# Ulam–Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays

Authors: Xue Wang; Danfeng Luo; Zhiguo Luo; Akbar Zada
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carathe´odory approximation. Furthermore, with the help of Ho¨lder’s inequality, Jensen’s inequality, Ito^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.


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## Body

## 1. Introduction

The fractional-order differential equations can better simulate many natural physical processes than integer-order differential equations, so it gradually becomes a powerful tool to analyze and solve problems in modern science and technology with the continuous development of natural science and production technology. It is mainly used in the fields of economy and insurance, the analysis of the quantitative structure of biological population, the control of diseases, and the research of genetic law, and we can see these monographs in [1–5]. For more notable achievements of this concept, the readers can also refer to [6–15].As it is well known, stochastic disturbance is inevitable in practical systems, and it has an important influence on the stability of systems. In [16], dut=kutdt was unstable when k>0, but it increased the stochastic feedback control rutdWt to become dut=kutdt+rutdWt. Apparently, dut=kutdt+rutdWt was stable if and only if r2>2k. This fact indicated that the stochastic control rutdWt can stabilize the unstable system dut=kutdt. Therefore, it is significant and challenging to study stochastic stabilization of deterministic systems. More relevant results can be found in [17–19].The research on the existence and uniqueness of solutions to fractional differential equations is an important content of differential equations. At the same time, the existence and uniqueness have made rapid development in the field of applied mathematics. In [20], the authors studied the existence and uniqueness of positive solutions of some nonlinear fractional differential equations by using mixed monotone operators on cones. Under a number of new conditions and combined with the generalized Gronwall inequality, the uniqueness of solution for fractional ψ-Hilfer differential equation with time delays was investigated in [21]. In addition, for many other relevant conclusions, readers can refer to [22–25].In 1940, S. M. Ulam proposed the stability to functional equations in a speech at the Wisconsin University [26]. Hyers [27] was the first to answer the question in 1941. From then, the Ulam–Hyers stability was produced. At the same time, more and more people were interested in exploring the Ulam–Hyers stability. In [28], by using fractional calculus, the properties of classical and generalized Mittag–Leffler functions and the Ulam–Hyers stability of linear fractional differential equations were proved by utilizing the Laplace transform method. The authors investigated the Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions in [29]. For more researched results, we can pay attention to [30–34].Inspired by the abovementioned, in this article, we are concerned with the existence and Ulam–Hyers stability of Caputo-type FSDEs with time delays:(1)D0+αCXt=ft,Xt,Xt−τ+gt,Xt,Xt−τdWtdt,t∈J=0,T,Xt=Φt,t∈−τ,0,where X0=Φ0, 1/2<α<1,f:0,T×ℝd×ℝd⟶ℝd, and g:0,T×ℝd×ℝd⟶ℝd×m are measurable continuous functions, Wt is an m-dimensional Brownian motion on a complete probability space Ω,ℱ,ℙ, Φt:−τ,0⟶ℝd is a continuous function, EΦt2<∞, and E is the mathematical expectation.Compared with the research results of [12, 20, 21, 24, 25, 28, 34], the major contributions of this paper include at least the following three aspects:(1)
In contrast to [20, 21, 28], the system we study is more generalized because it has not only the stochastic term but also the delay term.(2)
In the methods we investigate the existence and uniqueness of solutions to FSDEs are more novel than [24, 25]. In [24, 25], to explore the existence and uniqueness, Krasnoselskii’s fixed point theorem and Mo¨nch’s fixed point theorem, respectively, were used. However, in this paper, we adopt the Carathe´odory approximation to investigate the existence and uniqueness.(3)
In the study of various stability or existence and uniqueness of FSDEs, many literatures (see [12, 21, 34]) have used a stronger Lipschitz condition. However, in this paper, we used the weak non-Lipschitz condition to discuss the Ulam–Hyers stability of stochastic differential equations. This is a breakthrough in the exploration of the stability to FSDEs.The structure of this article is arranged as follows. We present some basic definitions and necessary assumptions in Section2. In Section 3, by Carathe´odory approximation, a number of assumed conditions are established for existence and uniqueness of solutions. Section 4 is devoted to testify stability results for the FSDEs with time delays. Examples are given to certify the application of our findings in Section 5.


## 2. Preliminaries

In this section, we intend to recommend a few basic definitions, lemmas, and some necessary assumptions that will play a key role in the paper.Letc,d−∞<c<d<+∞ be a finite interval, and we define the norm of Xt=X1t,X2t,…,XdtT∈ℝd on c,d as follows:(2)Xt=∑i=1dXi2t1/2.Definition 1 (see [35]).
The Riemann–Liouville integral operator of fractional-orderα>0 is defined by(3)I0+αft=1Γα∫0tt−vα−1fvdv,where t>0 and Γ:0,∞⟶ℝ is the well-known Eulers Gamma function.Definition 2 (see [1]).
For any continuous functionf:0,T×ℝd×ℝd⟶ℝd, the Caputo derivative of fractional-orderα>0is defined by(4)D0+αCft=1Γn−α∫0tt−vn−α−1fnvdv,where n−1<α<n, n∈ℕ.
In particular, forα∈0,1,(5)I0+αD0+αCft=ft−f0.Definition 3.
Anℝd-value stochastic process Xt−τ≤t≤T is called a solution to equation (1) if it satisfies the following conditions:(1)
Xt is t-continuous and ℱt adapted.(2)
ft,Xt,Xt−τ∈L0,T×ℝd×ℝd;ℝdF and gt,Xt,Xt−τ∈L20,T×ℝd×ℝd;ℝd×m.(3)
For∀t∈−τ,T,(6)Xt=Φ0+1Γα∫0tt−vα−1fv,Xv,Xv−τdv+1Γα∫0t∫0tt−vα−1gv,Xv,Xv−τdWv,t∈J,Φt,t∈−τ,0,where E∫−τTXt2dt<∞.(4)
For any other solutionX˜t, we obtain PXt=X˜t,−τ≤t≤T=1.Definition 4 (see [36]).
System (1) is Ulam–Hyers stable if there exists a real number δ>0 such that ∀ε>0 and for each continuously differentiable function Zt∈0,T,ℝd satisfying(7)Esup0≤t≤TD0+αCZt−ft,Zt,Zt−τ−gt,Zt,Zt−τdWtdt2≤ε,and there exists a solution Xt∈0,T,ℝd of (1) satisfying(8)Esup0≤t≤TZt−Xt2<εδ.Remark 1 (see [21]).
A functionZt∈0,T,ℝd is a solution of equation (7) if and only if there exists a function ht∈0,T,ℝd, such that(i)
Esupht20≤t≤T≤ε(ii)
D0+αCZt=ft,Zt,Zt−τ+gt,Zt,Zt−τdWt/dt+htHypothesis 1 (Lipschitz condition).
As for anyf,g∈ℝd, there is a constant l>0 such that, for all X1,X2,Y1,Y2∈ℝd,t∈0,T,(9)ft,X1,Y1−ft,X2,Y2∨gt,X1,Y1−gt,X2,Y2≤lX1−X2+Y1−Y2,where fandg are uniformly continuous functions and ∨ is defined as Y1∨Y2=maxY1,Y2.Hypothesis 2 (non-Lipschitz condition).
There is a functionGt,U,V, 0,+∞×ℝ+×ℝ+⟶ℝ+, such that(1)
For allX1,X2,Y1,Y2∈ℝd and 0≤t≤T,(10)ft,X1,Y1−ft,X2,Y22∨gt,X1,Y1−gt,X2,Y22≤Gt,X1−X22,Y1−Y22,wherefandg are continuous as well as bounded functions, and for any fixed t≥0, Gt,U,V is monotone, nondecreasing, continuous, and concave function with Gt,0,0=0.(2)
For everyt∈ℝ+ and any nonnegative function Y(t) such that(11)Yt≤m∫0tGv,Yvdv,where m>0 is a constant and Gv,Yv,Yv=Gv,Yv, we get Yt≡0.Hypothesis 3.
There exist three functionsat,bt,andqt, such that(12)Gt,U,V≤at+btU+qtV,U,V>0,∫0Tatdt<∞,∫0Tbtdt<∞,∫0Tqtdt<∞.Lemma 1 (see [37]).
Suppose Hypothesis2 and Hypothesis 3 are fulfilled. Then, there exists constant c>0 such that, for any t,X,Y∈0,T×ℝd×ℝd,(13)ft,X,Y2∨gt,X,Y2≤c1+X2+Y2.Proof.
Applying Jensen’s inequality and Hypothesis2 and Hypothesis 3, we have(14)ft,X,Y2=ft,X,Y−ft,0,0+ft,0,02≤2ft,X,Y−ft,0,02+2ft,0,02≤2Gt,X2,Y2+2sup0≤t≤Tft,0,02≤2at+2btX2+2qtY2+2sup0≤t≤Tft,0,02≤2sup0≤t≤Tat+2sup0≤t≤Tft,0,02+2sup0≤t≤TbtX2+2sup0≤t≤TqtY2≤k11+X2+Y2,where k1=max2sup0≤t≤Tat+2sup0≤t≤Tft,0,02,2sup0≤t≤Tbt,2sup0≤t≤Tqt<∞. In a similar way, we obtain(15)gt,X,Y2≤k21+X2+Y2.
Let us setc=maxk1,k2. The proof is therefore complete.


## 3. Existence and Uniqueness

Utilizing Carathe´odory approximation [35, 38], the existence and uniqueness of solutions to SFDEs can be obtained. So, let us define the Carathe´odory approximation as follows. For any integer n≥1,0≤t≤T define(16)Xnt=Φ0+1Γα∫0tt−vα−1fv,Xnv−1n,Xnv−τ−1ndv+1Γα∫0tt−vα−1gv,Xnv−1n,Xnv−τ−1ndWv,and Xnt=X0=Φ0 for all −1+τ≤t≤0.Theorem 1.
Suppose that Hypothesis 2 and Hypothesis 3hold and33/2cT2α−1<α−3/41/2Γ2α,3/4<α<1; then, system (1) has a unique solutionXt,t∈−τ,T.Proof.
The proof will be divided into three steps, whent∈0,T.Step 1.
The boundedness of the sequenceXnt,n≥1.
By (16) and Jensen’s inequality, we have(17)Esup0≤s≤tXns2≤3EΦ02+3Γ2αEsup0≤s≤t∫0ss−vα−1fv,Xnv−1n,Xnv−τ−1ndv2+3Γ2αEsup0≤s≤t∫0ss−vα−1gv,Xnv−1n,Xnv−τ−1ndWv2.
According to Ito^ isometry, Cauchy–Schwarz inequality, and Lemma 1, it is easy to obtain(18)Esup0≤s≤tXns2≤3EΦ02+3Γ2αE∫0tt−vα−1fv,Xnv−1n,Xnv−τ−1ndv2+3Γ2αE∫0tt−v2α−2gv,Xnv−1n,Xnv−τ−1n2dv≤3EΦ02+3T2α−12α−1Γ2αE∫0tc1+Xnv−1n2+Xnv−τ−1n2dv+3Γ2αE∫0tt−v2α−2c1+Xnv−1n2+Xnv−τ−1n2dv=J1+J2+J3.
Lettingv1=v−1/nandv2=v−τ−1/n. It is obvious that v1∈−1,v,v2∈−1+τ,v, and(19)J2≤3cT2α−12α−1Γ2α∫0t1+Esup−1≤v1≤vXnv12+Esup−1+τ≤v2≤vXnv22dv≤3cT2α−12α−1Γ2α∫0t1+2Esup−1+τ≤v2≤vXnv22dv≤c1T+2∫0tmaxEsup−1+τ≤V2≤0Xnv22,Esup0≤v2≤vxnv22dv=c1T+2c1∫0tmaxEΦ02,Esup0≤v2≤vXnv22dv,where c1=3cT2α−1/2α−1Γ2α.
Using Ho¨lders inequality and Jensen’s inequality, we obtain(20)J3≤3cΓ2α∫0tt−v4α−4dv1/2E∫0t1+Xnv−1n2+Xnv−τ−1n22dv1/2≤3cT2α−3/24α−31/2Γ2αE∫0t3+3Xnv−1n4+3Xnv−τ−1n4dv1/2,where 3+3Xnv−1/n4+3Xnv−τ−1/n4 is continuous function on 0,t. By the mean value theorem of integrals, there exists y∈0,t such that(21)∫0t3+3Xnv−1n4+3Xnv−τ−1n4dv=t3+3Xny−1n4+3Xny−τ−1n4≤T3+3Xny−1n4+3Xny−τ−1n4.
Using the inequality(22)L1+L2+L3≤L1+L2+L3,we derive(23)J3≤3cT2α−14α−31/2Γ2αE3+3Xny−1n4+3Xny−τ−1n41/2≤33/2cT2α−14α−31/2Γ2αE1+Xny−1n2+Xny−τ−1n2.
Let us setv3=y−1/n,v4=y−τ−1/n,v3∈−1,t,andv4∈−1+τ,t, and we have(24)J3≤33/2cT2α−14α−31/2Γ2αE1+Xnv32+Xnv42≤33/2cT2α−14α−31/2Γ2α1+Esup−1≤v3≤tXnv32+Esup−1+τ≤v4≤tXnv42≤33/2cT2α−14α−31/2Γ2α+33/2cT2α−14α−31/2Γ2α2Esup−1+τ≤V4≤tXnv42=c2+2c2maxE‖Φ0‖2,Esup0≤v4≤t‖Xnv4‖2,where c2=33/2cT2α−1/4α−31/2Γ2α.
IfmaxEΦ02,Esup0≤v4≤tXnv42=EΦ02, we obtain(25)Esup0≤s≤tXns2≤J1+J2+J3≤3EΦ02+c1T+2c1∫0tEΦ02dv+c2+2c2EΦ02≤3+2c1T+2c2EΦ02+c1T+c2=β1.
IfmaxEΦ02,Esup0≤v2≤vXnv22=Esup0≤v2≤vXnv22, we obtain(26)Esup0≤s≤tXns2≤J1+J2+J3≤3EΦ02+c1T+2c1∫0tEsup0≤v2≤vXnv22dv+c2+2c2Esup0≤v4≤tXnv42=3EΦ02+c1T+c2+2c1∫0tEsup0≤v2≤vXnv22dv+2c2Esup0≤s≤tXns2,and then,(27)Esup0≤s≤tXns2≤31−2c2EΦ02+c1T+c21−2c2+2c11−2c2∫0tEsup0≤v2≤vXn2dv.
By Gronwall’s inequality, we can conclude that(28)Esup0≤s≤tXns2≤31−2c2EΦ02+c1T+c21−2c2e2c1T/1−2c2=β2.
Lettingβ=maxβ1,β2, we obtain(29)Esup0≤s≤tXns2≤β,where β is a positive constant. So, we have proved that the sequence Xnt,n≥1 is bounded.Step 2.
For0≤s<t≤T and any integer n≥1, we obtain by (16)(30)Xnt−Xns=1Γα∫0st−vα−1−s−vα−1fv,Xnv−1n,Xnv−τ−1ndv+1Γα∫stt−vα−1fv,Xnv−1n,Xnv−τ−1ndv+1Γα∫0st−vα−1−s−vα−1gv,Xnv−1n,Xnv−τ−1ndWv+1Γα∫stt−vα−1gv,Xnn−1n,Xnv−τ−1ndWv.
By using Jensen’s inequality, Cauchy–Schwarz inequality, and Ito^ isometry, we conclude(31)EXnt−Xns2≤4Γ2αE∫0st−vα−1−s−vα−1fv,Xnv−1n,Xnv−τ−1ndv2+4Γ2αE∫stt−vα−1fv,Xnv−1n,Xnv−τ−1ndv2+4Γ2αE∫0st−vα−1−s−vα−1gv,Xnv−1n,Xnv−τ−1ndWv2+4Γ2αE∫stt−vα−1gv,Xnv−1n,Xnv−τ−1ndWv2≤4TΓ2αE∫0st−vα−1−s−vα−1fv,Xnv−1n,Xnv−τ−1n2dv+4TΓ2αE∫stt−vα−1fv,Xnv−1n,Xnv−τ−1n2dv+4Γ2αE∫0st−vα−1−s−vα−1gv,Xnv−1n,Xnv−τ−1n2dv+4Γ2αE∫stt−vα−1gv,Xnv−1n,Xnv−τ−1n2dv.
Recalling Lemma1, we obtain(32)EXnt−Xns2≤4c1+TΓ2α∫0st−vα−1−s−vα−12E1+Xnv−1n2+Xnv−τ−1n2dv+4c1+TΓ2α∫stt−v2α−2E1+Xnv−1n2+Xnv−τ−1n2dv=c3J4+J5,where c3=4c1+T/Γ2α.
Applying Ho¨lder’s inequality and Step 1, we obtain(33)J4≤∫0st−vα−1−s−vα−14dv1/2∫0s1+EXnv−1n2+EXnv−τ−1n22dv1/2≤∫0st−vα−1−s−vα−14dv1/2∫0s1+Esup0≤u1≤vXnu1−1n2+Esup0≤u2≤vXnu2−τ−1n22dv1/2≤∫0st−vα−1−s−vα−14dv12∫0s1+2β2dv1/2≤1+2βT1/2∫0st−vα−1−s−vα−14dv1/2=1+2βT1/2∫0st−v2α−2+s−v2α−22+4t−v2α−2s−v2α−2−4t−v3α−3s−vα−1−4s−v3α−3t−vα−1dv1/2≤1+2βT1/2∫0s2t−v4α−4+2s−v4α−4+4s−v4α−4−8t−v4α−4dv1/2=1+2βT1/2∫0s6s−v4α−4−6t−v4α−4dv1/2=1+2β6T1/2t−s4α−34α−3+s4α−34α−3−t4α−34α−31/2≤1+2β6T1/2t−s2α−1/24α−31/2=c4t−s2α−1/2,where c4=1+2β6T1/2/4α−31/2.
Using Ho¨lder’s inequality and Step 1 again,(34)J5≤∫stt−v4α−4dv1/2∫st1+2β2dv1/2=t−s1/21+2β∫stt−v4α−4dv1/2=t−s2α−11+2β4α−31/2=c5t−s2α−1,where c5=1+2β/4α−31/2.
Then,(35)EXnt−Xns2≤c3J4+J5≤c3c4t−s2α−3/2+c3c5t−s2α−1=r1t−s2α−3/2+r2t−s2α−1,where r1=c3c4 and r2=c3c5.Step 3.
We claim thatXnt,n≥1 is a Cauchy sequence. For integer m>n≥1, one can obtain(36)Xms−Xns=1Γα∫0ss−vα−1fv,Xmv−1m,Xmv−τ−1m−fv,Xnv−1n,Xnv−τ−1ndv+1Γα∫0ss−vα−1gv,Xmv−1m,Xmv−τ−1m−gv,Xnv−1n,Xnv−τ−1ndWv.
By Jensen’s inequality, Ho¨lder’s inequality, and Ito^ isometry, we can obtain(37)Esup0≤s≤tXms−Xns2≤2Γ2αE∫0tt−vα−1fv,Xmv−1m,Xmv−τ−1m−fv,Xnv−1n,Xnv−τ−1ndv2+2Γ2αE∫0tt−vα−1gv,Xmv−1m,Xmv−τ−1m−gv,Xnv−1n,Xnv−τ−1ndWv2≤2Γ2α∫0tt−v2α−2dv∫0tEfv,Xmv−1m,Xmv−τ−1m−fv,Xnv−1n,Xnv−τ−1n2dv+2Γ2α∫0tt−v2α−2Egv,Xmv−1m,Xmv−τ−1m−gv,Xnv−1n,Xnv−τ−1n2dv=J7+J8.
By applying Jensen’s inequality and Hypothesis2, we obtain(38)J7≤2T2α−12α−1Γ2α∫0tEfv,Xmv−1m,Xmv−τ−1m−fv,Xnv−1m,Xnv−τ−1m+fv,Xnv−1m,Xnv−τ−1m−fv,Xnv−1n,Xnv−τ−1n2dv≤2c6∫0tEfv,Xmv−1m,Xmv−τ−1m−fv,Xnv−1m,Xnv−τ−1m2dv,+2c6∫0tEfv,Xnv−1m,Xnv−τ−1m−fv,Xnv−1n,Xnv−τ−1n2dv≤2c6∫0tGv,EXmv−1m−Xnv−1m2,EXmv−τ−1m−Xnv−τ−1m2dv+2c6∫0tGv,EXnv−1m−Xnv−1n2,EXnv−τ−1m−Xnv−τ−1n2dv,where c6=2T2α−1/2α−1Γ2α.
Using Jensen’s inequality and Hypothesis2 again, we acquire(39)J8≤2Γ2α∫0tsup0≤v<tt−v2α−2Egv,Xmv−1m,Xmv−τ−1m−gv,Xnv−1m,Xnv−τ−1m+gv,Xnv−1m,Xnv−τ−1m−gv,Xnv−1n,Xnv−τ−1n2dv≤4sup0≤v<tt−v2α−2Γ2α∫0tEgv,Xmv−1m,Xmv−τ−1m−gv,Xnv−1m,Xnv−τ−1m2dv+4sup0≤v<tt−v2α−2Γ2α∫0tEgv,Xnv−1m,Xnv−τ−1m−gv,Xnv−1n,Xnv−τ−1n2dv≤4sup0≤v<tt−v2α−2Γ2α∫0tGv,EXmv−1m−Xnv−1m2,EXmv−τ−1m−Xnv−τ−1m2dv+4sup0≤v<tt−v2α−2Γ2α∫0tGv,EXnv−1m−Xnv−1n2,EXnv−τ−1m−Xnv−τ−1n2dv.
In terms of Step2, we can conclude that(40)Esup0≤s≤tXms−Xns2≤J7+J8≤2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,EXmv−1m−Xnv−1m2,EXmv−τ−1m−Xnv−τ−1m2dv+2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,EXnv−1m−Xnv−1n2,EXnv−τ−1m−Xnv−τ−1n2dv≤2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,Esup0≤u3≤vXmu3−Xnu32,Esup0≤u4≤vXmu4−Xnu42dv+2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,r11n−1m2α−3/2+r21n−1m2α−1,r11n−1m2α−3/2+r21n−1m2α−1dv=2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,Esup0≤u≤vXmu−Xnu2dv+2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,r11n−1m2α−3/2+r21n−1m2α−1dv.
Let(41)Yt=limm,n⟶∞Esup0≤s≤tXms−Xns2.
Then,(42)Yt≤2c6+4sup0≤v<tt−v2α−2Γ2α∫0tGv,Yvdv.
Thus, by Hypothesis2, we have(43)Yt=limm,n⟶∞Esup0≤s≤tXms−Xns2=0,indicating that Xnt,n≥1 is a Cauchy sequence. The Borel-Cantelli lemma makes clear, as n⟶∞,Xnt⟶Xt,t∈0,T holds uniformly. So, if we take the limit of both sides of (16), we get that Xt is a solution to (1), with the property(44)Esup0≤s≤tXs2<∞,0≤t≤T.
Now, we have proved the existence. The uniqueness of the solutions can be proved in the same way as Step3. When t∈−τ,0,Xt=Φt, obviously, there is a unique solution to FSDEs. The proof is complete.Remark 2.
IfGt,U,V=a¯U+V,a¯is a constant, then Hypothesis 2 and Hypothesis 3are equivalent to Hypothesis 1. Therefore, under Hypothesis 1and some proper conditions, there will exist a unique solutionXtto FSDEs (1).


## 4. Ulam–Hyers Stability Analysis of FSDEs

We are going to research the solutionXt,t∈0,T of system (1) is Ulam–Hyers stable and prove the stability theory of solutions to FSDEs (1) with Lipschitz and non-Lipschitz coefficients in this section.Theorem 2.
Assume that Hypothesis 1holds and12l2T2α−1/2<4α−31/2Γ2α,3/4<α<1. The FSDE (1) is Ulam–Hyers which is stable at0,T.Proof.
From Definition3 and Remark 1, we know(45)Zt=Φ0+1Γα∫0tt−vα−1fv,Zv,Zv−τdv+1Γα∫0tt−vα−1gv,Zv,Zv−τdWv+1Γα∫0tt−vα−1hvdv.
According to Definition3 and equation (45), we have(46)Zt−Xt=1Γα∫0tt−vα−1fv,v,Zv−τ−fv,Xv,Xv−τdv+1Γα∫0tt−vα−1gv,Zv,Zv−τ−gv,Zv,Zv−τdWv+1Γα∫0tt−vα−1hvdv,and then using Jensen’s inequality, we obtain(47)Esup0≤t≤TZt−Xt2≤3Γ2αEsup0≤t≤T∫0tt−vα−1fv,Zv,Zv−τ−fv,Xv,Xv−τdv2+3Γ2αEsup0≤t≤T∫0tt−vα−1gv,Zv,Zv−τ−gv,Xv,Xv−τdWv2+3Γ2αEsup0≤t≤T∫0tt−vα−1hvdv2=I1+I2+I3.
Now, we use Ho¨lder’s inequality and Hypothesis 1, and one can obtain(48)I1≤3Γ2αsup0≤t≤T∫0tt−v2α−2dv⋅E∫0Tfv,Zv,Zv−τ−fv,Xv,Xv−τ2dv≤3l2T2α−12α−1Γ2αE∫0TZv−Xv+Zv−τ+Xv−τ2dv≤6l2T2α−12α−1Γ2α∫0TEZv−Xv2+EZv−τ−Xv−τ2dv=b1T2α−1∫0TEZv−Xv2dv+b1T2α−1∫0TEZv−τ−Xv−τ2dv,where b1=6l2/2α−1Γ2α.
Then, by Ito^ isometry and Ho¨lder’s inequality, we obtain(49)I2≤3Γ2αE∫0TT−vα−1gv,Zv,Zv−τ−gv,Xv,Xv−τdWv2=3Γ2αE∫0TT−vα−1gv,Zv,Zv−τ−gv,Xv,Xv−τ2dv≤3Γ2α∫0TT−v4α−4dv1/2⋅E∫0Tgv,Zv,Zv−τ−gv,Xv,Xv−τ4dv1/2≤3T2α−3/24α−31/2Γ2αE∫0Tgv,Zv,Zv−τ−gv,Xv,Xv−τ4dv1/2=b2T2α−3/2E∫0Tgv,Zv,Zv−τ−gv,Xv,Xv−τ4dv1/2,where b2=3/4α−31/2Γ2α. Since gv,Zv,Zv−τ−gv,Xv,Xv−τ4 is a continuous function on 0,T, according to the mean value theorem of integrals, there exists y^∈0,T, such that(50)∫0Tgv,Zv,Zv−τ−gv,Xv,Xv−τ4dv=Tgy^,Zy^,Zy^−τ−gy^,Xy^,Xy^−τ4.
By Hypothesis1 and Jensen’s inequality, we obtain(51)I2≤b2T2α−1/2Egy^,Zy^,Zy^−τ−gy^,Xy^,Xy^−τ2≤2l2b2T2α−1/2EZy^−Xy^2+EZy^−τ−Xy^−τ2.
Finally, we use Cauchy–Schwarz inequality and Remark1 to yield(52)I3≤3Γ2αEsup0≤t≤T∫0tt−v2α−2dv⋅∫0thv2dv≤3T2α−12α−1Γ2α∫0TEsup0≤v≤Thv2dv≤3T2αε2α−1Γ2α=b3T2αε,where b3=3/2α−1Γ2α.
Hence, we obtain(53)Esup0≤t≤TZt−Xt2≤b1T2α−1∫0TEZv−Xv2dv+∫0TEZv−τ−Xv−τ2dv+2l2b2T2α−1/2EZy^−Xy^2+EZy^−τ−Xy^−τ2+b3T2αε≤b1T2α−1∫0TEsup0≤s1≤vZs1−Xs12dv+b1T2α−1∫0TEsup0≤s1≤vZs1−τ−Xs1−τ2dv+2l2b2T2α−1/2Esup0≤y^1≤y^Zy^1−Xy^12+2l2b2T2α−1/2Esup0≤y^1≤y^Zy^1−τ−Xy^1−τ2+b3T2αε.
Different from the approach of dealing with the delay in [18, 39, 40], we obtain(54)VT=Esup0≤t≤TZt−Xt2,Esup−τ≤t≤0Zt−Xt2=0,and then, we acquire(55)Esup0≤s1≤vZs1−τ−Xs1−τ2=Vv−τ.
Hence,(56)VT≤b1T2α−1∫0TVvdv+∫0TVv−τdv+2l2b2T2α−1/2Vy^+Vy^−τ+b3T2αε.
Let us setUT=supθ∈−τ,TVθ, then Vv≤Uv, and Vv−τ≤Uv. Thus,(57)VT≤2b1T2α−1∫0TUvdv+4l2b2T2α−1/2Uy^+b3T2αε.
For∀θ∈0,T, we obtain(58)Vθ≤2b1θ2α−1∫0θUvdv+4l2b2θ2α−1/2Uy^+b3θ2αε≤2b1T2α−1∫0TUvdv+4l2b2T2α−1/2Uy^+b3T2αε.
Then, we can obtain(59)UT=supθ∈−τ,TVθ≤maxsupθ∈−τ,0Vθ,supθ∈0,TVθ≤2b1T2α−1∫0TUvdv+4l2b2T2α−1/2UT+b3T2αε.
Then,(60)UT≤2b1T2α−11−4l2b2T2α−1/2∫0TUvdv+b3T2αε1−4l2b2T2α−1/2.
Using Gronwall’s inequality, we obtain(61)UT≤b3T2αε1−4l2b2T2α−1/2e2b1T2α/1−4l2b2T2α−1/2.
Therefore,(62)Esup0≤t≤TZt−Xt2≤b3T2αε1−4l2b2T2α−1/2e2b1T2α/1−4l2b2T2α−1/2,consequently, ∀ε>0; there exists δ=b3T2α/1−4l2b2T2α−1/2e2b1T2α/1−4l2b2T2α−1/2, such that(63)Esup0≤t≤TZt−Xt2≤εδ.
Therefore, this theorem is proved.Theorem 3.
Assume that Hypothesis 2 and Hypothesis 3 hold,6k˜T2α−1/2<4α−31/2Γ2α,k˜=maxsup0≤t≤Tbt,sup0≤t≤Tqt, and there exists a constant γsatisfying34α−31/2T2α+32α−1T2α−1/2/2α−14α−31/2Γ2α−62α−1k˜T2α−1/2sup0≤t≤Tat≤γε, 3/4<α<1. The FSDE (1) is Ulam–Hyers which is stable at 0,T.Proof.
From inequality (48), we obtain(64)Esup0≤t≤TZt−Xt2≤3Γ2αEsup0≤t≤T∫0tt−vα−1fv,Zv,Zv−τ−fv,Xv,Xv−τdv2+3Γ2αEsup0≤t≤T∫0tt−vα−1gv,Zv,Zv−τ−gv,Xv,Xv−τdWv2+3Γ2αEsup0≤t≤T∫0tt−vα−1hvdv2=I1+I2+I3.
Taking inequality (48) and Hypothesis 2 into account to achieve,(65)I1≤3T2α−12α−1Γ2αE∫0Tfv,Zv,Zv−τ−fv,Xv,Xv−τ2dv≤b4T2α−1∫0TEGv,Zv−Xv2,Zv−τ−Xv−τ2dv,where b4=3/2α−1Γ2α.
By inequalities (50)–(52) and Hypothesis 2, one can obtain(66)I2≤b2T2α−1/2Egy^,Zy^,Zy^−τ−gy^,Xy^,Xy^−τ2≤b2T2α−1/2EGy^,Zy^−Xy^2,Zy^−τ−Xy^−τ2.
Using Hypothesis3, it is immediate to obtain(67)I1+I2≤b4T2α−1∫0TEGv,Zv−Xv2,Zv−τ−Xv−τ2dv+b2T2α−1/2EGy^,Zy^−Xy^2,Zy^−τ−Xy^−τ2≤b4T2α−1∫0TEav+bvZv−Xv2+qvZv−τ−Xv−τ2dv+b2T2α−1/2Eay^+by^Zy^−Xy^2+qy^Zy^−τ−Xy^−τ2≤b4T2α+b2T2α−1/2sup0≤v≤Tav+b4k˜T2α−1∫0TEZv−Xv2+EZv−τ−Xv−τ2dv+b2k˜T2α−1/2EZy^−Xy^2+EZy^−τ−Xy^−τ2.
Now, through inequality (52), we have(68)I3≤b3T2αε.
Then,(69)Esup0≤t≤TZt−Xt2≤b4T2α+b2T2α−1/2sup0≤v≤Tav+b4k˜T2α−1∫0TEsup0≤s1≤vZs1−Xs12dv+b4k˜T2α−1∫0TEsup0≤s1≤vZs1−τ−Xs1−τ2dv+b2k˜T2α−1/2Esup0≤s2≤y^Zs2−Xs22+b2k˜T2α−1/2Esup0≤s2≤y^Zs2−τ−Xs2−τ2+b3T2αε.
Let us setVT=Esup0≤t≤TZt−Xt2 and Esup−τ≤t≤0supZt−Xt2=0, we can obtain(70)Esup0≤s1≤vZs1−τ−Xs1−τ2=Vv−τ.
Indeed, we can conclude that(71)VT≤b4k˜T2α−1∫0TVvdv+∫0TVv−τdv+b2k˜T2α−1/2Vy^+Vy^−τ+b4T2α+b2T2α−1/2sup0≤v≤Tav+b3T2αε.
LettingUT=supθ∈−τ,TVθ, then Vv≤Uv and Vv−τ≤Uv. Thus,(72)VT≤2b4k˜T2α−1∫0TU+2b2k˜T2α−1/2Uy^+b4T2α+b2T2α−1/2sup0≤v≤Tav+b3T2αε.
For∀θ∈0,T, we get that(73)Vθ≤2b4k˜θ2α−1∫0θUvdv+2b2k˜θ2α−1/2Uy^+b4θ2α+b2θ2α−1/2sup0≤v≤θav+b3θ2αε≤2b4k˜T2α−1∫0TUvdv+2b2k˜T2α−1/2UT+b4T2α+b2T2α−1/2sup0≤v≤Tav+b3T2αε.
Moreover, we have(74)UT=supθ∈−τ,TVθ≤maxsupθ∈−τ,0Vθ,supθ∈0,TVθ≤2b4k˜T2α−1∫0TUvdv+2b2k˜T2α−1/2UT+b4T2α+b2T2α−1/2sup0≤v≤Tav+b3T2αε.
Furthermore,(75)UT≤2b4k˜T2α−11−2b2k˜T2α−1/2∫0TUvdv+b4T2α+b2T2α−1/21−2b2k˜T2α−1/2sup0≤v≤Tav+b3T2αε1−2b2k˜T2α−1/2.
In view of Gronwall’s inequality, we get that(76)UT≤b4T2α+b2T2α−1/21−2b2k˜T2α−1/2sup0≤v≤Tav+b3T2αε1−2b2k˜T2α−1/2e2b4k˜T2α/1−2b2k˜T2α−1/2.
Therefore,(77)Esup0≤t≤TZt−Xt2≤b4T2α+b2T2α−1/21−2b2k˜T2α−1/2sup0≤v≤Tav+b3T2αε1−2b2k˜T2α−1/2e2b4k˜T2α/1−2b2k˜T2α−1/2≤γ+b3T2α1−2b2k˜T2α−1/2εe2b4k˜T2α+1/1−2b2k˜T2α−1/2,which implies that there exists δ=γ+b3T2α/1−2b2k˜T2α−1/2e2b4k˜T2α/1−2b2k˜T2α−1/2>0, ∀ε>0, satisfying(78)Esup0≤t≤TZt−Xt2≤εδ.
This completes the proof.


## 5. Examples

Example 1.
Consider the Ulam–Hyers stability and existence and uniqueness of the solution to the following equation:(79)D0+4/5CXt=115sinXt+223cosXt−τ+Γ1/224πcos2Xt+ln3/2πe4Xt−τdWtdt,where α=4/5, t∈0,8, and uniformly continuous functions(80)ft,Xt,Xt−τ=115sinXt+223cosXt−τ,gt,Xt,Xt−τ=Γ1/224πcos2Xt+ln3/2πe4Xt−τ.
Due to(81)ft,Zt,Zt−τ−ft,Xt,Xt−τ=115sinZt+223cosZt−τ−115sinXt−223cosXt−τ≤115sinZt−sinXt+223cosZt−τ−cosXt−τ=1152cosZt+Xt2sinZt−Xt2+223−2sinZt−τ+Xt−τ2sinZt−τ−Xt−τ2≤1152sinZt−Xt2+2232sinZt−τ−Xt−τ2≤115Zt−Xt+223Zt−τ−Xt−τ≤223Zt−Xt+Zt−τ−Xt−τ,gt,Zt,Zt−τ−gt,Xt,Xt−τ=Γ1/224πcos2Zt+ln3/2πe4Zt−τ−Γ1/224πcos2Xt−ln3/2πe4Xt−τ≤Γ1/224πcos2Zt−cos2Xt+ln3/2πe4Zt−τ−Xτ−τ=Γ1/224πcosZt+cosXtcosZt−cosXt+ln3/2πe4Zt−τ−Xt−τ≤Γ1/212πcosZt−cosXt+ln3/2πe4Zt−τ−Xt−τ≤223Zt−Xt+Zt−τ−Xt−τ,ft,Xt,Xt−τ and gt,Xt,Xt−τ satisfy Hypothesis 1, 33/2cT2α−1=33/2×2×2/232×53/5≈0.09<α−3/41/2Γ2α=1/20×Γ23/4≈0.30 and 12l2T2α−1/2=12×2/232×511/10≈0.54<4α−31/2Γ2α=1/5Γ24/5≈0.60. Therefore, according to Remark 2 and Theorem 2, we can see that there is a unique solution to equation (80) and the solution is Ulam–Hyers stable.
Now, a numerical simulation will be carried out to find the solution of (79) is Ulam–Hyers stable, and we can see it in Figure 1.Figure 1
X0=1,ht=ε,Esup0≤t≤8ht2=Esup0≤t≤8ε2=ε,ε=0.001.Example 2.
Consider the Ulam–Hyers stability of the following especial FSDEs with time delays:(82)D0+5/6CXt=1nXt+ε˜2Xt−τ+1n2Xt+ε˜24sin1Xt−τdWtdt,where α=5/6, t∈0,2,n≥10,ft,Xt,Xt−τ=1/nXt+ε˜Xt−τ, and gt,Xt,Xt−τ=1/n2Xt+ε˜2/4sin1/Xt−τ are measurable continuous functions, ε˜ is an arbitrary number, and 0<ε˜<2/8.
Because(83)ft,Zt,Zt−τ−ft,Xt,Xt−τ2+gt,Zt,Zt−τ−gt,Xt,Xt−τ2=1nZt+ε˜2Zt−τ−1nXt−ε˜2Xt−τ2+1n2Zt+ε˜24sin1Zt−τ−1n2Xt−ε˜24sin1Xt−τ2≤4n2Zt−Xt2+ε˜2Zt−τ−Xt−τ2+ε˜48sin1Zt−τ−sin1Xt−τ2=4n2Zt−Xt2+ε˜2Zt−τ−Xt−τ2+ε˜482cos1/Zt−τ+1/Xt−τ2sin1/Zt−τ−1/Xt−τ22≤ε˜42+4n2Zt−Xt2+ε˜2Zt−τ−Xt−τ2=Gt,Zt−Xt2,Zt−τ−Xt−τ2,obviously, Gt,Zt−Xt2,Zt−τ−Xt−τ2 is nondecreasing, continuous, and concave function and k˜=maxsup0≤t≤Tbt,sup0≤t≤Tqt=max4/n2,ε˜2=2/50,6k˜T2α−1/2≈0.54<4α−31/2Γ2α≈0.57. From the arbitrariness of ε˜, we know Gt,0,0=0. And, ∀ε>0, ∃γ=1/5ε>0, such that 34α−31/2T2α+32α−1T2α−1/2/2α−14α−31/2Γ2α−62α−1k˜T2α−1/2⋅sup0≤t≤Tat<0.20=1/5εε=γε. That satisfies all the conditions of Theorem 2. Therefore, we can conclude that system (82) is Ulam–Hyers stable on [0, 2].
Next, we will use a numerical simulation to verify the solution of (82) is Ulam–Hyers stable, and we can see it in Figure 2.Figure 2
X0=3,n=12,ε˜=2/9,ht=ε/2,Esup0≤t≤2ht2=Esup0≤t≤2ε/22=ε/4<ε, ε=0.007.


## 6. Conclusion

In this work, the objective is to research the existence and uniqueness of FSDEs with time delays using the novel Carathe´odory approximation and the weaker non-Lipschitz condition. Furthermore, different assumptions are used to prove the Ulam–Hyers stability of the solutions. Finally, we present two examples to test the validity of the proposed theory. Our future work will focus on exploring Ulam–Hyers stability of various types of fractional differential equations with weaker conditions, and the explored conditions can be applied to a wider range of differential equations.



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