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 Carathodory approximation. Furthermore, with the help of Hlder’s inequality, Jensen’s inequality, It 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.

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], was unstable when , but it increased the stochastic feedback control to become . Apparently, was stable if and only if . This fact indicated that the stochastic control can stabilize the unstable system . 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 -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:where , , and are measurable continuous functions, is an -dimensional Brownian motion on a complete probability space , is a continuous function, , and 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 Mnch’s fixed point theorem, respectively, were used. However, in this paper, we adopt the Carathodory 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 Section 2. In Section 3, by Carathodory 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.

Let be a finite interval, and we define the norm of on as follows:

Definition 1 (see [35]). The Riemann–Liouville integral operator of fractional-order is defined bywhere and is the well-known Eulers Gamma function.

Definition 2 (see [1]). For any continuous function, the Caputo derivative of fractional-orderis defined bywhere , .
In particular, for ,

Definition 3. An -value stochastic process is called a solution to equation (1) if it satisfies the following conditions:(1) is -continuous and adapted.(2) and .(3)For ,where .(4)For any other solution , we obtain .

Definition 4 (see [36]). System (1) is Ulam–Hyers stable if there exists a real number such that and for each continuously differentiable function satisfyingand there exists a solution of (1) satisfying

Remark 1 (see [21]). A function is a solution of equation (7) if and only if there exists a function , such that(i)(ii)

Hypothesis 1 (Lipschitz condition). As for any , there is a constant such that, for all ,where are uniformly continuous functions and is defined as .

Hypothesis 2 (non-Lipschitz condition). There is a function , , such that(1)For all and , where are continuous as well as bounded functions, and for any fixed , is monotone, nondecreasing, continuous, and concave function with .(2)For every and any nonnegative function Y(t) such thatwhere is a constant and , we get .

Hypothesis 3. There exist three functions , such that

Lemma 1 (see [37]). Suppose Hypothesis 2 and Hypothesis 3 are fulfilled. Then, there exists constant such that, for any ,

Proof. Applying Jensen’s inequality and Hypothesis 2 and Hypothesis 3, we havewhere . In a similar way, we obtainLet us set . The proof is therefore complete.

3. Existence and Uniqueness

Utilizing Carathodory approximation [35, 38], the existence and uniqueness of solutions to SFDEs can be obtained. So, let us define the Carathodory approximation as follows. For any integer defineand for all .

Theorem 1. Suppose that Hypothesis 2 and Hypothesis 3hold and,; then, system (1) has a unique solution.

Proof. The proof will be divided into three steps, when .

Step 1. The boundedness of the sequence .
By (16) and Jensen’s inequality, we haveAccording to It isometry, Cauchy–Schwarz inequality, and Lemma 1, it is easy to obtainLetting . It is obvious that , andwhere .
Using Hlders inequality and Jensen’s inequality, we obtainwhere is continuous function on . By the mean value theorem of integrals, there exists such thatUsing the inequalitywe deriveLet us set , and we havewhere .
If , we obtainIf , we obtainand then,By Gronwall’s inequality, we can conclude thatLetting , we obtainwhere is a positive constant. So, we have proved that the sequence is bounded.