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

Authors: Abdellatif Ben Makhlouf; Lassaad Mchiri; Mohamed Rhaima
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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 byH1 the group, and H2 the metric group with a metric δ~ and a constant ϑ>0. The question is to study if there exists λ>0 satisfies for every h:H1⟶H2 such that
(1)δ~hσυ,hσhυ≤λ,∀σ,υ∈H1,there exists a homomorphism f:H1⟶H2 satisfies
(2)δ~hσ,fσ≤ϑ,∀σ∈H1.In 1941, Hyers [6] presented a partial solution to the question of S. Ulam assuming that D1, D2 be two Banach spaces in the case of λ-linear transformations, that isLetD1, D2 be two Banach spaces and set h:D1⟶D2 be a linear transformation satisfying
(3)hσ+υ−hσ−hυ≤λ,∀σ,υ∈D1λ>0.There exists a unique linear transformationΔ:D1⟶D2 such that the limit Δσ=limn⟶+∞h2nσ/2n exists for each σ∈D1 and hσ−Δσ≤λ for all σ∈D1, 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 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 hσ+υ−hσ−hυ 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 Section2), 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 byΩ,F,Fσσ≥0,P the complete probability space where Fσσ≥0 is a filtration satisfying the usual conditions. Wσ is an m-dimensional Brownian motion defined on the probability space. Denote by L2v,w,ℝb the set of ℝb-valued Fσ-adapted processes ψσv≤σ≤w such that ∫vwψσ2dσ<∞ a.s. and M2v,w,ℝbthe set of processes ψσv≤σ≤w in L2v,w,ℝb satisfies E∫vwψσ2dσ<∞. Let C−ϱ,0;ℝb denote the set of functions φ from −ϱ,0 to ℝb that are right-continuous and have limits on the left. C−ϱ,0;ℝb is equipped with the norm φ=sup−ϱ≤s≤0φs and x=xTx for any x∈ℝb. Denote by CF0b~−ϱ,0;ℝb the set of all F0-measurable bounded C−ϱ,0;ℝb-valued random variables χ=χκ:−ϱ≤κ≤0. Let LFt2−ϱ,0;ℝb, t≥0, denote the set of all Ft-measurable, C−ϱ,0;ℝb-valued random variables ς=ςκ:−ϱ≤κ≤0 satisfies sup−ϱ≤κ≤0Eςκ2<∞.Consider the following SFDE for0≤ω0<T fixed:
(4)dζω=fω,ζωdω+gω,ζωdWω,ω0≤ω≤T,with the initial condition
(5)ζω0=χ∈LFω02−ϱ,0;ℝb,and recall that, given ζ∈Cω0,T;ℝb, for each ω∈ω0,T, we denote by ζω· the function in Cω0−ϱ,0;ℝb defined as ζωp=ζω+p,−ϱ≤p≤0. We assume that
(6)f:ω0,T×C−ϱ,0;ℝb⟶ℝb,g:ω0,T×C−ϱ,0;ℝb⟶ℝb×q.Using the definition of Itô’s stochastic differential and integrating the two sides of equation (4) from ω0 to ω, we have
(7)ζω=ζω0+∫ω0ωfν,ζνdν+∫ω0ωgν,ζνdWν,ω0≤ω≤T.We consider the following assumption:A1: (Uniform Lipschitz condition): Suppose that there is a constant K>0 satisfies
(8)fω,υ1−fω,υ22∨gω,υ1−gω,υ22≤K2υ1−υ22,∀ω∈ω0,T and υ1, υ2∈C−ϱ,0;ℝb, where z1∨z2 define the maximum of z1 and z2.


## 3. Stability Results

In this part, we discuss the HUS and the HURS of equation (4) under the assumption A1.Definition 1.
Equation (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 solution ζ~∈M2ω0−ϱ,T,ℝb, with ζ~ω0=χ, of the following inequation:
(9)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤ε,∀ω∈ω0,T,there exists a solution ζ∈M2ω0−ϱ,T,ℝb of (4), with ζω0=χ, such that Eζ~ω−ζω2≤Mε,
(10)∀ω∈ω0,T.Definition 2.
Equation (4) is called HURS w.r.t ε,θω, with θ·∈Cω0,T;ℝ+, if there exists a constant M>0 such that for each ε>0 and for each solution
ζ~∈M2ω0−ϱ,T,ℝb, with ζ~ω0=χ, satisfying
(11)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤εθω,∀ω∈ω0,T,there exists a solution ζω∈M2ω0−ϱ,T,ℝb of (4), with ζω0=χ, such that ∀ω∈ω0,T, Eζ~ω−ζω2≤εMθω.Definition 3.
Equation (4) is generalized HURS w.r.t θω, with θ·∈Cω0−ϱ,T;ℝ+, if there exists a constant M>0 such that for each solution.
ζ~∈M2ω0−ϱ,T,ℝb, with ζ~ω0=χ, satisfying
(12)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤εθω,∀ω∈ω0,T,there exists a solution ζω∈M2ω0−ϱ,T,ℝb of (4), with ζω0=χ, such that ∀ω∈ω0,T, Eζ~ω−ζω2≤Mθω.Lemma 4 (see [2]).
SetM=M2ω0−ϱ,T,ℝb. Let d:M×M⟶ℝ+ be the function such that
(13)d2ζ1,ζ2=infΛ∈0,+∞,Eζ1ω−ζ2ω2h1ω≤Λh2ω,∀ω∈ω0−ϱ,T,where h1, h2∈Cω0−ϱ,T,ℝ+∗. Then, M,d is a complete metric space.Theorem 5 (see [29]).
SupposeF,d is a complete metric space and L:F⟶F is a contraction (withτ∈0,1). Suppose that v∈F, λ>0 and dv,Lv≤λ. So, there exists a unique β∈F satisfies β=Lβ. Moreover,
(14)dv,β≤λ1−τ.Theorem 6.
Suppose thatA1 hold. Let ζ~∈M2ω0−ρ,T,ℝb, with ζ~ω0=χ, be a stochastic process satisfies
(15)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤εθω,∀ω∈ω0,T,where ε>0 and θ·∈Cω0,T;ℝ+∗ is a nondecreasing function. Then, there is a solution ζ∈M2ω0−ρ,T,ℝb of (4), with ζω0=χ, such that ∀ω∈ω0,T,
(16)Eζ~ω−ζω2≤11−α/α+δ2expα+δT−ω0εθω,where α=2K2T−ω0+1 and δ is any positive constant.Proof.
Considerd:M×M⟶ℝ+ such that
(17)d2ζ1,ζ2=infΛ∈0,+∞,Eζ1ω−ζ2ω2hω≤Λθ~ω,∀ω∈ω0−ρ,T,with hω=eγω−ω0 for ω∈ω0,T and hω=1 for ω∈ω0−ϱ,ω0, where γ=α+δ, and θ~ω=θω for ω∈ω0,T and θ~ω=θω0 for ω∈ω0−ρ,ω0.
Let the operatorR:M⟶M such that Rζω=ζ~ω, for ω∈ω0−ϱ,ω0, and
(18)Rζω=ζ~ω0+∫ω0ωfν,ζνdν+∫ω0ωgν,ζνdWν,for ω∈ω0,T.
It is easy to prove thatR is well defined.
Letζ1, ζ2∈M, for ω∈ω0−ϱ,ω0, we get Rζ1ω−Rζ2ω=0.
Forω∈ω0,T, we have
(19)Rζ1ω−Rζ2ω2≤2∫ω0ωfν,ζ1ν−fs,ζ2νdν2+2∫ω0ωgν,ζ1ν−gν,ζ2νdWν2.
Taking the expectation on both sides and using assumptionA1, we have
(20)ERζ1ω−Rζ2ω2≤2K2T−ω0∫ω0ωEζ1ν−ζ2ν2dν+∫ω0ωEζ1ν−ζ2ν2dν.
Then,(21)ERζ1ω−Rζ2ω2≤α∫ω0ωEζ1ν−ζ2ν2dν.
Forν∈ω0,ω, we have Eζ1ν−ζ2ν2=Eζ1ν+σ−ζ2ν+σ2 where σ∈−ϱ,0. Then,
(22)Eζ1ν−ζ2ν2=Eζ1ν+σ−ζ2ν+σ2hν+σθ~ν+σhν+σθ~ν+σ≤d2ζ1,ζ2hνθ~ν.
Therefore,(23)ERζ1ω−Rζ2ω2≤αd2ζ1,ζ2θω∫ω0ωhνdν≤αγd2ζ1,ζ2θωhω.
Then,d2Rζ1,Rζ2≤α/γd2ζ1,ζ2. Thus, dRζ1,Rζ2≤α/γdζ1,ζ2. Therefore, R is strictly contractive for c=α/α+δ.
Forω∈ω0−ϱ,ω0, we get Rζ~ω−ζ~ω=0.
From (15), we get
(24)Eζ~ω−Rζ~ω2θωhω=Eζ~ω−ζ~ω0−∫ω0ωfν,ζνdν−∫ω0ωgν,ζνdWν2θωhω≤ε,for all ω∈ω0,T.
Then,d2Rζ~,ζ~≤ε. Therefore, dRζ~,ζ~≤ε. Using the fixed point theorem, there is a solution ζ∗ of (4) such that dζ~,ζ∗≤1/1−cε. Then,
(25)Eζ~ω−ζ∗ω2θωhω≤11−c2ε,∀ω∈ω0,T.
Therefore,(26)Eζ~ω−ζ∗ω2≤eα+δT−ω01−c2εθω,∀ω∈ω0,T,as desired.Remark 7.
In our analysis of the HURS, we do not suppose any condition onK unlike the case of the Theorem 6 in [29].Theorem 8.
Assume thatA1 hold. Let ζ~∈M2ω0−ϱ,T,ℝb, with ζ~ω0=χ, be a stochastic process satisfies
(27)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤ε,∀ω∈ω0,T,where ε>0. Then, there is a solution ζ∈M2ω0−ρ,T,ℝb of (4), with ζω0=χ, such that ∀ω∈ω0,T,
(28)Eζ~ω−ζω2≤11−α/α+δ2expα+δT−ω0ε,where α=2K2T−ω0+1 and δ is a positive constant.Proof.
The proof of this theorem is similar to Theorem6.Theorem 9.
Assume thatA1 hold. Let ζ~∈M2ω0−ϱ,T,ℝb, with ζ~ω0=χ, be a stochastic process satisfies
(29)Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤θω,∀ω∈ω0,T,
whereθ·∈Cω0,T;ℝ+∗ is a nondecreasing function. Then, there is a solution ζ∈M2ω0−ρ,T,ℝb of (4), with ζt0=χ, such that ∀ω∈ω0,T,
(30)Eζ~ω−ζω2≤11−α/α+δ2expα+δT−ω0θω,where α=2K2T−ω0+1 and δ>0.Proof.
The proof of this theorem is similar to Theorem6.


## 4. Examples

Two examples are studied to show the interest of the main results.Example 10.
Consider the following SFDE for eachε>0(31)dζω=fω,ζωdω+gω,ζωdWω,Eζ~ω−ζ~ω0−∫ω0ωfν,ζ~νdν−∫ω0ωgν,ζ~νdWν2≤εω+1,where
(32)χ∈LFω02−ρ,0;ℝ,ζω∈M2ω0−ϱ,T,ℝ,fω,ξ1=sinωξ10+cosωξ1−ϱ,ξ1∈C−ϱ,0;ℝ,gω,ξ1=11+ω2ξ10+e−ω1+ω2ξ1−ϱ,ξ1∈C−ϱ,0;ℝ,with ϱ>0. Then, replacing now ξ1 by the segment of a solution ζω, we get
(33)gω,ζω=11+ω2ζω+e−ω1+ω2ζω−ϱ,fω,ζω=sinωζω+cosωζω−ϱ.
Letξ1,ξ2∈C−ϱ,0;ℝ, then
(34)fω,ξ1−fω,ξ22=sinωξ10−ξ20+cosωξ1−ϱ−ξ2−ϱ2≤2sinω2ξ10−ξ202+2cosω2ξ1−ϱ−ξ2−ϱ2≤4ξ1−ξ22,gω,ξ1−gω,ξ22=11+ω2ξ10−ξ20+e−ω1+ω2ξ1−ϱ−ξ2−ϱ2≤21+ω2ξ10−ξ202+2e−2ω1+ω2ξ1−ϱ−ξ2−ϱ2≤4ξ1−ξ22.
Hence, the uniform Lipschitz condition is satisfied.
Therefore, by Theorem6, there is a solution ζ∈M2ω0−ϱ,T,ℝ of (31), with ζω0=χ, such that ∀ω∈ω0,T,
(35)Eζ~ω−ζω2≤11−α/α+δ2expα+δT−ω0εω+1,where α=32T−ω0+1 and δ>0.For System (31), we conduct a simulation based on the Euler-Maruyama scheme with step size 10−3, for which we set ω0=0, ϱ=0.5 and the initial, data χ as a map, namely, χ=ω2 for all −0.5≤ω≤0. In Figure 1, we give a sequence of computer simulations of the exact solution path ζω and the rough solution path ζ~ω for System (31) on the interval −0.5,8. Choosing ε=10−4, δ=15, α=35.2, and T=0.1 one obtain M=1/1−α/α+δ2expα+δT−ω0=5.72×103. We use the time step 10−4 of the interval 0,0.1 and 10000 realizations for this discretisation; we give in Figure 2 the trajectory of ε×M×ω+1 and simulation of the mean square of ζ~ω−ζω on the interval 0,0.1. It is clear that the convergence plot verifies the theoretical findings.Example 11.
Consider the following SFDE for eachε>0(36)dζω=fω,ζωdω+gω,ζωdWω,Eζ~ω−ζ~0−∫0ωfν,ζ~νdν−∫0ωgν,ζ~νdWν2≤ε,where
(37)χ∈LF02−ρ,0;ℝ,ζω∈M2−ϱ,3,ℝ,fω,ξ1=ωξ10+ω2ξ1−ϱ,ξ1∈C−ϱ,0;ℝ,gω,ξ1=ω2ξ10+ωξ1−ϱ,ξ1∈C−ϱ,0;ℝ,with ρ>0. Then, replacing now ξ1 by the segment of a solution ζω, we have
(38)fω,ζω=ωζω+ω2ζω−ϱ,gω,ζω=ω2ζω+ωζω−ϱ.
Letξ1,ξ2∈C−ϱ,0;ℝ, then
(39)fω,ξ1−fω,ξ22=ωξ10−ξ20+ω2ξ1−ϱ−ξ2−ϱ2≤2ω2ξ10−ξ202+2ω4ξ1−ϱ−ξ2−ϱ2≤180ξ1−ξ22,gω,ξ1−gω,ξ22=ω2ξ10−ξ20+ωξ1−ϱ−ξ2−ϱ2≤2ω4ξ10−ξ202+2ω2ξ1−ϱ−ξ2−ϱ2≤180ξ1−ξ22.
Hence, the uniform Lipschitz condition is satisfied.
Therefore, by Theorem8, there is a solution ζ∈M2−ϱ,3,ℝ of (36), with ζ0=χ, such that ∀ω∈0,3,
(40)Eζ~ω−ζω2≤11−α/α+δ2exp3α+δε,where α=1440 and δ>0.Figure 1
Simulation ofζω and ζ~ω are trajectory in System (31) with ϱ=0.5 and χω=ω2 for ω∈−0.5,8.Figure 2
HURS with respect toε,θω of ζω on the interval 0,0.1 with χω=ω2.We use again Euler-Maruyama scheme with step size10−5 to conduct a simulation for System (36). We fix ρ=0.5 and the initial data χ as a linear mapping, namely, χ=−ω+1 for all −0.5≤ω≤0. In Figure 3, we plot the path of the exact solution ζω and the rough solution path ζ~ω for System (36) on the interval −0.5,2. Choosing ε=10−9, δ=10, α=1440, and T=0.06 one obtain a large value of M=1/1−α/α+δ2exp3α+δ. We use the time step 10−4 of the interval 0,0.06 and 10000 realizations for this discretisation; we give in Figure 4 the trajectory of the constant function ε×M and simulation of the mean square of ζ~ω−ζω on the interval 0,0.06. It is clear that the convergence plot verifies the theoretical findings.Figure 3
Simulation ofζω and ζ~ω are trajectory in System (36) with ϱ=0.5 and χω=−ω+1 for ω∈−0.5,2.Figure 4
HUS with respect toε of ζω on the interval 0,0.06 with χω=−ω+1.


## 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.



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