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

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 H€ older’s inequality, Jensen’s inequality, It􏽢o 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.


Introduction
e fractional-order differential equations can better simulate many natural physical processes than integerorder 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][2][3][4][5]. For more notable achievements of this concept, the readers can also refer to [6][7][8][9][10][11][12][13][14][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], du(t) � ku(t)dt was unstable when k > 0, but it increased the stochastic feedback control ru(t)dW(t) to become du(t) � ku(t)dt + ru(t)dW(t). Apparently, du(t) � ku(t)dt + ru(t)dW(t) was stable if and only if r 2 > 2k. is fact indicated that the stochastic control ru(t)dW(t) can stabilize the unstable system du(t) � ku(t) dt. erefore, it is significant and challenging to study stochastic stabilization of deterministic systems. More relevant results can be found in [17][18][19]. e 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][23][24][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. e 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][31][32][33][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: 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 M€ onch'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. is is a breakthrough in the exploration of the stability to FSDEs. e structure of this article is arranged as follows. We present some basic definitions and necessary assumptions in Section 2. 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.

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.
In particular, for α ∈ (0, 1), Definition 3. An R d -value stochastic process X(t) { } − τ≤t≤T is called a solution to equation (1) if it satisfies the following conditions: Mathematical Problems in Engineering 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 and there exists a solution Remark 1 (see [21] Hypothesis 1 (Lipschitz condition). As for any f, g ∈ R d , there is a constant l > 0 such that, for all X 1 , X 2 , Y 1 , where f and g are uniformly continuous functions and ∨ is defined as Hypothesis 2 (non-Lipschitz condition). ere is a function where f and g are continuous as well as bounded functions, and for any fixed t ≥ 0, G(t, U, V) is monotone, nondecreasing, continuous, and concave function with G(t, 0, 0) � 0. (2) For every t ∈ R + and any nonnegative function such that where m > 0 is a constant and

Hypothesis 3.
ere exist three functions a(t), b(t), and q(t), such that Lemma 1 (see [37]). Suppose Hypothesis 2 and Hypothesis 3 are fulfilled. en, there exists constant c > 0 such that, for any Proof. Applying Jensen's inequality and Hypothesis 2 and Hypothesis 3, we have Mathematical Problems in Engineering where In a similar way, we obtain Let us set c � max(k 1 , k 2 ). e proof is therefore complete.

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 and

Theorem 1. Suppose that Hypothesis 2 and Hypothesis 3 hold and 3
Proof. e proof will be divided into three steps, when t ∈ [0, T].

Using H€ olders inequality and Jensen's inequality, we obtain
Using the inequality ���������� we derive where and then, By Gronwall's inequality, we can conclude that where β is a positive constant. So, we have proved that the sequence X n (t), n ≥ 1 is bounded.
Applying H€ older's inequality and Step 1, we obtain 8 Mathematical Problems in Engineering where Using H€ older's inequality and Step 1 again, where where r 1 � c 3 c 4 and r 2 � c 3 c 5 .
Using Jensen's inequality and Hypothesis 2 again, we acquire In terms of Step 2, we can conclude that Let en, us, by Hypothesis 2, we have indicating that X n (t), n ≥ 1 is a Cauchy sequence. e Borel-Cantelli lemma makes clear, as n ⟶ ∞, X n (t) ⟶ X (t), t ∈ [0, T] holds uniformly. So, if we take the limit of both sides of (16), we get that X(t) is a solution to (1), with the property Mathematical Problems in Engineering 13 Now, we have proved the existence. e uniqueness of the solutions can be proved in the same way as Step 3. When t ∈ [− τ, 0], X(t) � Φ(t), obviously, there is a unique solution to FSDEs. e proof is complete. G(t, U, V) � a(U + V), a is a constant, then Hypothesis 2 and Hypothesis 3 are equivalent to Hypothesis 1. erefore, under Hypothesis 1 and some proper conditions, there will exist a unique solution X(t) to FSDEs (1).

Ulam-Hyers Stability Analysis of FSDEs
We are going to research the solution X(t), 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.

FSDE (1) is Ulam-Hyers which is stable at [0, T].
Proof. From Definition 3 and Remark 1, we know (45) According to Definition 3 and equation (45), we have and then using Jensen's inequality, we obtain Now, we use H€ older's inequality and Hypothesis 1, and one can obtain en, by Ito isometry and H€ older's inequality, we obtain

is a continuous function on
[0, T], according to the mean value theorem of integrals, there exists y ∈ [0, T], such that
Hence, we obtain Different from the approach of dealing with the delay in [18,39,40], we obtain and then, we acquire E sup Hence, Let us set U(T) � sup
Next, we will use a numerical simulation to verify the solution of (82) is Ulam-Hyers stable, and we can see it in

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.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.