Journal of Function Spaces
Volume 2021 (2021), Article ID 5544847, 7 pages
https://doi.org/10.1155/2021/5544847

Research Article

Ulam-Hyers-Rassias Stability of Stochastic Functional Differential Equations via Fixed Point Methods

Abdellatif Ben Makhlouf,1,2 Lassaad Mchiri,³ and Mohamed Rhaima3,4

¹Department of Mathematics, College of Science, Jouf University, Saudi Arabia
²Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia
³Department of Statistics and Operations Research, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
⁴University of Tunis Manar, Faculty of Sciences of Tunis, Department of Mathematics, Tunisia

Correspondence should be addressed to Abdellatif Ben Makhlouf; benmakhloufabdellatif@gmail.com

Received 8 February 2021; Revised 28 February 2021; Accepted 6 March 2021; Published 21 April 2021

Academic Editor: Zoran Mitrovic

Copyright © 2021 Abdellatif Ben Makhlouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The Ulam-Hyers-Rassias stability for stochastic systems has been studied by many researchers using the Gronwall-type inequalities, but there is no research paper on the Ulam-Hyers-Rassias stability of stochastic functional differential equations via fixed point methods. The main goal of this paper is to investigate the Ulam-Hyers Stability (HUS) and Ulam-Hyers-Rassias Stability (HURS) of stochastic functional differential equations (SFDEs). Under the fixed point methods and the stochastic analysis techniques, the stability results for SFDE are investigated. We analyze two illustrative examples to show the validity of the results.

1. Introduction

In recent years, SFDEs play an important role in different areas such as physics, mechanics, population dynamics, ecology, medicine biology, and other areas of sciences. SFDEs have great applications and have been developed very fast, see for example [1–5]. Stability investigation is conducted for stochastic nonlinear differential equations with constant delay. The Lyapunov method is used for stability investigation of different mathematical models such as predator-prey relationships and inverted controlled pendulum.

The HUS problem of functional systems began from a question of S. Ulam, queried in 1940, about the stability of functional differential equations for homomorphism as follows. The question regarding the stability problem of homomorphisms is as follows:

Denote by the group, and the metric group with a metric and a constant . The question is to study if there exists satisfies for every such that

there exists a homomorphism satisfies

In 1941, Hyers [6] presented a partial solution to the question of S. Ulam assuming that , be two Banach spaces in the case of -linear transformations, that is

Let , be two Banach spaces and set be a linear transformation satisfying

There exists a unique linear transformation such that the limit exists for each and for all , which was the first step towards more answers in this area. Many researchers have analyzed the HUS of various classes of differential systems (see, for instance, [1, 6–21]). In 1978, Rassias [22] provided a generalized answer to the Ulam question for approximate -linear transformations. In [23], Rassias obtained an extension of the Hyers’s answer.

In 1994, Gavruta [24] gave a generalization form of Rassias’s Theorem for the unbounded Cauchy difference and introduced the notion of generalized HURS in the sense of Rassias approach.

In the last decades, there is an increasing interest and work on the Ulam stability and the Ulam-Hyers stability of some deterministic systems using the Banach contraction principle and Schaeferâ€™s fixed point theorem (see [25, 26]).

In the literature, there are a few papers about the HUS and the HURS of stochastic systems (see [13, 27–30]). The stability of SFDEs has attracted much more attention (see [2, 19] etc.). Consequently, it is interesting to extend the research results on the deterministic functional systems to the stochastic case.

Let us outline the framework of this paper. After some basic notions and assumptions (see Section 2), in Section 3, the HUS and HURS of the solution of the system are proved by using the fixed point methodology. In the last section, two numerical examples are presented to illustrate the main results.

2. Preliminary

Denote by the complete probability space where is a filtration satisfying the usual conditions. is an -dimensional Brownian motion defined on the probability space. Denote by the set of -valued -adapted processes such that a.s. and the set of processes in satisfies . Let denote the set of functions from to that are right-continuous and have limits on the left. is equipped with the norm and for any . Denote by the set of all -measurable bounded -valued random variables . Let , , denote the set of all -measurable, -valued random variables satisfies .

Consider the following SFDE for fixed:

with the initial condition

and recall that, given , for each , we denote by the function in defined as . We assume that

Using the definition of Itô’s stochastic differential and integrating the two sides of equation (4) from to , we have

We consider the following assumption:

: (Uniform Lipschitz condition): Suppose that there is a constant satisfies

and , , where define the maximum of and .

3. Stability Results

In this part, we discuss the HUS and the HURS of equation (4) under the assumption .

Definition 1. Equation (4) is called HUS with respect to (w.r.t) if there exists a constant such that for each and for each solution , with , of the following inequation:

there exists a solution of (4), with , such that ,

Definition 2. Equation (4) is called HURS w.r.t , with , if there exists a constant such that for each and for each solution
, with , satisfying

there exists a solution of (4), with , such that , .

Definition 3. Equation (4) is generalized HURS w.r.t , with , if there exists a constant such that for each solution.
, with , satisfying

there exists a solution of (4), with , such that , .

Lemma 4 (see [2]). Set . Let be the function such that

where , . Then, is a complete metric space.

Theorem 5 (see [29]). Suppose is a complete metric space and is a contraction (with). Suppose that , and . So, there exists a unique satisfies . Moreover,

Theorem 6. Suppose that hold. Let , with , be a stochastic process satisfies

where and is a nondecreasing function. Then, there is a solution of (4), with , such that ,

where and is any positive constant.

Proof. Consider such that

with for and for , where , and for and for .
Let the operator such that , for , and

for .
It is easy to prove that is well defined.
Let , , for , we get .
For , we have

Taking the expectation on both sides and using assumption , we have

Then,

For , we have where . Then,

Therefore,

Then, . Thus, . Therefore, is strictly contractive for .
For , we get .
From (15), we get

for all .
Then, . Therefore, . Using the fixed point theorem, there is a solution of (4) such that . Then,

Therefore,

as desired.

Remark 7. In our analysis of the HURS, we do not suppose any condition on unlike the case of the Theorem 6 in [29].

Theorem 8. Assume that hold. Let , with , be a stochastic process satisfies

where . Then, there is a solution of (4), with , such that ,

where and is a positive constant.

Proof. The proof of this theorem is similar to Theorem 6.

Theorem 9. Assume that hold. Let , with , be a stochastic process satisfies

where is a nondecreasing function. Then, there is a solution of (4), with , such that ,

where and .

Proof. The proof of this theorem is similar to Theorem 6.

4. Examples

Two examples are studied to show the interest of the main results.

Example 10. Consider the following SFDE for each

where

with . Then, replacing now by the segment of a solution , we get

Let , then

Hence, the uniform Lipschitz condition is satisfied.
Therefore, by Theorem 6, there is a solution of (31), with , such that ,

where and .

For System (31), we conduct a simulation based on the Euler-Maruyama scheme with step size , for which we set , and the initial, data as a map, namely, for all . In Figure 1, we give a sequence of computer simulations of the exact solution path and the rough solution path for System (31) on the interval . Choosing , , , and one obtain . We use the time step of the interval and realizations for this discretisation; we give in Figure 2 the trajectory of and simulation of the mean square of on the interval . It is clear that the convergence plot verifies the theoretical findings.

Example 11. Consider the following SFDE for each

where

with . Then, replacing now by the segment of a solution , we have

Let , then

Hence, the uniform Lipschitz condition is satisfied.
Therefore, by Theorem 8, there is a solution of (36), with , such that ,

where and .

Figure 1: Simulation of and are trajectory in System (31) with and for .

Figure 2: HURS with respect to of on the interval with .

We use again Euler-Maruyama scheme with step size to conduct a simulation for System (36). We fix and the initial data as a linear mapping, namely, for all . In Figure 3, we plot the path of the exact solution and the rough solution path for System (36) on the interval . Choosing , , , and one obtain a large value of . We use the time step of the interval and realizations for this discretisation; we give in Figure 4 the trajectory of the constant function and simulation of the mean square of on the interval . It is clear that the convergence plot verifies the theoretical findings.

Figure 3: Simulation of and are trajectory in System (36) with and for .

Figure 4: HUS with respect to of on the interval with .

5. Conclusion

In this paper, we investigate the Ulam-Hyers-Rassias stability of stochastic functional differential equations. To obtain the main results, we used the fixed point theorem and the classical stochastic calculus techniques. Moreover, we extend the Ulam-Hyers-Rassias stability for a generalization version. An example is presented to show the applicability of our results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through Research Group No (RG-1441-328).

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