# Ulam–Hyers Stability for a Class of Caputo-Type Fractional Stochastic System with Delays

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

---

## Abstract

In this study, we explore the Caputo-type fractional stochastic differential equations (FSDEs) with delays. Firstly, the existence and uniqueness of the solutions for the considered system are obtained under the non-Lipschitz condition by using the Carathe´odory approximation method with the aid of Fatous Lemma, Ito^ isometry, Cauchy–Schwarz inequality, and Gronwall’s inequality. Secondly, we study the Ulam–Hyers (U–H) stability of the addressed equations with the aid of the Cauchy–Schwarz inequality and Gronwall’s inequality. Finally, an example with numerical simulation is presented to support the validity of our theory.


---

## Body

## 1. Introduction

In 1695, the German mathematician Leibniz and the French mathematician L’Hospital first discussed the one-half derivative in their correspondence. Since then, the fractional derivative has appeared in the sight of many scholars. With the research of Bernoulli, Euler, Lagrange, Riemann, Liouville, Lacroix, Abel, Caputo, and other scholars, the definition of fractional calculus is more and more perfect, and its properties, applications are gradually discovered. So far, the fractional calculus has experienced more than three hundred years of development. The study of fractional calculus was focused on pure mathematics at first and then gradually developed in the field of practical application. Nowadays, we can see the application of fractional calculus to fractional physics, viscous elasticity, automatic control theory, chemical physics, anomalous diffusion, and chain unlinking of polymer materials. For more details on fractional calculus, please refer to [1–8]. Authors of [1–7] introduced, in detail, the origin and definition of various types of fractional calculus and elaborated the boundary value problem of fractional differential equations (FDEs) and its research methods and introduced the application fields and application models of fractional calculus. Agarwal et al. [8] introduced a new extension of the Caputo fractional derivative operator involving the generalized hypergeometric-type function Fpa,b;c;z;k.Due to the development of FDEs in various fields, some experts found that the traditional differential equations cannot accurately describe the long tail phenomenon and power law phenomenon, as well as polymer materials, all kinds of anomalous diffusion in the field of porous media percolation mechanics phenomenon, but FSDEs were considered better able to describe these phenomena. So, FSDEs were proposed to solve those problems that were affected by random factors in real life. Therefore, FSDEs are considered to be very useful, and more and more scholars have carried out research on FSDEs. Refer to [9–11] for some results related to FSDEs. Mishura and Zili [9] were devoted to general information from the theory of Gaussian processes, fractional and subfractional Brownian motions, and mixed fractional and mixed subfractional Brownian motions. Mishura [10] was devoted to the Wiener integration with respect to fractional Brownian motion, stochastic integration with respect to fractional Brownian motion, and other aspects of stochastic calculus of fBm and filtering problems in the mixed fractional models. Meerschaert and Sikorskii [11] covered basic limit theorems for random variables and random vectors with heavy tails and the basic ideas of fractional calculus and anomalous diffusion were introduced in the context of probability theory.In order to facilitate the further development and application of FSDEs, it is necessary to study some important properties of FSDEs. On the one hand, we should explore the existence and uniqueness of solutions. Many scholars had made a lot of important contributions in this area. In [12], the authors employed two methods to prove the existence and uniqueness of fractional stochastic neutral differential equations, in which the contraction mapping principle was used under the Lipschitz condition at first, and then, the Picard approximation was applied under the non-Lipschitz condition. In [13], the author proved existence and uniqueness of ψ-Caputo fractional stochastic evolution equations with varying-time delay driven by fBm using the Schauder fixed-point theorem [14] and the Banach contraction principle, separately. In [15], Picard iterations were used to explore the existence and uniqueness of multidimensional FSDEs with variable order under non-Lipschitz condition. Although [13, 15] both used Picard approximation, the mild solution in [13] was solved by Laplace transform and its inverse, while the mild solution in [15] was obtained by integrating both sides of the differential equations. In addition, Yang [13] investigated the constant fractional order, while Moualkia [15] explored variable fractional order. In [16], the authors studied the existence of fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space by using fractional calculus, operator semigroup, and fixed-point theorem. For more types of FDEs and the methods used to study the existence and uniqueness of solutions, one can refer to [17–21].On the contrary, it is necessary to study the stability of solution to FSDEs, and U–H stability is an important content. In 1940, Ulam [22] first proposed the stability of functional equations in a speech at Wisconsin University. In 1941, Hyers [23] gave the answer to this question, and then, U-H stability came into being. The study of U-H stability for FDEs can not only provide an important theoretical basis for the existence and uniqueness of solutions but also provide a reliable theoretical basis for approximate solution of FSDEs. For the system with U-H stability, we can substitute an approximate solution for an exact one when it is relatively difficult to obtain the exact solution and the U-H stability can guarantee the reliability of the approximated solution to a certain extent. At present, many scholars had studied the U-H stability of various types of FDEs. For example, in [24], the researchers used the technique of Picard operator and Gronwall’s integral inequalities to explore U-H stability for conformable FDEs with delay. The authors applied the tools of fractional calculus and Ho¨lder’s inequality of integration to derive the U-H stability of nonlinear differential equations with fractional integrable impulses in [25]. For further contributions on this topic, see [26–30] and its references. The stability of stochastic differential equations has also been studied in [31–34], but there are relatively few papers on the U-H stability of FSDEs. Among them, Ahmadova et al. [35, 36] have made important contributions for U-H stability of FSDEs, and they studied the U-H stability of Caputo-type fractional stochastic neutral differential equations with weighted maximum norm and Ito^ isometry in [36].Inspired by the above papers, in this study, we consider the following system:(1)cD0+λas=Bas+Ks,as,as−τ+Ns,as,as−τdwsds,s∈0,U,as=bs,s∈−τ,0,where cD0+λ is the Caputo fractional derivative, 1/2<λ<1, B∈ℝn×n, K:0,U×ℝn×ℝn⟶ℝn and N:0,U×ℝn×ℝn⟶ℝn×m are measurable functions, ws is an m-dimensional Brownian motion on a complete probability space Ω,ℱ,ℙ, b:−τ,0⟶ℝn is a continuous function, a0=b0=b0, and Ea02<∞. In this study, we use the Carathe´odory approximation method to study the existence and uniqueness and use the direct method to prove the U-H stability of the solution to system (1).The main innovations of this study at least include the following aspects:(1)
The system in which we study the existence and uniqueness of solutions by the Carathe´odory approximation method is more generalized. In [37], the system is FSDEs, and in [38, 39], the system is first-order stochastic differential equations with delay while the system in this study is FSDEs with delay.(2)
The U-H stability and the existence and uniqueness of solutions are proved by using weaker non-Lipschitz condition. This is a contribution to the study of FSDEs.(3)
We give a numerical simulation to verify the U-H stability of the system, and this is the deficiency in [24–28, 36].The structure of this study is as follows. In Section2, we give some definitions, lemmas, and assumptions to promote the progress of the following paper. In Section 3, we prove the existence and uniqueness of system (1) under non-Lipschitz conditions by using Caratheodory approximation. The main purpose of section 4 is to explore the U-H stability of FSDEs. In Section 5, we test the effectiveness of our theory through an example with numerical simulation.


## 2. Preliminaries

In the section, we give some definitions, lemmas, and assumptions which will be used throughout the study. Let· be the standard Euclidean norm of a vector x∈ℝn or a matrix A∈ℝn×m be defined by(2)x=∑i=1nxi21/2,i=1,2,…,n,A=λ,where λ is the maximum eigenvalue of ATA.Definition 1 (see [1]).
Forω∈0,U,ℝn, the Caputo derivative of fractional order λ>0 is defined as(3)cD0+λωη=1Γn−α∫0ηn−αi−α−1ωiσdσ,where i−1<λ<i, i∈ℕ.Definition 2 (see [37]).
Forω:0,U⟶ℝn, the Riemann–Liouville integral operator of fractional order λ>0 is defined as(4)I0+λωη=1Γλ∫0ηη−σλ−1ωσdσ,where η>0 and Γ:0,∞⟶ℝ is the well-known Euler’s Gamma function. In particular, for λ∈0,1, we have(5)I0+λcD0+λωη=ωη−ω0.Lemma 1 (see [40]).
Suppose thatrx,tx,fx∈C0,U,ℝ+,sx∈C−τ,0,ℝ+, and rxandsx are nondecreasing and r0=s0. If μx∈C−τ,U,ℝ+ and(6)μx≤rx+∫0xtvμv+fvμv−τdv,x∈0,v,μx≤sx,x∈−τ,0,then(7)μx≤rxexp∫0xtv+fvdv,x∈0,v.Lemma 2 (see [12]).
For eachU>0,2λ>1, if g· is an ℝn-value stochastic process and ∫0Ugη2dη<∞, then the following inequality holds:(8)sup0≤η≤UE∫0ηη−vλ−1gvdwv2≤U2λ−12λ−1E∫0Ugv2dv.Lemma 3.
Let us consider the solution for the following system:(9)cD0+λas=Bas+Ks,s∈0,U,as=bs,s∈−τ,0,where b0=b0.Whens∈0,U, system (9) is equivalent to the integral equation as follows:(10)as=b0+1Γλ∫0ss−vλ−1Bavdv+1Γλ∫0ss−vλ−1Kvdv.Applying Laplace transformation, it is easy to obtain(11)a˜u=b0u+Buλa˜u+1uλk˜u.Then,(12)a˜u=b0uλ−1uλ−B+k˜uuλ−B=b0LMλsλB;u+k˜uLsλ−1Mλ,λsλB;u.Utilizing inverse Laplace transformation, we get the solution to system (9):(13)as=b0MλsλB+∫0ss−vλ−1Mλ,λs−vλBKvdv,s∈0,U,where MλandMλ,λ are matrix Mittag–Leffler functions and Mλ,Mλ,λ:ℝn×n⟶ℝn×n are defined as [36](14)MλsλB=∑j=0∞BjsjλΓjλ+1,Mλ,λsλB=∑j=0∞BjsjλΓjλ+λ.SinceMλ· and Mλ,λ· are bounded on ∈0,U, for convenience, we assume(15)σ1:=sup0≤s≤UMλsλB2,σ2:=sup0≤s≤UMλ,λsλB2.Definition 3 (see [37]).
If anℝn-value stochastic process as−τ≤s≤U is satisfied,(1)
as is ℱs-adapted and continuous.(2)
Ks,as,as−τ∈L0,U×ℝn×ℝn;ℝn and Ns,as,as−τ∈L20,U×ℝn×ℝn;ℝn×m.(3)
Fors∈−τ,0, as=bs and for s∈0,U,(16)as=b0MλsλB+∫0ss−vλ−1Mλ,λs−vλBKv,av,av−τdv+∫0ss−vλ−1Mλ,λs−vλBNv,av,av−τdwv,s∈0,U,bs,s∈−τ,0,where E∫−τUas2ds<∞.(4)
For each other solutiona˜s, we obtain Pas=a˜s,−τ≤s≤U=1; then, as−τ≤s≤U is a solution to (1), where Ks,as,as−τ∈L0,U×ℝn×ℝn;ℝn represents the Lebesgue integrability of the function K, Ns,as,as−τ∈L20,U×ℝn×ℝn;ℝn×m represents the Lebesgue integrability of the function N squared.Assumption 1.
(non-Lipschitz condition). Suppose that there exists a functionH:0,+∞×ℝ+⟶ℝ+ is satisfied:(i)
For each fixedφ≥0,θ∈0,+∞⟶Hθ,φ∈ℝ+ is locally integrable, and for each fixed θ≥0,φ∈R+⟶Hθ,φ∈ℝ+ is nondecreasing, concave, and continuous, and Hθ,0=0,∀θ∈0,+∞,∫0+1/Hθ,φdφ=+∞.(ii)
Forξ1,ξ2,ρ1,ρ2∈ℝn and each fixed θ≥0, the inequality(17)Kθ,ξ1,ρ1−Kθ,ξ2,ρ22∨Nθ,ξ1,ρ1−Nθ,ξ2,ρ22≤Hθ,ξ1−ξ22+ρ1−ρ22holds.(iii)
Forθ∈ℝ+ and Qθ≥0 such that(18)Qθ≤l∫0θHu,Qudu,where l>0 is a constant, we obtain Qθ≡0.Assumption 2.
LetKθ,0,0,Nθ,0,0∈L20,U, and ∀θ∈0,U; it follows that(19)Kθ,0,02∨Nθ,0,02≤σ3,where σ3 is a constant.Lemma 4 (see [37]).
Given that the functionHv,u:0,U×ℝ+⟶ℝ+ is concave, there exist c1v,c2v:0,U⟶ℝ+ such that(20)Hv,u≤c1v+c2vu,where(21)∫0Uc1vdv<∞,∫0Uc2vdv<∞.Assumption 3.
In equation (1), for any s∈−τ,0, there exists a nondecreasing function T·∈C−τ,0,ℝ+ and(22)T0=3b0σ1+12σ2U2λ−12λ−1Uσ3+sup0≤v≤Uc1v+Usup0≤v≤Uc2vsup−τ≤v≤0b2v+2b2τ,such that(23)sup−τ≤y≤sb2y≤Ts,s∈−τ,0.


## 3. Existence and Uniqueness

This section explores the existence and uniqueness of the addressed system. Let us consider Carathe´odory successive approximations as follows: for each integer m≥1, define am on −1+τ,U by(24)ams=bτ,s∈−1+τ,−τ,(25)ams=bs,s∈−τ,0,(26)ams=b0MλsλB+∫0ss−vλ−1Mλ,λs−vλBKv,amv−1m,amv−τ−1mdv+∫0ss−vλ−1Mλ,λs−vλBNv,amv−1m,amv−τ−1mdwv.Theorem 1.
Suppose that assumptions1-3 hold; then, equation (1) has a unique solution at,t∈−τ,U.Proof 1.
For−τ≤s≤0, obviously ams,m≥1 is bounded, equicontinuous, and a Cauchy sequence. For 0≤s≤U, the proof will be divided into the following steps.Step 1
:ams,m≥1 is bounded.
By (26), we have(27)sup0≤y≤sEamy2≤3sup0≤y≤sEb0MλyλB2+3sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBKv,amv−1m,amv−τ−1mdv2+3sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBNv,amv−1m,amv−τ−1mdwv2,where we use the following inequality:(28)∑i=1nxi2≤n∑i=1nxi2.
Applying Cauchy–Schwarz inequality, Lemma2, inequality (28), and assumption 1 (ii), we attain(29)sup0≤y≤sEamy2≤3b0σ1+3σ2sup0≤y≤sE∫0yy−v2λ−2dv,∫0yKv,amv−1m,amv−τ−1m2dv+3σ2U2λ−12λ−1E∫0sNv,amv−1m,amv−τ−1m2dv≤3b0σ1+3σ2U2λ−12λ−1E∫0sKv,amv−1m,amv−τ−1m−Kv,0,0+Kv,0,02dv+3σ2U2λ−12λ−1E∫0sNv,amv−1m,amv−τ−1m−Nv,0,0+Nv,0,02dv≤3b0σ1+12σ2σ3U2λ2λ−1+6σ2U2λ−12λ−1,E∫0sKv,amv−1m,amv−τ−1m−kv,0,02dv+6σ2U2λ−12λ−1E∫0sNv,amv−1m,amv−τ−1m−Nv,0,02dv≤3b0σ1+12σ2σ3U2λ2λ−1+12σ2U2λ−12λ−1E∫0sHv,amv−1m2+amv−τ−1m2dv.
From Lemma4, it follows that(30)sup0≤y≤sEamy2≤3b0σ1+12σ2σ3U2λ2λ−1+12σ2U2λ−12λ−1E∫0SC1v+C2vamv−1m2+C2vamv−τ−1m2dv+3b0σ1+12σ2σ3U2λ2λ−1+12σ2U2λ2λ−1sup0≤v≤Uc1v+12σ2U2λ−12λ−1sup0≤v≤UC2v∫0ssup0≤v1≤vEamv1−1m2dv+12σ2U2λ−12λ−1sup0≤v≤UC2v∫0ssup0≤v1≤vEamv1−τ−1m2dv≤f1+f2∫0ssup0≤v1≤1/mEamv1−1m2+sup1/m≤v1≤vEamv1−1m2dv+f2∫0ssup0≤v1≤1/mEamv1−τ−1m2+sup1/m≤v1≤vEamv1−τ−1m2dv≤f1+f2∫0Sb2τ+sup−t≤v1≤0Eamv12+sup0≤v1≤vEamv12dv+f2∫0sb2τ+sup0≤v1≤vEamv1−τ2dv≤f1+f2∫0sb2τ+sup−τ≤v1≤0b2v1+sup0≤v1≤vEamv12dv+f2Ub2τ+f2∫0ssup0≤v1≤0Eamv1−τ2dv≤f1+f2Usupτ≤v1≤02v1+2f2Ub2τ+∫0sf2sup0≤v1≤vEamv12+f2sup0≤v1≤vEamv1−τ2dv=T0+∫0sf2sup0≤v1≤vEamv12+f2sup0≤v1≤vEamv1−τ2dv,where f1=3b0σ1+12σ2σ3U2λ/2λ−1+12σ2U2λ/2λ−1sup0≤v≤Uc1vandf2=12σ2U2λ−1/2λ−1sup0≤v≤Uc2v.
Then, Lemma1 and assumption 3 read(31)sup0≤y≤sEamy2≤0exp∫0sf2+f2dv≤T0e2f2U.=c,where c=T0e2f2U is a positive constant. Therefore, ams,m≥1 is bounded.Step 2:
ams,m≥1 is equicontinuous.
For0≤s1<s2≤U and each integer m≥1, from (26), we obtain(32)=ams2−ams1b0Mλs2λB−b0Mλs1λB+∫0s1s2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλBKv,amv−1m,amv−τ−1mdv+∫s1s2s2−vλ−1Mλ,λs2−vλBKv,amv−1m,amv−τ−1mdv+∫0s1s2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλBNv,amv−1m,amv−τ−1mdwv+∫s1s2s2−vλ−1Mλ,λs2−vλBNv,amv−1m,amv−τ−1mdwv,in which, with the aid of (28), Cauchy–Schwarz inequality and It o^ isometry yield(33)Eams2−ams12≤5b0Mλs2λB−b0Mλs1λB2+5∫0s1s2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλB2dv,∫0s1EKv,amv−1m,amv−τ−1m2dv+5∫s1s2s2−vλ−1Mλ,λs2−vλB2dv,∫s1s2EKv,amv−1m,amv−τ−1m2dv+5∫0s1Es2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλBNv,amv−1m,amv−τ−1m2dv+5∫s1s2Es2−vλ−1Mλ,λs2−vλB,Nv,amv−1m,amv−τ−1m2dv,where(34)EKv,amv−1m,amv−τ−1m2=Ekv,amv−1m,amv−τ−1m−kv,0,0+kv,0,02≤2Ekv,amv−1m,amv−τ−1m−kv,0,02+2kv,0,02≤2EHvamv−1m2+amv−τ−1m2+2σ3≤2Ec1v+c2vamv−1m2+c2vamv−τ−1m2+2σ3≤2sup0≤v≤Uc1v+2sup0≤v≤Uc2vsup−1≤v1≤0Eamv12+sup0≤v1≤vEamv12+2sup0≤v≤Uc2vsup−1+τ≤v2≤0Eamv22+sup0≤v2≤vEamv22+2σ3≤2sup0≤v≤Uc1v+2sup0≤v≤Uc2vb2τ+sup−τ≤v1≤0b2v1+c+2sup0≤v≤Uc2vb2τ+sup−τ≤v2≤0b2v2+c+2σ3=p1.
Thus,(35)Eams2−ams12≤5b0Mλs2λB−b0Mλs1λB2+5q1U∫0s1s2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλB2dv+5q1s2−s1∫s1s2s2−vλ−1Mλ,λs2−vλB2dv+5q1∫0s1s2−vλ−1Mλ,λs2−vλB−s1−vλ−1Mλ,λs1−vλB2dv+5q1∫s1s2s2−vλ−1Mλ,λs2−vλB2dv.
Lets1⟶s2; we obtain Eams2−ams1⟶0 which implies that ams,m≥1 is equicontinuous.Step 3
:ams,m≥1 is a Cauchy sequence. For integer j>m≥1, based on (28), Cauchy–Schwarz inequality and Lemma 2, we can derive(36)sup0≤y≤sEajy−amy2≤2sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBKv,ajv−1j,ajv−τ−1j−Kv,amv−1m,amv−τ−1mdv2+2sup0≤v≤sE∫0yy−vλ−1Mλ,λy−vλBNv,ajv−1j,ajv−τ−1j−Nv,amv−1m,amv−τ−1mdwv2≤2σ2U2λ−12λ−1E∫0sKv,ajv−1j,ajv−τ−1j−Kv,amv−1m,amv−τ−1m2dv+2σ2U2λ−12λ−1E∫0sNv,ajv−1j,ajv−τ−1j−Nv,amv−1m,amv−τ−1m2dv=2σ2U2λ−12λ−1E∫0sKv,ajv−1j,ajv−τ−1j−Kv,ajv−1m,ajv−τ−1m+Kv,ajv−1m,ajv−τ−1m−Kv,amv−1m,amv−τ−1m2dv+2σ2U2λ−12λE∫0sNv,ajv−1j,ajv−τ−1j−Nv,ajv−1m,ajv−τ−1m+Nv,ajv−1m,ajv−τ−1m−Nv,amv−1m,amv−τ−1m2dv≤4σ2U2λ−12λ−1∫0sEKv,ajv−1j,ajv−τ−1j−Kv,ajv−1m,ajv−τ−1m2dv+4σ2U2λ−12λ−1∫0sEKv,ajv−1m,ajv−τ−1m−Kv,amv−1m,amv−τ−1m2dv+4σ2U2λ−12λ−1∫0sENv,ajv−1j,ajv−τ−1j−Nv,ajv−1m,ajv−τ−1m2dv+4σ2U2λ−12λ−1∫0sENv,ajv−1m,ajv−τ−1m−Nv,amv−1m,amv−τ−1m2dv,where(37)EKv,ajv−1m,ajv−τ−1m−Kv,amv−1m,amv−τ−1m2≤EHv,ajv−1m−amv−1m2+ajv−τ−1m−amv−τ−1m2≤Hv,Eajv−1m−amv−1m2+Eajv−τ−1m−amv−τ−1m2≤Hv,sup0≤v1≤vEajv1−amv12+sup0≤v2≤vEajv2−amv22=Hv,2sup0≤v1≤vEajv1−amv12,ENv,ajv−1m,ajv−τ−1m−Nv,amv−1m,amv−τ−1m2≤Hv,2sup0≤v1≤vEajv1−amv12.
Then, we have(38)sup0≤y≤sEajy−amy2≤8σ2U2λ−12λ−1∫0sEHv,ajv−1j−ajv−1m2+ajv−τ−1j−ajv−τ−1m2dv+8σ2U2λ−12λ−1∫0sHv,2sup0≤v1≤vEajv1−amv12dv.
Noticing the Fatous lemma and continuity ofHθ,·, we obtain(39)limj,m⟶∞sup2sup0≤y≤sEajy−amy2≤16σ2U2λ−12λ−1∫0sEHv,limj,m⟶∞supajv−1j−ajv−1m2+ajv−τ−1j−ajv−τ−1m2dv+16σ2U2λ−12λ−1∫0sEHv,limj,m⟶∞sup2sup0≤v1≤vEajv1−amv12dv.
Asj,m⟶∞, we get v−1/j⟶v−1/m and v−τ−1/j⟶v−τ−1/m. Then, by assumption 1 (i) and Step 2, we obtain(40)Hv,limj,m⟶∞supajv−1j−ajv−1m2+ajv−τ−1j−ajv−τ−1m2=Hv,0=0.
Let(41)Xs=limj,m⟶∞sup2sup0≤y≤sEajy−amv2.
We derive(42)Xs≤16σ2U2λ−12λ−1∫0sHv,Xvdv.
Last through inequality (42) and assumption 1 (iii), it is easy to verify(43)Xs=limj,m⟶∞sup2sup0≤y≤sEajy−amy2=0,which implies that ams,m≥1 is a Cauchy sequence. Then, we know from the Borel–Cantelli lemma that ams⟶asn⟶∞ holds uniformly for s∈−τ,U. So, if we take the limits of both sides of equations (24), (25), and (26), we get that as is a solution to (1) and(44)sup0≤y≤sEay2≤∞,0≤s≤U.
Hence, the existence has been proved. The uniqueness can be proved by the same method as Step 3, and then, the proof is completed.


## 4. Ulam–Hyers Stability Analysis of FSDEs

Definition 4 (see [41]).
For system (1), if there exists a constant δ>0 such that ∀ε>0 and for each solution r·∈C0,U,ℝn of the following inequality,(45)sup0≤s≤UED0λrs−Brs−Ks,rs,rs−τ−Ns,rs,rs−τdwsds≤ε.
There exists a solutiona·∈C0,U,ℝn of (1) with(46)sup0≤s≤UErs−as≤εδ.
Then, system (1) is called U-H stable.Remark 1 (see [41]).
A functionr·∈0,U,ℝn is a solution of (45) if there is a function z·∈0,U,ℝn, such that(1)
sup0≤s≤UEzs2≤ε(2)
cD0+λrs=Brs+ks,rs,rs−τ+Ns,rs,rs−τ+zsTheorem 2.
If assumption1 (i) (ii) hold and there exists a constant β>0 such that sup0≤v≤Uc1v≤βε, then the FSDEs (1) is U-H stable on 0,U.Proof 2.
Taking Lemma3 and Remark 1 into account, we find(47)rs=b0MλsλB+∫0ss−vλ−1Mλ,λs−vλBKv,rv,rv−τdv+∫0ss−vλ−1Mλ,λs−vλBNv,rv,rv−τdwv+∫0ss−vλ−1Mλ,λs−vλBzvdv.
In view of Definition3 and (47), we obtain(48)rs−as=∫0ss−vλ−1Mλ,λs−vλBKv,rv,rv−τ−Kv,av,av−τdv+∫0ss−vλ−1Mλ,λs−vλBNv,rv,rv−τ−Nv,av,av−τdwv+∫0ss−vλ−1Mλ,λs−vλBzvdv.
Then, using (28) Cauchy–Schwarz inequality and Lemma 2, we have(49)sup0≤y≤sEry−ay2≤3sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBKv,vv,rv−τ−kv,av,av−τdv2+3sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBNv,rv,rv−τ−Nv,av,av−τdwv2+3sup0≤y≤sE∫0yy−vλ−1Mλ,λy−vλBzvdv2≤3σ2U2λ−12λ−1∫0sEKv,rv,rv−τ−Kv,av,av−τ2dv+3σ2U2λ−12λ−1∫0sENv,rv,rv−τ−Nv,av,av−τ‖2dv+3σ2U2λ2λ−1ε.
By assumption1 (ii) and Lemma 4, we obtain(50)sup0≤y≤sEry−ay2≤6σ2U2λ−12λ−1∫0sEHv,rv−av2+rv−τ−av−τ2dv+3σ2U2λ2λ−1ε≤6σ2U2λ−12λ−1∫0sEc1v+c2vrv−av2+c2v‖rv−τ−av−τ‖2dv+3σ2U2λ2λ−1ε≤6σ2U2λ−12λ−1∫0sEc1v+c2vsup0≤v1≤vrv1−av12+c2vsup−τ≤v2≤0rv2−av22+c2vsup0≤v2≤vrv2−av22dv+3σ2U2λ2λ−1ε=6σ2U2λ−12λ−1∫0sEc1v+2c2vsup0≤v1≤vrv1−av12dv+3σ2U2λ2λ−1ε≤6σ2U2λ−12λ−1∫0sEsup0≤v≤Uc1v+2sup0≤v≤Uc2vsup0≤v1≤vrv1−av12dv+3σ2U2λ−12λ−1ε≤6σ2βU2λ2λ−1ε+3σ2U2λ2λ−1ε+∫0s12σ2U2λ−12λ−1sup0≤v≤Uc2vsup0≤v1≤vErv1−av12dv.
Hence, employing Gronwall’s inequality, we may obtain(51)sup0≤y≤sEry−ay2≤2β+13σ2U2λ2λ−1εexp∫0s12σ2U2λ−12λ−1sup0≤v≤Uc2vdv≤2β+13σ2U2λ2λ−1exp12σ2U2λ2λ−1sup0≤v≤Uc2vε=εδ,where δ=2β+13σ2U2λ/2λ−1exp12σ2U2λ/2λ−1sup0≤v≤Uc2v is a positive constant. Therefore, this theorem is proved.


## 5. Example

Example 1.
Consider the U-H stability of the following system:(52)cD0+34as=1π4as+17cosas+as−1+1e4as+as−12dwsds,s∈0,8,as=1,s∈−1,0,where B=1/π4, and measurable functions are(53)Ks,as,as−τ=17cosas+as−1,Ns,as,as−τ=1e4as+as−12.
We have(54)Ks,rs,rs−1−Ks,as,as−12=17cosrs+rs−1−cosas+as−12≤17rs+rs−1−as+as−12=17rs−as+rs−1−as−1rs+rs−1+as+as−12≤27rs−as2+rs−1−as−12min0≤s≤8rs+rs−1+as+as−12≤k^rs−as2+rs−1−as−12≔Hs,rs−as2+rs−1−as−12=0+k^rs−as2+rs−1−as−12≔c1s+c2srs−as2+rs−1−as−12,Ns,rs,rs−1−Ns,as,as−12=1e4rs+rs−12−as+as−122=1e4rs+rs−1+as+as−1rs−as+rs−1−as−12≤2e4max0≤s≤8rs+rs−1+as+as−12rs−as2+rs−1−as−12≤k^rs−as2+rs−1−as−12≔Hs,rs−as2+rs−1−as−12=0+k^rs−as2+rs−1−as−12≔c1s+c2srs−as2+rs−1−as−12,where(55)k^=max27min0≤s≤8rs+rs−1+as−as−12,2e4max0≤s≤8rs+rs−1+as+as−12.
c1s=0andc2s=k^. Therefore, K and N satisfy assumption 1 (i) and (ii) and we can find constant ∀β>0 such that sup0≤v≤8c1s=0≤βε, where ε=0.09. According to Theorem 2, equation (52) is stable in U-H sense on 0,8.
We can see from Figure1 that the distance between as and rs is less than a constant, and according to Definition 4, the solution of (52) is U-H stable.Figure 1
The U-H stable of the solution for system (52) with ε=0.09.


## 6. Conclusion

The original intention of this study is to provide some more generalized results for the study of stability to FSDEs. We mainly prove the U-H stability, existence, and uniqueness of solutions of system (1). To achieve this goal, we utilize the technique of the Carathe´odory approximation method, Fatous Lemma, Ito^ isometry, Cauchy–Schwarz inequality, and Gronwall’s inequality. Finally, the theoretical results are verified by an example with numerical simulation.



---

*Source: 5898922-2022-04-26.xml*