Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
Special Issue

Fixed Point Theory and Applications for Function Spaces

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 5544847 | https://doi.org/10.1155/2021/5544847

Abdellatif Ben Makhlouf, Lassaad Mchiri, Mohamed Rhaima, "Ulam-Hyers-Rassias Stability of Stochastic Functional Differential Equations via Fixed Point Methods", Journal of Function Spaces, vol. 2021, Article ID 5544847, 7 pages, 2021. https://doi.org/10.1155/2021/5544847

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

Academic Editor: Zoran Mitrovic
Received08 Feb 2021
Revised28 Feb 2021
Accepted06 Mar 2021
Published21 Apr 2021

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 [15]. 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, 621]). 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 Schaefer’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, 2730]). 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 .

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.

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).

References

  1. X. Ma, X. B. Shu, and J. Mao, “Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay,” Stochastics and Dynamics, vol. 20, pp. 1–31, 2020. View at: Google Scholar
  2. X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, U. K, 1997.
  3. X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, 1994.
  4. X. Mao, “Robustness of exponential stability of stochastic differential delay equations,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 442–447, 1996. View at: Publisher Site | Google Scholar
  5. P. H. A. Ngoc, “A new approach to mean square exponential stability of stochastic functional differential equations,” IEEE Control Systems Letters, vol. 5, no. 5, pp. 1645–1650, 2021. View at: Publisher Site | Google Scholar
  6. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941. View at: Publisher Site | Google Scholar
  7. Y. Başcı, A. Mısır, and S. Öğrekçi, “On the stability problem of differential equations in the sense of Ulam,” Results in Mathematics, vol. 75, no. 1, article 6, 2020. View at: Publisher Site | Google Scholar
  8. S. Boulares, A. Ben Makhlouf, and H. Khellaf, “Generalized weakly singular integral inequalities with applications to fractional differential equations with respect to another function,” The Rocky Mountain Journal of Mathematics, vol. 50, no. 6, pp. 2001–2010, 2020. View at: Publisher Site | Google Scholar
  9. M. Choubin and H. Javanshiri, “A new approach to the Hyers-Ulam-Rassias stability of differential equations,” Results in Mathematics, vol. 76, no. 1, article 11, 2021. View at: Publisher Site | Google Scholar
  10. Y. Gao and Y. Li, “Initial value problems of semilinear supdiffusion equations,” Mathematics, vol. 9, no. 1, pp. 1–10, 2021. View at: Google Scholar
  11. P. Gavruta and L. Gavruta, “A new method for the generalized Hyers-Ulam-Rassias stability,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 11–18, 2010. View at: Google Scholar
  12. Y. Guo, X. B. Shu, Y. Li, and F. Xu, “The existence and Hyers-Ulam stability of solution for an impulsive Riemann Liouville fractional neutral functional stochastic differential equation with infinite delay of order ,” Boundary Value Problems, vol. 2019, no. 1, Article ID 59, 2019. View at: Google Scholar
  13. Y. Guo, M. Chen, X. B. Shu, and F. Xu, “The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm,” Stochastic Analysis and Applications, pp. 1–24, 2020. View at: Publisher Site | Google Scholar
  14. D. H. Hyers, G. Isac, and T. H. M. Rassias, Stability of functional equation in several variables, Birkhäuser, Basel, 1998.
  15. T. Li and A. Zada, “Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces,” Advances in Difference Equations, vol. 2016, no. 1, Article ID 153, 2016. View at: Publisher Site | Google Scholar
  16. L. Liu, Q. Dong, and G. Li, “Exact solutions and Hyers-Ulam stability for fractional oscillation equations with pure delay,” Applied Mathematics Letters, vol. 112, p. 106666, 2021. View at: Publisher Site | Google Scholar
  17. O. Saifia, D. Boucenna, and A. Chidouh, “Study of Mainardi’s fractional heat problem,” Journal of Computational and Applied Mathematics, vol. 378, p. 112943, 2020. View at: Publisher Site | Google Scholar
  18. A. Taieb and Z. Dahmani, “Triangular system of higher order singular fractional differential equations,” Kragujevac Journal of Mathematics, vol. 45, no. 1, pp. 81–101, 2021. View at: Publisher Site | Google Scholar
  19. L. Tan, W. Jin, and Y. Suo, “Stability in distribution of neutral stochastic functional differential equations,” Statistics & Probability Letters, vol. 107, pp. 27–36, 2015. View at: Publisher Site | Google Scholar
  20. C. Tunç and E. Biçer, “Hyers-Ulam-Rassias stability for a first order functional differential equation,” Journal of Mathematical and Fundamental Sciences, vol. 47, no. 2, pp. 143–153, 2015. View at: Publisher Site | Google Scholar
  21. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ, New York, 1960.
  22. T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978. View at: Publisher Site | Google Scholar
  23. T. M. Rassias, “On a modified Hyers-Ulam sequence,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 106–113, 1991. View at: Publisher Site | Google Scholar
  24. P. Gavruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994. View at: Publisher Site | Google Scholar
  25. M. Benchohra, S. Bouriah, and J. Henderson, “Ulam stability for nonlocal differential equations involving the Hilfer-Katugampola fractional derivative,” Afrika Matematika, 2021. View at: Publisher Site | Google Scholar
  26. A. Boutiara, S. Etemad, A. Hussain, and S. Rezapour, “The generalized U-H and U-H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving -Caputo fractional operators,” Adv. Difference Equ., vol. 2021, no. 1, article 95, 2021. View at: Publisher Site | Google Scholar
  27. A. Ahmadova and N. I. Mahmudov, “Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations,” Statistics & Probability Letters, vol. 168, article 108949, 2021. View at: Publisher Site | Google Scholar
  28. S. Li, L. Shu, X. B. Shu, and F. Xu, “Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays,” Stochastics, vol. 90, no. 5, pp. 663–681, 2018. View at: Google Scholar
  29. N. P. N. Ngoc, “Ulam-Hyers-Rassias stability of a nonlinear stochastic integral equation of Volterra type,” Differential Equations & Applications, vol. 9, no. 2, pp. 183–193, 2009. View at: Publisher Site | Google Scholar
  30. X. Zhao, “Mean square Hyers-Ulam stability of stochastic differential equations driven by Brownian motion,” Advances in Difference Equations, vol. 2016, no. 1, Article ID 271, 2016. View at: Publisher Site | Google Scholar

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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views446
Downloads408
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.