Mean square exponential stability of stochastic function differential equations in the G-framework

Guangjie Li; Zhipei Hu

doi:10.1515/math-2022-0582

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Mean square exponential stability of stochastic function differential equations in the G-framework

Guangjie Li and Zhipei Hu

Published/Copyright: June 27, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

This research focuses on the stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) with infinite delay. It is proved that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square. An example is also presented to illustrate the effectiveness of the obtained theory.

Keywords: mean square exponential stability; stochastic functional differential equations; G-Brownian motion; infinite delay

MSC 2010: 34K20; 60H10; 34K50

1 Introduction

Stochastic differential equations (SDEs), which are often used to describe some stochastic dynamical systems, have been encountered in many science and engineering problems (see, e.g., [1,2]). In reality, these stochastic dynamical systems depend not only on the present but also on the past states. For such systems, stochastic functional differential equations (SFDEs) are used to described them (see, e.g., [3–5]).

In the study of stochastic dynamical systems, stability analysis, as a hot topic of stochastic dynamical systems, has aroused great concern (please see monographs [6] and [7]). So far, there are numerous literature on the stability of SFDEs (e.g., [8–14]). Among them, Zhou et al. [11] investigated the exponential stability of criteria for SFDEs with infinite delay. Pavlovic and Jankovic [12] studied both the p th moment and the almost sure stability on a general decay for SFDEs with infinite delay by Razumikhin approach. Ngoc [13] presented the criteria for the mean square exponential stability of general SFDEs with infinite delay. Li and Xu [14] discussed the exponential stability in mean square of SFDEs and neutral SFDEs with infinite delay by a novel approach.

Motivated by describing measuring finance risk and volatility uncertainty, Peng [15] has developed a theoretical framework of G-expectation. Based on the framework of G-expectation, Peng [15,16] introduced the G-Brownian motion and set up its Itô integral. Hu and Peng [17] found that a weakly compact family of probability measures can be used to represent the G-expectation. Under the G-framework, many efforts have been made to study the stability of SDEs driven by G-Brownian motion (G-SDEs), (see [18–22] and references therein). Faizullah et al. [23] studied the mean square exponential stability of nonlinear neutral G-SFDEs. Pan et al. [24] derived the p th moment exponential stability and quasi-sure (q.s.) exponential stability of impulsive G-SFDEs. Li [25] investigated the mean square stability with general decay rate of nonlinear neutral G-SFDEs. However, there are few results on the exponential stability of G-SFDEs with infinite delay, which motivates the present research. This article proves that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square.

This article is organized as follows. Section 2 presents some preliminaries. Section 3 proves that the trivial solution of a G-SFDE is exponentially stable in mean square under some conditions. Finally, an example is presented to illustrate the obtained results.

2 Preliminaries

This section briefly recalls some preliminaries in G-framework. More relevant details can be seen in [15,16,26,27].

Let R = ( − ∞ , + ∞ ) , R + = [ 0 , + ∞ ) and R − = ( − ∞ , 0 ] . For ∀ c ∈ R n , denote by ∣ c ∣ = c T c . For ∀ a ∈ R , a + = max { a , 0 } and a − = ( − a ) + . On a non-empty basic space Ω , we can define a linear space H of real-valued functions. We suppose that H satisfies C ∈ H for each constant C , and if X ∈ H , ∣ X ∣ ∈ H . If X 1 , X 2 , … , X n ∈ H , then φ ( X 1 , X 2 , … , X n ) ∈ H for each φ ∈ C l , Lip ( R n ) , where C l , Lip ( R n ) is the space of linear function φ : R n → R :

C l , Lip ( R n ) = { φ ∣ ∃ C ∈ R + , m ∈ N such that ∣ φ ( b 1 ) − φ ( b 2 ) ∣ ≤ C ( 1 + ∣ b 1 ∣ m + ∣ b 2 ∣ m ) ∣ b 1 − b 2 ∣ } ,

for ∀ b 1 , b 2 ∈ R n .

Definition 2.1

[16] A functional E ˆ : H → R is called a sublinear expectations; if for all X , Y ∈ H , C ∈ R and λ ≥ 0 , it satisfies the following properties:

Monotonicity: If X ≥ Y , then E ˆ [ X ] ≥ E ˆ [ Y ] .
Constant preserving: E ˆ [ C ] = C .
Sub-additivity: E ˆ [ X + Y ] ≤ E ˆ [ X ] + E ˆ [ Y ] .
Positive homogeneity: E ˆ [ λ X ] = λ E ˆ [ X ] .

Also, if E ˆ [ X ] = E ˆ [ − X ] = 0 , then E ˆ [ C + λ X + Y ] = C + E ˆ [ Y ] .

The triple ( Ω , H , E ˆ ) is called a sublinear expectation space. If ( i ) and ( i i ) are satisfied, E ˆ [ ⋅ ] is called a nonlinear expectation and the triple ( Ω , H , E ˆ ) is relevantly called a nonlinear expectation space.

For the details of G-normal distribution, G-expectation, G-conditional expectation, and G-Brownian motion, please see Chapter 3 and Chapter 4 of Peng [16].

Denote by Ω T = { ω ( ⋅ ∧ T ) : ω ∈ Ω } , ∀ T ≥ 0 . For T ∈ R + , a partition μ T of [ 0 , T ] is a finite ordered subset μ T = { t 0 , t 1 , … , t N } such that 0 = t 0 < t 1 < t 2 < ⋯ < t N = T , π ( μ T ) = max { ∣ t i + 1 − t i ∣ : i = 0 , 1 , … , N − 1 } . Let

L i p ( Ω T ) = { φ ( ω ( t 1 ) , ω ( t 2 ) , … , ω ( t n ) ) : t 1 , t 2 , … , t n ∈ [ 0 , T ] , φ ∈ C l , Lip ( R n ) }

and its countably many union L i p ( Ω ) = ⋃ n = 1 ∞ L i p ( Ω n ) . Denote by L G p ( Ω T ) the completion of L i p ( Ω T ) under the norm ‖ X ‖ p = ( E ˆ ∣ X ∣ p ) 1 ∕ p , for any p ≥ 1 . Besides, the space is defined by:

M G p , 0 ( [ 0 , T ] ) = η t = ∑ j = 0 N − 1 ξ j I [ t j , t j + 1 ) ( t ) : ξ j ∈ L G p ( Ω t j ) , p ≥ 1 ,

and its completion M G p ( [ 0 , T ] ) equipped with the norm:

‖ η ‖ M G p ( [ 0 , T ] ) = 1 T ∫ 0 T E ˆ [ ∣ η t ∣ p ] d t 1 p ,

where E ˆ stands for the G-expectation.

We now show the representation theorem of G-expectation as follows.

Lemma 2.1

[17,28] Let E ˆ be the G-expectation on ( Ω , L G 1 ( Ω ) ) . Then, there is a weakly compact family of probability measures P on ( Ω , ℬ ( Ω ) ) such that E ˆ [ X ] = sup P ∈ P E P [ X ] ∀ X ∈ L G 1 ( Ω ) . Moreover, P is called a set that represents the G-expectation E ˆ .

From the aforementioned lemma, the weakly compact family of probability P characterizes the degree of Knightian uncertainty. If P is singleton, that is { P } , then the model has no ambiguity, and the G-expectation E ˆ is the classical expectation. Then, define G-upper capacity V ( ⋅ ) and G-lower capacity v ( ⋅ ) by:

V ( U ) = sup P ∈ P P ( U ) , ∀ U ∈ ℬ ( Ω ) , v ( U ) = inf P ∈ P P ( U ) , ∀ U ∈ ℬ ( Ω ) .

Definition 2.2

A set U ∈ ℬ ( Ω ) is a polar set provided that V ( U ) = 0 . A property is said to hold q.s. if it is true outside a polar set.

P -q.s. means that it holds P -a.s. for each P ∈ P . If an event U satisfies V ( U ) = 1 , then we claim that the event U occurs V -a.s.

Let ( ω ( t ) ) t ≥ 0 be a one-dimensional G-Brownian motion. Define G ( a ) = 1 2 E ˆ [ a 2 ( 1 ) ] = 1 2 ( σ ¯ 2 a + − σ ̲ 2 a − ) ( ∀ a ∈ R ), where E ˆ [ ω 2 ( 1 ) ] = σ ¯ 2 , − E ˆ [ − ω 2 ( 1 ) ] = σ ̲ 2 , 0 ≤ σ ̲ ≤ σ ¯ < ∞ . And ℱ t represents a filtration generated by G-Brownian motion ( ω ( t ) ) t ≥ 0 . In the following, we next show the stochastic integral with respect to the quadratic variation of G-Brownian motion.

Definition 2.3

The stochastic integral with respect to the quadratic variation of G-Brownian motion ( ⟨ ω ⟩ ( t ) ) t ≥ 0 is given by:

∫ 0 t η t d ⟨ ω ⟩ ( t ) = ∑ j = 0 N − 1 ξ j ( ⟨ ω ⟩ ( t j + 1 ) − ⟨ ω ⟩ ( t j ) ) , ∀ η t ∈ M 1 , 0 ( [ 0 , T ] ) ,

where ⟨ ω ⟩ ( t ) = lim N → ∞ ∑ j = 0 N − 1 ( ⟨ ω ⟩ ( t j + 1 N ) − ⟨ ω ⟩ ( t j N ) ) 2 = ω 2 ( t ) − 2 ∫ 0 t ω ( s ) d ω ( s ) .

Lemma 2.2

([29], Lemma 2.1) For each η ∈ M G p [ ( 0 , T ) ] ,

E ˆ ∫ 0 T ∣ η ( t ) ∣ p d t ≤ ∫ 0 T E ˆ ∣ η ( t ) ∣ p d t .

3 Main results

In this section, we first give the following notations. Denote by BC ≔ BC ( ( − ∞ , 0 ] ; R ) the family of bounded continuous functions ϕ : ( − ∞ , 0 ] → R with the norm ‖ ϕ ‖ = sup − ∞ ≤ s ≤ 0 ∣ ϕ ( s ) ∣ . For t ≥ 0 , let ( Ω , H , { Ω t } t ≥ 0 , E ˆ , V ) be a generalized sublinear expectation space. Let ( ω ( t ) ) t ≥ 0 be a one-dimensional G-Brownian motion defined on the sublinear expectation space.

Consider the following G-SFDE

(3.1) d x ( t ) = f ( x t , t ) d t + g ( x t , t ) d ⟨ ω ⟩ ( t ) + h ( x t , t ) d ω ( t ) , ∀ t ≥ t 0 > 0 ,

with the initial value

(3.2) x t 0 = ξ = { ξ ( θ ) : − ∞ ≤ θ ≤ 0 }

being ℱ t 0 -measurable and a BC -valued random variable and ξ ∈ M G 2 ( R − , R ) . Here, x ( t ) is the value of stochastic process at time t and x t = { x ( t + θ ) : − ∞ ≤ θ ≤ 0 } , f , g , h : R × R + → R are continuous functions. For the purpose of the stability, we assume that f ( 0 , t ) = g ( 0 , t ) = h ( 0 , t ) = 0 for ∀ t ∈ R . The equation is also assumed to satisfy the basic conditions for the existence and uniqueness of the solution, which are the local Lipschitz condition and the linear growth condition. As for the proof of the existence and uniqueness of the solution, we can adopt the technique of [3] analogously.

In the following, the definition of the exponential stability in mean square for the G-SFDE (3.1) with the initial value (3.2) is given.

Definition 3.1

The trivial solution of the G-SFDE (3.1) with the initial value (3.2) is said to be exponentially stable in mean square, if there exist positive numbers γ ˜ and C ˜ such that

E ˆ ∣ x ( t ) ∣ 2 ≤ C ˜ E ˆ ‖ ξ ‖ 2 e − γ ˜ t , ∀ t ≥ t 0 .

Theorem 3.1

Assume that there exist α > 0 and bounded Borel-measurable functions β ( ⋅ ) : R + → R , η ( ⋅ , ⋅ ) : R + × R − → R + satisfying ∫ − ∞ 0 e − α s < ∞ , and

(3.3) 2 φ ( 0 ) f ( φ , t ) + G ⟨ 4 φ ( 0 ) g ( φ , t ) + 2 h 2 ( φ , t ) ⟩ ≤ β ( t ) ∣ φ ( 0 ) ∣ 2 + ∫ − ∞ 0 η ( t , s ) ∣ φ ( s ) ∣ 2 d s ,

with ∀ t ∈ R + , φ ∈ BC . If

(3.4) β ( t ) + ∫ − ∞ 0 e − α s η ( t , s ) d s ≤ − α , ∀ t ∈ R + ,

then the trivial solution of equation (3.1) with the given initial value (3.2) is exponentially stable in mean square.

Proof

For any given initial value (3.2). Let K > 1 . We now prove that E ˆ ∣ x ( t ) ∣ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 for ∀ t ∈ R . For t ∈ ( − ∞ , t 0 ] , it is obvious that we can obtain E ˆ ∣ x ( t ) ∣ 2 ≤ E ˆ ‖ ξ ‖ 2 ≤ K E ˆ ‖ ξ ‖ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 .

We next prove that E ˆ ∣ x ( t ) ∣ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 for ∀ t ≥ t 0 . If it is not true, then there exists t ¯ > t 0 such that E ˆ ∣ x ( t ¯ ) ∣ 2 > K e − α ( t ¯ − t 0 ) E ˆ ‖ ξ ‖ 2 . Let t ∗ = inf { t > t 0 : E ˆ ∣ x ( t ) ∣ 2 > K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 } . Then, E ˆ ∣ x ( t ) ∣ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 for t ∈ [ t 0 , t ∗ ] and E ˆ ∣ x ( t ∗ ) ∣ K e − α ( t ∗ − t 0 ) E ˆ ‖ ξ ‖ 2 . Assume that γ is a constant and γ > 0 . Applying the G-Itô formula (see, e.g., [16,28]) to e γ t ∣ x ( t ) ∣ 2 , we can obtain

(3.5) d ( e γ t ∣ x ( t ) ∣ 2 ) = γ e γ t ∣ x ( t ) ∣ 2 d t + 2 e γ t x ( t ) f ( x t , t ) d t + 2 e γ t x ( t ) g ( x t , t ) d ⟨ ω ⟩ ( t ) + 2 e γ t x ( t ) h ( x t , t ) d ω ( t ) + e γ t h 2 ( x t , t ) d ⟨ ω ⟩ ( t ) = γ e γ t ∣ x ( t ) ∣ 2 d t + 2 e γ t x ( t ) f ( x t , t ) d t + G ⟨ 4 e γ t x ( t ) g ( x t , t ) + 2 e γ t h 2 ( x t , t ) ⟩ d t + 2 e γ t x ( t ) g ( x t , t ) d ⟨ ω ⟩ ( t ) + 2 e γ t x ( t ) h ( x t , t ) d ω ( t ) + e γ t h 2 ( x t , t ) d ⟨ ω ⟩ ( t ) − G ⟨ 4 e γ t x ( t ) g ( x t , t ) + 2 e γ t h 2 ( x t , t ) ⟩ d t .

Then, we can further gain

(3.6) e γ t ∣ x ( t ) ∣ 2 = e γ t 0 ∣ x ( t 0 ) ∣ 2 + γ ∫ t 0 t e γ s ∣ x ( s ) ∣ 2 d s + 2 ∫ t 0 t e γ s x ( s ) f ( x s , s ) d s + ∫ t 0 t e γ s G ⟨ 4 x ( s ) g ( x s , s ) + 2 h 2 ( x s , s ) ⟩ d s + M t 0 t ,

with

M t 0 t = 2 ∫ t 0 t e γ s x ( s ) h ( x s , s ) d ω ( s ) + 2 ∫ t 0 t e γ s x ( s ) g ( x s , s ) d ⟨ ω ⟩ ( s ) + ∫ t 0 t e γ s h 2 ( x s , s ) d ⟨ ω ⟩ ( s ) − ∫ t 0 t e γ s G ⟨ 4 x ( s ) g ( x s , s ) + 2 h 2 ( x s , s ) ⟩ d s

being a G-martingale [16] ([16, page 72, Proposition 1.4]). Taking expectation on the both sides of (3.6), we can obtain

(3.7) e γ t E ˆ ∣ x ( t ) ∣ 2 = e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + γ E ˆ ∫ t 0 t e γ s ∣ x ( s ) ∣ 2 d s + 2 E ˆ ∫ t 0 t e γ s x ( s ) f ( x s , s ) d s + E ˆ ∫ t 0 t e γ s G ⟨ 4 x ( s ) g ( x s , s ) + 2 h 2 ( x s , s ) ⟩ d s .

By (3.3), we can further obtain

(3.8) e γ t E ˆ ∣ x ( t ) ∣ 2 ≤ e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + γ E ˆ ∫ t 0 t e γ s ∣ x ( s ) ∣ 2 d s + E ˆ ∫ t 0 t e γ s β ( s ) ∣ x ( s ) ∣ 2 + ∫ − ∞ 0 η ( s , r ) ∣ x ( s + r ) ∣ 2 d r d s = e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + E ˆ ∫ t 0 t e γ s ( γ + β ( s ) ) ∣ x ( s ) ∣ 2 d s + E ˆ ∫ t 0 t e γ s ∫ − ∞ 0 η ( s , r ) ∣ x ( s + r ) ∣ 2 d r d s .

By Lemma 2.2, we can obtain

(3.9) e γ t E ˆ ∣ x ( t ) ∣ 2 ≤ e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + ∫ t 0 t e γ s ( γ + β ( s ) ) E ˆ ∣ x ( s ) ∣ 2 d s + ∫ t 0 t e γ s ∫ − ∞ 0 η ( s , r ) E ˆ ∣ x ( s + r ) ∣ 2 d r d s .

Thus,

(3.10) e γ t ∗ E ˆ ∣ x ( t ∗ ) ∣ 2 ≤ e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + ∫ t 0 t ∗ e γ s ( γ + β ( s ) ) E ˆ ∣ x ( s ) ∣ 2 d s + ∫ t 0 t ∗ e γ s ∫ − ∞ 0 η ( s , r ) E ˆ ∣ x ( s + r ) ∣ 2 d r d s .

For t ∈ [ t 0 , t ∗ ) , E ˆ ∣ x ( t ) ∣ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 , so

(3.11) e γ t ∗ E ˆ ∣ x ( t ∗ ) ∣ 2 ≤ e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + ∫ t 0 t ∗ K E ˆ ‖ ξ ‖ 2 e γ s e − α ( s − t 0 ) γ + β ( s ) + ∫ − ∞ 0 e − α r η ( s , r ) d r d s < e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + K E ˆ ‖ ξ ‖ 2 e α t 0 ∫ t 0 t ∗ e ( γ − α ) s γ + β ( s ) + ∫ − ∞ 0 e − α r η ( s , r ) d r d s .

By (3.4), we obtain

(3.12) e γ t ∗ E ˆ ∣ x ( t ∗ ) ∣ 2 ≤ e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + K E ˆ ‖ ξ ‖ 2 e α t 0 ∫ t 0 t ∗ e ( γ − α ) s ( γ − α ) d s = e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + K E ˆ ‖ ξ ‖ 2 e α t 0 ( e ( γ − α ) t ∗ − e ( γ − α ) t 0 ) = e γ t 0 E ˆ ∣ x ( t 0 ) ∣ 2 + K E ˆ ‖ ξ ‖ 2 ( e γ t ∗ e − α ( t ∗ − t 0 ) − e γ t 0 ) = e γ t 0 ( E ˆ ∣ x ( t 0 ) ∣ 2 − K E ˆ ‖ ξ ‖ 2 ) + K E ˆ ‖ ξ ‖ 2 e γ t ∗ ⋅ e − α ( t ∗ − t 0 ) .

For E ˆ ∣ x ( t 0 ) ∣ 2 ≤ E ˆ ‖ ξ ‖ 2 and K > 1 , thus E ˆ ∣ x ( t 0 ) ∣ 2 < K E ˆ ‖ ξ ‖ 2 . We can obtain

(3.13) e γ t ∗ E ˆ ∣ x ( t ∗ ) ∣ 2 < K E ˆ ‖ ξ ‖ 2 e γ t ∗ ⋅ e − α ( t ∗ − t 0 ) .

Namely, E ˆ ∣ x ( t ∗ ) ∣ 2 < K e − α ( t ∗ − t 0 ) E ˆ ‖ ξ ‖ 2 , which is contrary with E ˆ ∣ x ( t ∗ ) ∣ 2 = K e − α ( t ∗ − t 0 ) E ˆ ‖ ξ ‖ 2 . Therefore, E ˆ ∣ x ( t ) ∣ 2 ≤ K e − α ( t − t 0 ) E ˆ ‖ ξ ‖ 2 for ∀ t ≥ t 0 .□

Theorem 3.2

Assume that there exist real numbers l < 0 , m ≥ 0 , and a Borel-measurable function h ( ⋅ ) : R − → R + such that

(3.14) ∫ − ∞ 0 e − l s h ( s ) d s < ∞

and

(3.15) 2 φ ( 0 ) f ( φ , t ) + G ⟨ 4 φ ( 0 ) g ( φ , t ) + 2 h 2 ( φ , t ) ⟩ ≤ m ∣ φ ( 0 ) ∣ 2 + ∫ − ∞ 0 h ( s ) ∣ φ ( s ) ∣ 2 d s , ∀ t ∈ R + , φ ∈ BC .

If m + ∫ − ∞ 0 h ( s ) d s < 0 , then the trivial solution of (3.1) with the given initial value (3.2) is exponentially stable in mean square.

Proof

Let

H ( α ) = m + α + ∫ − ∞ 0 e − α s h ( s ) d s , α ∈ [ 0 , l ] .

(3.14) ensures H ( α ) is well-defined and continuous on [ 0 , l ] . By m + ∫ − ∞ 0 h ( s ) d s < 0 , we can obtain that H ( 0 ) < 0 . Then, we can obtain H ( α 0 ) < 0 for some α 0 ∈ ( 0 , l ] , that is, m + ∫ − ∞ 0 e − α 0 s h ( s ) d s < − α 0 . It follows from Theorem 3.1 that we can obtain that the trivial solution of equation (3.1) is exponentially stability in mean square.□

Remark 3.1

Though authors in [11– 14] investigated the exponential stability of SFDEs with infinite delay, they did not take G-Brownian motion into consideration. Faizullah et al. [23] and Li [25] studied the exponential stability of nonlinear neutral G-SFDEs, and Pan et al. [24] discussed the exponential stability of impulsive G-SFDEs, but these articles did not consider infinite delay. Therefore, results in this article are more rich and are different from the existing literature.

4 An example

In this section, an example is given to illustrate the obtained results. Let ω ( t ) be a one-dimensional G-Brownian motion with ω ( 1 ) ∼ N ( 0 ; [ 1 ∕ 2 , 1 ] ) .

Example 4.1

Consider the following scalar nonlinear G-SFDE:

(4.1) d x ( t ) = − 4 x ( t ) − x 3 ( t ) + ∫ − ∞ 0 e s sin ( x ( t + s ) ) d s d t + 2 ∫ − ∞ 0 e s sin ( x ( t + s ) ) d s d ⟨ ω ⟩ ( t ) + ∫ − ∞ 0 e s sin ( x ( t + s ) ) d s d ω ( t ) , ∀ t ≥ t 0 ≥ 0 .

By computing, we can have

(4.2) 2 φ ( 0 ) f ( φ , t ) = 2 φ ( 0 ) − 4 φ ( 0 ) − φ 3 ( 0 ) + ∫ − ∞ 0 e s sin φ ( s ) d s = − 8 φ 2 ( 0 ) − 2 φ 4 ( 0 ) + 2 φ ( 0 ) ∫ − ∞ 0 e s sin φ ( s ) d s ≤ − 8 φ 2 ( 0 ) + ∫ − ∞ 0 e s φ 2 ( 0 ) d s + ∫ − ∞ 0 e s sin 2 φ ( s ) d s ≤ − 8 φ 2 ( 0 ) + ∫ − ∞ 0 e s φ 2 ( 0 ) d s + ∫ − ∞ 0 e s sin 2 φ 2 ( s ) d s ≤ − 7 φ 2 ( 0 ) + ∫ − ∞ 0 e s φ 2 ( s ) d s

and

(4.3) G ⟨ 4 φ ( 0 ) g ( φ , t ) + 2 h 2 ( φ , t ) ⟩ = G 8 φ ( 0 ) ∫ − ∞ 0 e s sin φ ( s ) d s + 2 ∫ − ∞ 0 e s sin φ ( s ) d s 2 ≤ G 8 φ ( 0 ) ∫ − ∞ 0 e s sin φ ( s ) d s + 2 ∫ − ∞ 0 e s φ 2 ( s ) d s ≤ G 4 ∫ − ∞ 0 e s φ 2 ( 0 ) d s + 4 ∫ − ∞ 0 e s φ 2 ( s ) d s + 2 ∫ − ∞ 0 e s φ 2 ( s ) d s = G 4 φ 2 ( 0 ) + 6 ∫ − ∞ 0 e s φ 2 ( s ) d s ≤ 2 φ 2 ( 0 ) + 3 ∫ − ∞ 0 e s φ 2 ( s ) d s .

Then, by (4.2) and (4.3), we can obtain

(4.4) 2 φ ( 0 ) f ( φ , t ) + G ⟨ 4 φ ( 0 ) g ( φ , t ) + 2 h 2 ( φ , t ) ⟩ ≤ − 5 φ 2 ( 0 ) + 4 ∫ − ∞ 0 e s φ 2 ( s ) d s .

Therefore, m = 5 , h ( s ) = 4 e s . According to Theorem 3.2, we can assert that the trivial solution of equation (4.1) is exponentially stable in mean square.

Acknowledgement

The authors would like to thank anonymous referees and editors for their helpful comments and suggestions, which greatly improved the quality of this article.

Funding information: This research was partially supported by the National Natural Science Foundation of China (No. 11901398, No. 62003204), in part by the Basic and Applied Basic Research of Guangzhou Basic Research Program (No. 202201010250), in part by the Guangdong Basic and Applied Basic Research Foundation (2023A1515012781), and in part by Shantou University Scientific Research Foundation for Talents (NTF19031).
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Not applicable.

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Received: 2022-06-22

Revised: 2023-03-20

Accepted: 2023-03-30

Published Online: 2023-06-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0582

Keywords for this article

mean square exponential stability; stochastic functional differential equations; G-Brownian motion; infinite delay

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