBy2+μ2ψu−ψ02−2ρ+2pΓ2p+1ℒζxn−xBy+4ψu−ψ02−2ρ+2pTraQΓ2p+1ℒuxn−xBy2+ℒvlxn−xBy2+ℒwxn−xBy2+4ψu−ψ02−2pΓ2ρℒkxn−xBy⟶0,asn⟶∞,∀u∈uk,uk+1.
Thus,(71)Txn−TxB′y′⟶0,asn⟶∞,so that T is continuous.Step 3
: the operatorTℬr is equicontinuous, for all u∈J.
Foru=0, the class of functions Tx:x∈ℬr is equicontinuous. For any u1,u2∈Jk with u1<u2, k=1,2,…m, x∈ℬr, and C1u>0, we have(72)E(ψu−ψ0)1−ρTxu1−Txu22≤C12u1Eψu2−ψ01−ρTxu1−Txu2C12u1Eψu1−ψ01−ρTxu1−ψu2−ψ01−ρTxu22+C12u1Eψu2−ψ01−ρTxu2−ψu1−ψ01−ρTxu22≤C12u1Eψu1−ψ01−ρTxu1−ψu2−ψ01−ρTxu22+C12u1ETxu22ψu2−ψ01−ρ−ψu1−ψ01−ρ2.
The right side of the last inequality is independent ofx∈ℬr and approaches to zero as u1⟶u2, since (ψu−ψ0)1−ρTxu∈CJk,H1 and (ψu−ψ0)1−ρ−ψu1−ψ01−ρ⟶0 as u1⟶u2. Therefore, (Tx)u1−Txu2B′′⟶0 as u1⟶u2. Hence, the operator Tℬr is equicontinuous, for all u∈J.Step 4
: it remains to show that the Mönchs condition holds.
Let(73)(Tx)u=T1xu+T2xu+T3xu+T4xu,where(74)T1xu=gu,xu+ϕ˜u+(ψu−ψ0)1−ρΓρϕ,T2xu=μΓρ∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds,T3xu=(ψu−ψ0)ρ−1Γρ∑k=1mIkxuk+ϕ˜uk,T4xu=1Γρ∫0uψ′sψu−ψsp−1Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜sdωs.
Assume thatW⊆ℬr is countable and W⊆co¯0∪TW. We show that βW=0, where β is the Hausdorff measure of noncompactness. Without loss of generality, we may take W=xnn=1∞. Since TW is equicontinuous on the interval Jk, therefore, W⊆co¯0∪TW is equicontiuous on Jk as well.
Using Lemmas2 and 4 along with conditions A1−A3, we have(75)βT1xnun=1∞≤βgu,xu+ϕ˜u≤η∗sup−∞≤θ≤0βxnuθn=1∞,βT2xnun=1∞≤2μΓp∫0uψ′sψu−ψsp−1η5ssup−∞≤θ≤0βxnuθn=1∞ds≤2μψu−ψ0pΓp+1η5L1α5J,ℝ+sup−∞≤θ≤0βxnθn=1∞,βT3xnun=1∞≤βψu−ψ0ρ−1Γρ∑k=1mIkxuk+ϕ˜uk≤ψu−ψ0ρ−1Γρ∑k=1mKksup−∞≤θ≤0βxnθn=1∞,βT4xnun=1∞≤2T.TraqQΓp∫0uψ′sψu−ψsp−1[η1ssup−∞≤θ≤0βxnuθn=1∞+η2ssup−∞≤θ≤0βxnuθn=1∞+η3ssup−∞≤θ≤0βxnuθn=1∞]ds≤2ψu−ψ0p+1/2Γp+1η2L1α4J,ℝ+sup−∞≤θ≤0βxnθn=1∞.
Thus, we have(76)βTxnun=1∞≤βT1xnun=1∞+βT2xnun=1∞+βT3xnun=1∞+βT4xnun=1∞≤η∗sup−∞≤θ≤0βxnuθn=1∞+2μψu−ψ0pΓp+1η5L1α4J,ℝ+sup−∞≤θ≤0βxnθn=1∞+ψu−ψ0ρ−1Γρ∑k=1mKksup−∞≤θ≤0βxnθn=1∞+2ψu−ψ0p+1/2Γp+1η1L1α2J,ℝ++η2L1α2J,ℝ++η3L1α2J,ℝ+sup−∞≤θ≤0βxnθn=1∞=η∗+2μψT−ψ0pΓp+1η5L1α2J,ℝ++ψT−ψ0ρ−1Γρ∑k=1mKk+2ψT−ψ0p+1/2Γp+1η1L1α2J,ℝ++η2L1α2J,ℝ++η3L1α2J,ℝ+βxnθn=1∞=ℳβxnθn=1∞.
As bothW and TW are continuous, it follows from Lemma 2 that βTW≤ℳβW.
Thus, from Mönchs condition, we have(77)βW≤βco¯0∪TW=βTW≤ℳβW.
Sinceℳ<1, we get βW=0, from which it follows that W is compact. Thus, T has a fixed point x in W. Thanks to Lemma 4, this completes the proof.Theorem 3.
Let the assumptionsA0ii−A3ii hold, provided that ξ=maxξ1,ξ2<1, where(78)ξ1=3ψT−ψ02−2ρℒgl+μ2(ψT−ψ0)2pΓ2p+1ℒζ+(ψT−ψ0)2pTraQΓ2p+1ℒu+ℒvl+ℒw,ξ2=4ψT−ψ02−2ρℒgl+μ2(ψT−ψ0)2pΓ2p+1ℒζ+(ψT−ψ0)2pTraQΓ2p+1ℒu+ℒvl+ℒw+(ψu−ψ0)2ρ−2pΓ2ρℒ1.
Then, the operatorT:By′′⟶By′′ has a unique fixed point on J.Proof.
Define the operatorT:By′′⟶By′′ by(79)Txu=gu,xu+ϕ˜u+ψu−ψ0ρ−1Γρϕu+μΓρ∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γρ∫0uψ′sψu−ψsp−1Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜sdωs,u∈J0,gu,xu+ϕ˜u+ψu−ψ0ρ−1Γρϕu+∑k=1m(Ikxuk+ϕ˜uk+μΓρ∫0uψ′sψu−ψsp−1ζs,xs+ϕ˜sds+1Γρ∫0uψ′sψu−ψsp−1Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψ[xs+ϕ˜s]dωs,u∈uk,uk+1,k=1,2,…,m.
Foru∈(−∞,0], the proof is trivial. For any u∈(0,u1] and x,y∈CJ,H1, we consider(80)Eψu−ψ01−ρ(Txu−Tyu23Eψu−ψ01−ρgu,xu+ϕ˜u−gu,yu+ϕ˜u2+≤3Eψu−ψ01−ρΓp∫0uψ′sψu−ψ0p−1μζs,xs+ϕ˜s−ζ(s,ys+ϕ˜s)ds+3Eψu−ψ01−ρΓp∫0uψ′sψu−ψ0p−1Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜s−F(s,ys+ϕ˜s,ys+ϕ˜s,I0+p;ψys+ϕ˜sdωs2≤3ψu−ψ02−2ρℒglx−yBy2+3μ2ψu−ψ02−2ρ+2pΓ2p+1ℒζx−yBy+3ψT−ψ02−2ρ+2pTraQΓ2p+1ℒux−yBy2+ℒvlx−yBy2+ℒwx−yBy2=3ψT−ψ02−2ρℒgl+μ2ψT−ψ02pΓ2p+1ℒζ+ψT−ψ02pTraqQΓ2p+1ℒu+ℒvl+ℒwx−yBy=ξ1x−yBy≤ξx−yBy.
Also, for allu∈uk,uk+1, where k=1,2,…m, we have(81)Eψu−ψ01−ρTxu−Tyu2≤4Eψu−ψ01−ρgu,xu+ϕ˜t−gu,yu+ϕ˜t2+4Eψu−ψ01−ρΓp∫0uψ′sψu−ψ0p−1μζs,xs+ϕ˜s−ζs,ys+ϕ˜sds+4Eψu−ψ01−ρΓp∫0uψ′sψu−ψsp−1Fs,xs+ϕ˜s,xs+ϕ˜s,I0+p;ψxs+ϕ˜s−Fs,ys+ϕ˜s,ys+ϕ˜s,I0+p;ψxs+ϕ˜sdωs2+4E1Γp∑k=1mIkxtk−+ϕ˜uk−−Ikyuk−)+ϕ˜tk−2≤4ψT−ψ02−2ρℒgl+μ2ψT−ψ02pΓ2p+1ℒζ+ψT−ψ02pTraQΓ2p+1ℒu+ℒvl+ℒw+ψu−ψ02ρ−2pΓ2ρℒkx−yBy=ξ2x−yBy≤ξx−yBy.
From the conditionξ<1, the mapping T is contractional; therefore, T has a unique fixed point.


## 4. UH Stability Analysis

In this section, we are establishing UH stability result.For anyϵ>0, consider the following inequality:(82)ED0+p,q;ψℌxu−gu,xu−μζu,xu−Fu,xu,xu,I0+p;ψxudωudu2<ϵ.s.Definition 4.
The considered system (4) is UH stable if we can find constant λ>0 such that, for every ϵ>0, there exists a solution xu which satisfies inequality (82) and has the same initial value as yu, where yu is the solution of (4). Then, xu satisfies the relation(83)E(ψu−ψ0)1−ρxu−yu2≤λϵ.Theorem 4.
Let conditionsA0iv, A1iii, A2ii, and A3ii be satisfied; then, solution to (4) is UH stable.Proof.
Let inequality (82) is true, and consider the following system:(84)D0+p,q;ψℌxu−gu,xu=μζu,xs+Fu,xu,xu,I0+p;ψxudωudu,u∈J=]0,T]−u1,u2,…um,μ∈ℝ+,T>0,ΔI1−ρ;ψxuk=Ik(xuk−),ρ=p+q−pq,k=1,2,…m,ΔI1−ρ;ψguk,xuk=0,k=1,2,…m,I0+1−ρ;ψxu=φ∈By,u∈−∞,0..
Using the method of fundamental solution to (84), we obtain(85)xu=ϕ∈By,u∈(−∞,T],gu,xu+(ψu−ψ0)ρ−1Γρϕu+μΓρ∫0uψ′sψu−ψsp−1ζs,xsds+1Γρ∫0uψ′sψu−ψsp−1Fs,xs+ϕ˜s,xs,I0+p;ψxsdωs,u∈J0,gu,xu+(ψu−ψ0)ρ−1Γρϕu+∑k=1mIkxuk+ϕ˜uk+μΓρ∫0uψ′sψu−ψsp−1ζs,xsds+1Γ(p)∫0uψ′s(ψu−ψs)p−1F(s,xs,xs,I0+p;ψxs)dωs,u∈(uk,uk+1],k=1,2,…,m.
In the interval−∞,0, it is obvious that the solution to (84) is UH stable. Now, for u∈(0,u1] and 0<ϵ<1, we have(86)Eψu−ψ01−ρyu−xu2≤3Eψu−ψ01−ρgu,yu−gu,xu2+3Eψu−ψ01−ρΓp∫0uψ′sψu−ψsp−1μζs,ys−ζs,xsds2+3Eψu−ψ01−ρTraQΓp∫0uψ′sψu−ψsp−1×Fs,ys,ys,I0+p;ψys−Fs,xs,ys,I0+p;ψysdωs2≤3ℒglEψu−ψ01−ρyu−xu2+3μ2ψu−ψ01−ρ+2pΓ2p+1ϵ+3ψu−ψ01+2p−ρTraQΓ2p+1[ℒuEψu−ψ01−ρyu−xu2+lℒvEψu−ψ01−ρyu−xu2+ℒwEψu−ψ01−ρD0+p,q;ψℌI0+p;ψyu−D0+p,q;ψℌI0+p;ψxu2]≤3μ2ψu−ψ01−ρ+2pΓ2p+1ϵ+3lℒg+ψT−ψ01−ρTraQΓ2p+1ℒu+lℒv+ℒw×Eψu−ψ01−ρyu−xu2.
Thus,(87)E(ψu−ψ0)1−ρyu−xu2≤3μ2(ψu−ψ0)1−ρ+2p1−K1ϵ=K1ϵ,where(88)K1=3lℒg+(ψT−ψ0)1−ρTraQΓ2p+1ℒu+lℒv+ℒw
(89)K1=3(ψT−ψ0)1−ρ1−K1.
If we consider the intervalu∈(uk,uk+1], then(90)E(ψu−ψ0)1−ρyu−xu2≤≤4μ2(ψu−ψ0)1−ρ+2pΓ2p+1ϵ+4lℒg+(ψT−ψ0)1−ρTraQΓ2p+1ℒu+lℒv+ℒw×E(ψu−ψ0)1−ρyu−xu2+4E1Γρ∑k=1mIkyuk−−Ikxuk−2.
SinceIk is continuous and for u∈(0,u1], the conclusion yu−xu<K1ϵ implies that there exists some positive constant say ℒ2, such that(91)yu−xu<ℒ2ϵ.
Therefore,(92)E(ψu−ψ0)1−ρyu−xu2≤4μ2(ψT−ψ0)1−ρ+2pΓ2p+1ϵ+4lℒg+(ψT−ψ0)1−ρTraQΓ2p+1ℒu+lℒv+ℒw×E(ψu−ψ0)1−ρyu−xu2+4ℒ22k2Γ2ρϵ.
Thus,(93)E(ψu−ψ0)1−ρyu−xu2≤41−K1μ2(ψT−ψ0)1−ρ+2pΓ2p+1+ℒ12k2Γ2ρϵ=K2ϵ,where(94)K2=41−K1μ2(ψT−ψ0)1−ρ+2pΓ2p+1+ℒ12k2Γ2ρ.
Continuing the same process, we can prove that, for anyu∈(uk,uk+1],k=1,2,…m, there exists a constant K=maxK1,K2,…,Km such that(95)E(ψu−ψ0)1−ρyu−xu2≤Kϵ.
By Definition1, system (4) is UH stable.


## 5. Illustrative Example

In this section, we illustrate the above results by an example.Example 1.
Consider the following impulsive neutralψ-Hilfer stochastic integro-fractional differential system(96)D0+32,1;ψuℌyu−14∫−∞0ψ′se−2syu+sds=ψu−ψ02∫−∞0ψ′se−2sysds+ψu−ψ04[∫−∞0ψ′se−2sy(u+s)ds+I0+23;ψuyu]dωudu,u∈J=(0,12]∪(12,1]−12,ΔI0+1−ρ;ψy1−2=14,ρ=p+q−pq=23+1−231=1,T=m=1,ψu=u,I0+1−ρ;ψy0=0,yu=0,u∈(−∞,0].
Set the functions as(97)Fu,x,y,z=14∫−∞0e−2syu+sds+I0+23;ψuyudωudu,u∈J,gu,x=14∫−∞0e−2syu+sds,u∈J,ξu,x=12∫−∞0e−2sysds,u∈J.
For anyy∈By, we define(98)yBy=∫−∞0fssups≤θ≤0Ey(θ)212ds,fs=e−2s,l=12,thenyu∈By.
The spaceBy,.By is a Banach space having the properties given below.
ℋ1∗: if y:(−∞,1]⟶ℝ is continuous on J, y0∈By, then yu is also continuous on J and yt∈By.
ℋ2∗: By is a Banach space.
ℋ3∗:(99)yuBy=12supu∈0,TEyH1212+y0By.
Additionally, letℒu=ℒv=ℒw=ℒg=1/16, ℒξ=1/4, ℒ1=0, μ=1/2, and TraQ=1. Thus, assumptions A0 to A4 are satisfied.
Now,(100)ξ1=3ψT−ψ02−2ρℒgl+μ2(ψT−ψ0)2pΓ2p+1ℒζ(ψT−ψ0)2pTraQΓ2p+1ℒu+ℒvl+ℒw=0.7266,ξ2=4ψT−ψ02−2ρℒgl+μ2(ψT−ψ0)2pΓ2p+1ℒζ+(ψT−ψ0)2pTraQΓ2p+1ℒu+ℒvl+ℒw+(ψu−ψ)02ρ−2pΓ2ρℒ1=0.9687.
Asξ=max0.7266,0.9687=0.9687<1, thus, system (96) has a unique solution. Also, 1−0.7266≠0; therefore, system (96) is UH stable on J.


## 6. Conclusion

The fractional stochastic neutral impulsive differential equations have applications in various fields such as viscoelasticity, automatic control, and electrochemistry. Based on the some well-known fixed-point theorems of fractional calculus, technique of stochastic analysis existence results for the considered system is given. Likewise under specific assumptions and conditions, we have found the UH stability result for the solution of (4).



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*Source: 1490583-2022-03-19.xml*