# Ulam–Hyers Stability for a Class of Hilfer-Type Fractional Stochastic Differential Equations

Authors: Meiling Song; Zhiguo Luo
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this work, we study a class of Hilfer-type fractional stochastic differential equations (HFSDEs). By Laplace transform and its inverse, we obtain a mild solution to HFSDEs. Furthermore, we research the existence and uniqueness of solutions to the addressed equations. Subsequently, we explore the Ulam–Hyers (U–H) stability of the considered system by the Picard iteration. Finally, we verify the correctness of the theoretical results with an example.


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## Body

## 1. Introduction

Fractional calculus was born in 1695, when L’Hopital wrote a letter to Leibniz about thenth derivative of a simple linear function fx=x. What happened when n=1/2? Leibniz replied that this is a seemingly absurd but meaningful question that will one day lead to useful results. Since then, fractional calculus has become a topic of great interest to some mathematicians. The one of earliest systematic study of fractional calculus was done by Liouville, Riemann, and Holmgren in the early and mid-19th century. By the 20th century, Weyl, Hardy, Littlewood, Caputo, Hilfer, and Miller Ross et al. made important contributions to the development of fractional calculus [1–5]. Among them, Hilfer [5] introduced the definition of generalized fractional derivative called Hilfer fractional derivative, which is the interpolated cases of Riemann–Liouville fractional derivatives and Caputo fractional derivatives. Compared with integer calculus, fractional calculus can reflect the properties and objective facts of objects more specifically, so the application prospect of fractional differential equations (FDEs) is broader. Nowadays, fractional calculus has formed a relatively important branch in the field of mathematics and has achieved rapid development of theory and application, and it has been widely used in the fields of electroanalytical chemistry, viscoelasticity, signal processing, and the unzipping of polymer materials [6–14].There are various random disturbances and random phenomena in nature, and it is no longer appropriate to use a deterministic model to describe this system affected by random factors. However, it becomes more reasonable to describe the development law of things through stochastic differential equations (SDEs), which also promote the development of SDEs theory. More than half a century ago, the idea of Itô stochastic analysis established by Itô [15] and the theory of SDEs developed based on it enabled researchers to have a deeper understanding of many random phenomena in nature. Scholars are increasingly aware that SDEs are more accurate and objective in describing various random phenomena in real life. Since then, SDEs have been widely used in chemistry, physics, biology, finance, control, and other fields [16–18]. In recent years, many monographs have discussed the main achievements in the theory and application of SDEs [19–22].In general, it is not easy to solve the explicit solutions of FDEs, so it is necessary to research the existence and uniqueness of FDEs. Since Hilfer introduced the Hilfer fractional derivative, more and more scholars have begun to study various properties of HFSDEs. There are some articles that explore the existence and uniqueness of HFSDEs. Saravanakumar and Balasubramaniam [23] proved the existence of solutions to HFSDEs with noninstantaneous impulsive by utilizing the Mönch fixed-point theorem. In [24], the authors used the same method to study HFSDEs. However, the systems of [23, 24] are different. Lv and Yang [25] investigated the existence of HFSDEs by applying the fixed-point theorem and verified its uniqueness by using the contraction mapping principle. The authors employed Sadovskii fixed-point theorem to explore the existence of HFSDEs [26]. To the best of our knowledge, compared with other types of fractional differential equations, there are few articles about the existence and uniqueness of solutions to HFSDEs. Therefore, there is still a lot of works waiting for researchers to further explore the existence and uniqueness of HFSDEs. For the methods and results of the existence and uniqueness of other types to FDEs, readers can refer to [27–31] and its references. We will also draw on the methods in these articles to study the existence and uniqueness of HFSDEs.Moreover, for a U–H stable system, when it is difficult to get an exact solution, the system can be approximately solved, and U–H stability can ensure the reliability of approximate solution. The U–H stability originated from a lecture by Ulam [32] at Wisconsin University in 1940, and then, Hyers [33] gave the first affirmative answer to this question in Banach space in the following year. The U–H stability theory has evolved significantly over time. Nowadays, there are a lot of studies using many methods to study the U–H stability of FDEs. The authors obtained the U–H stability of FDEs with a generalized Caputo derivative by the direct method and Grönwall inequality in [34]. With the help of weighted maximum norm and Itô isometry, the researchers explored the U–H stability of Caputo-type fractional stochastic neutral differential equations in [35]. In [36], Vivek et al. investigated the U–H stability and U–H–Rassias stability of Hilfer fractional implicit differential equations with nonlocal conditions. However, there are still very few articles on the U–H stability of HFSDEs, so this field still has great prospects.In view of the fact that there are few articles about the existence, uniqueness, and U–H stability of the solutions to HFSDEs, most of them prove these theories by the fixed-point theorem. In this study, we intend to use Picard’s type successive approximations to obtain the existence and stability results of HFSDEs, and the system is different from those abovementioned studies. Motivated by the above articles, we aim to study the following system:(1)D0+a,bηω=Nηω+pω,ηω+qω,ηωdWωdω,ω∈0,V,I0+1−rη0=ξ0,r=a+b−ab,where D0+a,b is the Hilfer fractional derivative with 0≤a≤1 and 1/2<b<1, N is a matrix of dimensions m×m, the state vector η∈ℝm is a stochastic process, p:0,V×ℝm⟶ℝm and q:0,V×ℝm⟶ℝm×j are the continuous functions, and W⋅ is an j-dimensional Brownian motion on a complete probability space Ω,ℱ,ℙ and Eη02<∞.The rest of this article is divided into the following sections. We will give some definitions and properties of fractional integrals and derivatives, as well as the lemmas and conditions in Section2. The purpose of Section 3 is to investigate the existence and uniqueness of HFSDEs by Picard iteration and contradiction, respectively. With the aid of Picard iteration and the properties of convergent series, we derive the U–H stability of system (1) in Section 4. In Section 5, we give an example to test the validity of our theoretical results.


## 2. Preliminaries

For a vector functionη=η1,η2,…,ηm∈L20,V×ℝm and a matrix N∈ℝm×j, we define the standard Euclidean norm as follows:(2)η=∑k=1mηi21/2,i=1,2,…,m;N=α.where L2 denote the function square Lebesgue integrable and α represents the largest eigenvalue of NTN.Definition 1.
(see [37]). For the function u:0,+∞⟶ℝ, the Riemann–Liouville fractional integral of order β with the lower limit 0 is defined as(3)I0+βuω=1Γβ∫0ωω−xβ−1uxdx,ω>0,where Γ⋅ is the Gamma function.Definition 2.
(see [37]). The Riemann–Liouville fractional derivative of order β∈0,1 of the function u:0,+∞⟶ℝ is defined by(4)RLD0+βuω=1Γ1−βddω∫0ωuxω−xβdx.
In particular, forr∈0,1,(5)I0+rRLD0+ruω=uω−I0+1−ru0Γrωr−1.Definition 3.
(see [37]). For the function u:0,+∞⟶ℝ, the Hilfer fractional derivative of order a∈0,1 and b∈0,1 with the lower limit 0 is defined as(6)D0+a,buω=I0+a1−bddωI0+1−βuω,β=1−a1−b.Definition 4.
(see [38]). The Mittag–Leffler function is defined as(7)Ma,by=∑j=0∞yjΓja+b,y∈ℂ,where ℂ stands for the set of complex numbers.
The Laplace integral ofωb−1Ma,b±Nωa is defined as(8)ℒωb−1Ma,b±Nωa;s=∫0∞e−sωωb−1Ma,b±Nωadω=sa−bsaI∓N,where I is an identity matrix.Lemma 1.
We consider the system of the following form:(9)D0+a,bηω=Nηω+fω,ω∈0,V,I0+1−rη0=ξ0,r=a+b−ab.
Simultaneously integrating both sides of system (9) yields(10)I0+bD0+a,bηω=I0+bNηω+I0+bfω=1Γb∫0ωNω−θb−1ηθdθ+1Γb∫0ωω−θb−1fθdθ.
Based on(11)I0+bD0+a,bηω=I0+bI0+a1−bddωI0+1−rηω=I0+a+b−abddω1Γ1−r∫0ωω−θ−rηθdθ=I0+r1Γ1−rddω∫0ωηθω−θrdθ=I0+rRLD0+rηω=ηω−I0+1−rη0Γrωr−1=ηω−ξ0Γrωr−1,we get(12)ηω=ξ0Γrωr−1+1Γb∫0ωNω−θb−1ηθdθ+1Γb∫0ωω−θb−1fθdθ.
By using the Laplace transform, we have(13)ηℒs=ξ0sr+Nηℒssb+fℒssb,where ηℒs and fℒs represent the Laplace transform of ηω and fω, respectively. Furthermore, we get(14)ηℒs=ξ0sb−rsbI−N+fℒssbI−N.
Through the inverse Laplace transform, the solution of equation (9) is(15)ηω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbfθdθ.
Similarly, the mild solution of equation (1) can be obtained as follows:(16)ηω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,ηθdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηθdWθ,ω∈0,V,where Mb,r,Mb,b:ℝm×m⟶ℝm×m are the matrix Mittag–Leffler functions. Since Mb,r⋅ and Mb,b⋅ are bounded, for convenience, we assume(17)λ1:=sup0≤ω≤VMb,rNωb2,λ2:=sup0≤ω≤VMb,bNωb2.Lemma 2.
(see [39]). (Grönwall–Bellman inequality) For continuous functions g,h:ℝ+⟶ℝ+ and positive constant K, if(18)gω≤K+∫0ωhθgθdθ,ω∈ℝ+,holds, then(19)gω≤Kexp∫0ωhθdθ,ω∈ℝ+.Lemma 3.
(see [40]). For any V>0,2b>1, h⋅ is an ℝm-value stochastic process and ∫0Vgθ2dθ<∞. Then, we can get(20)sup0≤ω≤VE∫0ωω−θb−1gθdWθ2≤V2b−12b−1E∫0Vgθ2dθ.Definition 5.
(see [41]). For an ℝm-value stochastic process ηω0≤ω≤V, if the following conditions hold, then ηω0≤ω≤V is a unique solution to equation (1).(1)
pω,ηω∈L0,V×ℝm;ℝm and qω,ηω∈L20,V×ℝm;ℝm×j(2)
ηω is ℱω-adapted and continuous(3)
Forω∈0,V, we have(21)ηω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,ηθdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηθdWθ,ω∈0,V.(4)
For other solutionsη˜ω, we get ℙηω=η˜ω,0≤ω≤V=1Assumption 1.
Forx1,x2∈ℝm and each fixed ω≥0, the inequality(22)pω,x1−pω,x22∨qω,x1−qω,x22≤lx1−x22,holds and “∨” is defined as p∨q=maxp,q.Assumption 2.
Letpω,0,qω,0∈C0,V,ℝm, it follows that(23)pω,02∨qω,02≤λ3,where λ3 is a constant and C0,V,ℝm represents the set of continuous functions on interval 0,V.


## 3. Existence and Uniqueness

In this section, the existence and uniqueness results of system (1) are established.Theorem 1.
Suppose that Assumptions1-2 and λ2lV2b<2b−1/16 are satisfied, then system (1) has a unique solution in L20,V,ℝm.Proof.Step 1.
We prove the existence of equation (1). To achieve this, we consider the following Picard iteration:(24)ηnω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,ηn−1θdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηn−1θdWθ.
Forω∈0,V,n∈ℕ+, and η0ω=ξ0/ω1−rMb,rNωb∈L20,V,ℝm, obviously, ηn⋅∈L20,V,ℝm. By Jensen’s inequality, we have(25)Eηnω2≤3Eξ0ω1−rMb,rNωb2+3E∫0ωω−θb−1Mb,bNω−θbpθ,ηn−1θdθ2+3sup0≤y≤ωE∫0yy−θb−1Mb,bNy−θbqθ,ηn−1θdWθ2,which with the aid of Lemma 3, Jensen’s inequality, Cauchy–Schwarz inequality, and Assumptions 1-2 gives(26)Eηnω2≤3λ1ξ02ω2−2r+3λ2ω2b−12b−1E∫0ωpθ,ηn−1θ2dθ+3λ2ω2b−12b−1E∫0ωqθ,ηn−1θ2dθ≤3λ1ξ02V2−2r+3λ2V2b−12b−1E∫0ωpθ,ηn−1θ−pθ,0+pθ,02dθ+3λ2V2b−12b−1E∫0ωqθ,ηn−1θ−qθ,0+qθ,02dθ≤3λ1ξ02V2−2r+6λ2V2b−12b−1∫0ωEpθ,ηn−1θ−pθ,02+pθ,02dθ+6λ2V2b−12b−1∫0ωE‖qθ,ηn−1θ−qθ,0‖2+qθ,02dθ≤3λ1ξ02V2−2r+12λ2lV2b−12b−1∫0ωEηn−1θ2dθ+12λ2λ3V2b2b−1.
For an arbitrary integeri≥1, we can conclude that(27)max1≤n≤iEηnω2≤3λ1ξ02V2−2r+12λ2λ3V2b2b−1+12λ2lV2b−12b−1∫0ωmax1≤n≤iEηn−1θ2dθ.
We should notice that(28)max1≤n≤iEηn−1θ2=maxEη0θ2,Eη1θ2,…,Eηi−1θ2≤maxEξ0θ1−rMb,rNθb2,Eη1θ2,…,Eηi−1θ2,Eηiθ2=maxEξ0θ1−rMb,rNθb2,max1≤n≤iEηnθ2≤ξ0θ1−rMb,rNθb2+max1≤n≤iEηnθ2.
Hence,(29)max1≤n≤iEηnω2≤ρ1+ρ2∫0ωξ0θ1−rMb,rNθb2+max1≤n≤iEηnθ2dθ≤ρ1+ρ2λ1ξ02V2r−12r−1+ρ2∫0ωmax1≤n≤iEηnθ2dθ,where ρ1=3λ1ξ02/V2−2r+12λ2λ3V2b/2b−1 and ρ2=12λ2lV2b−1/2b−1.
Based on Grönwall–Bellman inequality, we obtain(30)max1≤n≤iEηnω2≤ρ1+ρ2λ1ξ02V2r−12r−1exp∫0ωρ2dθ≤ρ1+ρ2λ1ξ02V2r−12r−1eρ2V.
Since the integeri>0 is arbitrary, it is easy to get(31)Eηnω2≤ρ1+ρ2λ1ξ02V2r−12r−1eρ2V,for ∀n≥1 and ω∈0,V.
Next, forn=1, it can be seen from equation (24) that(32)η1ω−η0ω=η1ω−ξ0ω1−rMb,rNωb=∫0ωω−θb−1Mb,bNω−θbpθ,η0θdθ+∫0ωω−θb−1Mb,bNω−θbqθ,η0θdWθ.
Based on Lemma3, Jensen’s inequality, Cauchy–Schwarz inequality, and Assumptions 1-2, we infer that(33)Eη1ω−η0ω2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,η0θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNy−θbqθ,η0θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,η0θ2dθ+2λ2V2b−12b−1E∫0ωqθ,η0θ2dθ=2λ2V2b−12b−1E∫0ωpθ,η0θ−pθ,0+pθ,02dθ+2λ2V2b−12b−1E∫0ωqθ,η0θ−qθ,0+qθ,02dθ≤8λ2lV2b−12b−1E∫0ωη0θ2dθ+8λ2λ3V2b2b−1=8λ2lV2b−12b−1E∫0ωξ0θ1−rMb,rNθb2dθ+8λ2λ3V2b2b−1≤8λ1λ2lξ02V2b+2r−12b−12r−1+8λ2λ3V2b2b−1≔l1.
Then,(34)Esup0≤ω≤Vη1ω−η0ω2≤l1.
Forn=2, by Lemma 3, Jensen’s inequality, Cauchy–Schwarz inequality and Assumptions 1-2, we get(35)Eη2ω−η1ω2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,η1θ−pθ,η0θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNω−θbqθ,η1θ−qθ,η0θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,η1θ−pθ,η0θ2dθ+2λ2V2b−12b−1E∫0ωqθ,η1θ−qθ,η0θ2dθ≤4λ2lV2b−12b−1E∫0ωη1θ−η0θ2dθ≤4λ2lV2b2b−1Esup0≤ω≤Vη1ω−η0ω2.
Similarly, forn=3, we can get(36)Eη3ω−η2ω2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,η2θ−pθ,η1θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNy−θbqθ,η2θ−qθ,η1θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,η2θ−pθ,η1θ2dθ+2λ1V2b−12b−1E∫0ωqθ,η2θ−qθ,η1θ2dθ≤4λ2lV2b−12b−1∫0ωEη2θ−η1θ2dθ≤4λ2lV2b2b−12Esup0≤ω≤Vη1ω−η0ω2.
Assume for somen,(37)Eηnω−ηn−1ω2≤4λ2lV2b2b−1n−1Esup0≤ω≤Vη1ω−η0ω2.
Then,(38)Eηn+1ω−ηnω2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,ηnθ−pθ,ηn−1θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNy−θbqθ,ηnθ−qθ,ηn−1θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,ηnθ−pθ,ηn−1θ2dθ+2λ2V2b−12b−1E∫0ωqθ,ηnθ−qθ,ηn−1θ2dθ≤4λ2lV2b−12b−1∫0ωEηnθ−ηn−1θ2dθ≤4λ2lV2b2b−1nEsup0≤ω≤Vη1ω−η0ω2.
Therefore, according to the above induction, for eachn=1,2,…, we have(39)Eηnω−ηn−1ω2≤4λ2lV2b2b−1n−1Esup0≤ω≤Vη1ω−η0ω2.
Furthermore,(40)sup0≤ω≤VEηnω−ηn−1ω2≤4λ2lV2b2b−1n−1Esup0≤ω≤Vη1ω−η0ω2≤4λ2lV2b2b−1nl1.
Indeed, use Doob’s martingale inequality to yield(41)ℙsup0≤ω≤Vηnω−ηn−1ω>12n≤22nEsup0≤ω≤Vηnω−ηn−1ω2≤22n4λ2lV2b2b−1nl1.
Note the series(42)∑n=1∞22n4λ2lV2b2b−1nl1<∞.
Then,(43)ℙsup0≤ω≤Vηnω−ηn−1ω>12ninfinitely often=0.
From above,(44)ηnω=η0ω+∑k=1nηkω−ηk−1ω,converges uniformly on 0,V. Denote the limit of ηn⋅ by η⋅. Note that η⋅ is ℱω-adapted and continuous. From (41), we know that ηn⋅,n≥1 is a Cauchy sequence in L20,V,ℝm, and limn⟶∞Eηnω−ηω2=0.
Letn⟶∞ on both sides of inequality (31), we can get(45)Eηω2≤ρ1+ρ2λ1ξ02V2r−12r−1eρ2V,which means that(46)E∫0Vηω2dω≤∫0Vρ1+ρ2λ1ξ02V2r−12r−1eρ2Vdω<∞.
This shows thatη⋅∈L20,V,ℝm.
Next, we prove thatη⋅ satisfies system (1) on interval 0,V. Note that(47)E∫0ωω−θb−1Mb,bNω−θbpθ,ηn−1θ−pθ,ηθdθ2≤λ2V2b−12b−1E∫0ωpθ,ηn−1θ−pθ,ηθ2dθ≤λ2lV2b−12b−1E∫0ωηn−1θ−ηθ2dθ≤λ2lV2b2b−1Esup0≤ω≤Vηn−1θ−ηθ2⟶0,n⟶∞.
Similarly, we have(48)E∫0ωω−θb−1Mb,bNω−θbqθ,ηn−1θ−qθ,DθdWθ2≤λ2lV2b2b−1Esup0≤ω≤Vηn−1θ−ηθ2⟶0,n⟶∞.
Therefore, we obtain that(49)limn⟶∞∫0ωω−θb−1Mb,bNω−θbpθ,ηn−1θdθ=∫0ωω−θb−1Mb,bNω−θbpθ,ηθdθ,limn⟶∞∫0ωω−θb−1Mb,bNω−θbqθ,ηn−1θdwθ=∫0ωω−θb−1Mb,bNω−θbqθ,ηθdWθ.
Then, taking the limit on both sides of (24), we can get(50)ηω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,ηθdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηθdWθ,ω∈0,V,which implies that η⋅ is a solution of equation (1).Step 2.
We prove the uniqueness of system (1). Suppose η⋅ and η^⋅ are two different solutions of system (1). For all ω∈0,V, by Jensen’s inequality Lemma 3 and Assumption 1, we have(51)Eηω−η^ω2=E‖∫0ωω−θb−1Mb,bNω−θbpθ,ηθ−pθ,η^θdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηθ−qθ,η^θdWθ‖2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,ηθ−pθ,η^θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNω−θbqθ,ηθ−qθ,η^θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,ηθ−pθ,η^θ2dθ+2λ2V2b−12b−1E∫0ωqθ,ηθ−qθ,η^θ2dθ≤4λ2lV2b−12b−1∫0ωEηθ−η^θ2dθ,which, with the aid of Grönwall–Bellman inequality, yields Eηω−η^ω2=0. Therefore, the solution of equation (1) is unique. The proof of this theorem is completed.


## 4. Ulam–Hyers Stability Analysis of HFSDEs

In this section, we consider the U–H stability of the HFSDEs:(52)D0+a,bηω=Nηω+pω,ηω+qω,ηωdWωdω.Letε>0, and we consider the following inequality:(53)ED0+a,bσω−Nσω−pω,σω−qω,σωdWωdω2<ε.Definition 6.
(see [34]). Equation (1) is called U–H stable, if there exists a real number δ>0, such that ∀ε>0, and for each solution σ⋅∈L20,V,ℝm of the inequality (53), there exists a solution η⋅∈L20,V,ℝm of equation (1) with(54)Eσω−ηω2≤δε.Remark 1.
(see [34]). A function σ⋅∈L20,V,ℝm is a solution of the inequality (53) if there exists a function μ⋅∈L20,V,ℝm, such that(i)
sup0≤ω≤VEμω2≤ε(ii)
D0+a,bσω=Nσω+pω,σω+qω,σωdWω/dω+μω
Therefore, the solution to inequality (53) is(55)σω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,σθdθ+∫0ωω−θb−1Mb,bNω−θbqθ,σθdWθ,+∫0ωω−θb−1Mb,bNω−θbμθdθ,ω∈0,V.Theorem 2.
If Assumption1 and λ2lV2b<2b−1/4 hold, then HFSDEs (1) is U–H stable in L20,V,ℝm.Proof.
We considerηn,n≥0 in L20,V,ℝm given by η0ω=σω and(56)ηnω=ξ0ω1−rMb,rNωb+∫0ωω−θb−1Mb,bNω−θbpθ,ηn−1θdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηn−1θdWθ.
Forn=1 and ω∈0,V, by Cauchy–Schwarz inequality and Jensen’s inequality, we can conclude that(57)Eη1ω−η0ω2=E‖∫0ωω−θb−1Mb,bNω−θbpθ,η0θ−pθ,σθdθ+∫0ωω−θb−1Mb,bNω−θbqθ,η0θ−qθ,σθdWθ−∫0ωω−θb−1Mb,bNω−θbμθdθ‖2≤3E∫0ωω−θb−1Mb,bNω−θbpθ,η0θ−pθ,σθdθ2+3E∫0ωω−θb−1Mb,bNω−θbqθ,η0θ−qθ,σθdWθ2+3E∫0ωω−θb−1Mb,bNω−θbMθdθ2≤3∫0ωEω−θb−1Mb,bNω−θb2dθ⋅∫0ωEμθ2dθ≤3λ2V2b2b−1ε.
Then,(58)Esup0≤ω≤Vη1ω−η0ω2≤3λ2V2b2b−1ε.
Forn≥1, by basing on Assumption 1 and mathematical induction, we derive that(59)Esup0≤ω≤Vηnω−ηn−1ω2≤4λ2lV2b2b−1n−1Esup0≤ω≤Vη1ω−η0ω2.
In order to check the correctness of the above inequality, we assume that it is true forn=k, and for n=k+1, by Jensen’s inequality, Cauchy–Schwarz inequality, Assumption 1, and Lemma 3, we have(60)Eηk+1ω−ηkω2=E‖∫0ωω−θb−1Mb,bNω−θbpθ,ηkθ−pθ,ηk−1θdθ+∫0ωω−θb−1Mb,bNω−θbqθ,ηkθ−qθ,ηk−1θdWθ‖2≤2E∫0ωω−θb−1Mb,bNω−θbpθ,ηkθ−pθ,ηk−1θdθ2+2sup0≤y≤ωE∫0yy−θb−1Mb,bNy−θbqθ,ηkθ−qθ,ηk−1θdWθ2≤2λ2V2b−12b−1E∫0ωpθ,ηkθ−pθ,ηk−1θ2dθ+2λ2V2b−12b−1E∫0ωqθ,ηkθ−qθ,ηk−1θ2dθ≤4λ2lV2b2b−1Esup0≤ω≤Vηkω−ηk−1ω2≤4λ2lV2b2b−1kEsup0≤ω≤Vη1ω−η0ω2.
Thus,(61)Esup0≤ω≤Vηk+1ω−ηkω2≤4λ2lV2b2b−1kEsup0≤ω≤Vη1ω−η0ω2,which means that (59) holds for all n≥1. Then, from (59), it is easy to get(62)∑k=1∞Esup0≤ω≤Vηkω−ηk−1ω2≤∑k=1∞4λ2lV2b2b−1k−1Esup0≤ω≤Vη1ω−η0ω2≤3λ2V2b2b−1ε∑k=1∞4λ2lV2b2b−1k−1.
Since∑k=1∞4λ2lV2b/2b−1k−1 is a geometric series and 4λ2lV2b/2b−1<1, the series is convergent, and we assume that it converges to λ. Therefore, we have(63)∑k=1∞Esup0≤ω≤Vηkω−ηk−1ω2≤3λλ2V2b2b−1ε.
With the same manner as in the proof of Theorem1, we can able to show that ηn,n≥0 converges to the unique solution η⋅ of equation (1) in C0,V,ℝm as n⟶∞, i.e.,(64)limn⟶∞Esup0≤ω≤Vηnω−ηω2=0.
By using the triangular inequality of the norm⋅, we obtain(65)Esup0≤ω≤Vηnω−η0ω2≤Esup0≤ω≤Vηnω−ηn−1ω2+Esup0≤ω≤Vηn−1ω−η0ω2≤Esup0≤ω≤Vηnω−ηn−1ω2+Esup0≤ω≤Vηn−1ω−ηn−2ω2+Esup0≤ω≤Vηn−2ω−η0ω2≤Esup0≤ω≤Vηnω−ηn−1ω2+⋯+Esup0≤ω≤Vη1ω−η0ω2=∑k=1nEsup0≤ω≤Vηkω−ηk−1ω2.
Furthermore, we derive that(66)Esup0≤ω≤Vηω−σω2=Esup0≤ω≤Vηω−η0ω2=limn⟶∞Esup0≤ω≤Vηnω−η0ω2≤limn⟶∞∑k=1nEsup0≤ω≤Vηkω−ηk−1ω2=∑k=1∞Esup0≤ω≤Vηkω−ηk−1ω2≤3λλ2V2b2b−1ε.
Then,(67)Eηω−σω2≤3λλ2V2b2b−1ε,ω∈0,V.
Letδ=3λλ2V2b/2b−1, and it is immediate to obtain(68)Eηω−σω2≤δε,which implies that equation (1) is U–H stable. The proof is completed.


## 5. Example

Example 1.
Consider the U–H stability of the following HFSDEs on the interval0,6:(69)D0+1/2,3/4ηω=120ηω+sinηωπ2+1e2cosηωdWωdω,where a=1/2,b=3/4,N=1/20, and continuous functions(70)pω,ηω=sinηωπ2,qω,ηω=1e2cosηω.
Since(71)pω,ηω−pω,y^ω2=sinηωπ2−siny^ωπ22=2sin12ηωπ2−y^ωπ2cos12ηωπ2+y^ωπ22≤2sin12ηωπ2−y^ωπ22≤2×12ηωπ2−y^ωπ22=1π4ηω−y^ω2,(72)qω,ηω−qω,y^ω2=1e2cosηω−1e2⋅cosy^ω2=1e4cosηω−cosy^ω2=1e4−2sinηω+y^ω2sinηω−y^ω22≤1e42sinηω−y^ω22≤1e42ηω−y^ω22=1e4ηω−y^ω2,for ∀y^⋅∈ℝ, we can get(73)pω,ηω−pω,y^ω2∨qω,ηω−qω,y^ω2≤1e4ηω−y^ω2,which implies that assumption 1 is satisfied with l=1/e4.
Forω∈0,6, we obtain(74)M34,34112ω3/4=∑j=0∞1/20ω3/4jΓ3/4j+3/4<∑j=0∞1/20ω3/4j0.8=108∑j=0∞120ω3/4j≤108∑j=0∞120×63/4j<108∑j=0∞420j=108×1/51−1/5=516.
Then,(75)λ2=sup0≤ω≤6M34,34120ω3/42≤516.
Thus,(76)4λ2lV2b=4×516×1e4×63/2<0.4<2b−1=0.5,which shows that 4λ2lV2b<2b−1 holds. Hence, by Theorem 2, system (69) is U–H stable on 0,6.


## 6. Conclusion

In this study, we present two theories about HFSDEs. First, we give the existence and uniqueness theorem of equation (1) and use Picard iteration and contradiction to demonstrate our results, respectively. To prove this, we use the following technical methods: Grönwall–Bellman inequality, Jensen’s inequality, Doob’s martingale inequality, and Cauchy-Schwarz inequality. Second, with the aid of Picard iteration and the properties of convergent series, we derive the U–H stability of HFSDEs. Finally, the U–H stable result is verified by an example.



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*Source: 9844295-2022-06-25.xml*