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

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.


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][2][3][4][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 H 1 the group, and H 2 the metric group with a metricδ and a constant ϑ > 0. The question is to study if there exists λ > 0 satisfies for every h : there exists a homomorphism f : In 1941, Hyers [6] presented a partial solution to the question of S. Ulam assuming that D 1 , D 2 be two Banach spaces in the case of λ-linear transformations, that is Let D 1 , D 2 be two Banach spaces and set h : D 1 ⟶ D 2 be a linear transformation satisfying There exists a unique linear transformation Δ : D 1 ⟶ D 2 such that the limit ΔðσÞ = lim n⟶+∞ hð2 n σÞ/2 n exists for each σ ∈ D 1 and khðσÞ − ΔðσÞk ≤ λ for all σ ∈ D 1 , 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][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In 1978, Rassias [22] provided a generalized answer to the Ulam question for approximate λ-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 hðσ + υÞ − hðσÞ − hðυÞ 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][28][29][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.

Stability Results
In this part, we discuss the HUS and the HURS of equation (4) under the assumption A 1 . (4) is called HUS with respect to (w.r.t) ε if there exists a constant M > 0 such that for each ε > 0 and for each solutionζ ∈ M 2 ð½ω 0 − ϱ, T, ℝ b Þ, withζ ω 0 = χ, of the following inequation:
Remark 7. In our analysis of the HURS, we do not suppose any condition on K unlike the case of the Theorem 6 in [29].
Proof. The proof of this theorem is similar to Theorem 6.
Proof. The proof of this theorem is similar to Theorem 6.

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