Generalized Mean Square Exponential Stability for Stochastic Functional Differential Equations

Abstract

This work focuses on a class of stochastic functional differential equations and neutral stochastic differential functional equations. By using a new approach, some sufficient conditions are obtained to guarantee the generalized mean square exponential stability for the equation under consideration. Certain existing results are refined and extended. Lastly, the validity of the main results is confirmed through several simulation examples.

1. Introduction

Recently, models based on stochastic differential equations (SDEs) have played an important role across various fields of science and industry [1,2,3,4]. In particular, they are extensively applied in biology and neural networks, which has led to considerable attention being given to the stability analysis of differential equations [5,6,7,8,9,10,11,12,13,14]. Additionally, the process of synaptic transmission in nervous systems involves noise and can be regarded as stochastic perturbation [15]. Consequently, investigating the stochastic differential equations is essential.

Stability is a key focus in the research of SDEs and has been extensively explored from various perspectives, including stochastic stability, stochastic asymptotic stability, moment exponential stability, almost sure stability, and mean square polynomial stability [16,17,18,19,20,21,22,23,24,25].

A common method of analyzing the stability of SDEs is the Lyapunov function approach. This technique employs Lyapunov functions and functionals, which have proven highly effective in evaluating and ensuring the stability of SDEs. For example, Chen et al. [6] examined the asymptotic stability in the mean square of stochastic coupled partial differential equations, and their work contributes to a broader understanding of stability in systems influenced by both deterministic and random factors. Zhang et al. [26] utilized the Lyapunov technique to investigate the asymptotic stability of networks with distributed delays. Liu et al. [27] examined the Lyapunov function approach and stochastic technique to analyze the global mean square stability of neural network discrete-time nonlinear systems with random time-varying delays. Finding an appropriate Lyapunov function for SDEs is often challenging, and the stability criteria derived through the Lyapunov function approach are usually expressed using differential or matrix inequalities, among other forms. The conditions provided by Lyapunov functions (functionals) tend to be somewhat strict, generally implicit, and difficult to evaluate.

However, alternative methods exist for the analysis of the stability of SDEs. Chen et al. [6,28] employed the fixed point technique to examine the asymptotic stability of SDEs. Ngoc et al. [29,30] used proof by contradiction to examine the mean square exponential stability of SDEs and SDEs with delays. Huang et al. [31,32] used the Halanay inequality and Dini derivative to investigate the stability of SDEs.

Recently, generalized exponential stability has been introduced and studied by some researchers by using inequality techniques and Lyapunov methods. Generalized exponential stability is a broader concept of stability that encompasses traditional forms such as exponential, polynomial, and logarithmic stability. It provides a more comprehensive framework for the analysis of the stability of systems, allowing for a deeper understanding of how systems behave under different conditions and timescales. This generalized approach offers greater flexibility in studying complex dynamic systems, enabling researchers to account for a wider range of stability scenarios. Hien et al. [33] established clear conditions for achieving generalized exponential stability in nonlinear stochastic systems with non-autonomous time delays, utilizing generalized Halanay inequalities. These criteria provide a systematic way to assess the stability of such complex systems, particularly in scenarios where time delays and stochastic perturbations introduce additional challenges. Lu et al. [34] investigated the generalized exponential stability within a set of non-autonomous cellular neural networks via a Halanay-type inequality. Ruan et al. [35] studied the generalized exponential stability of stochastic functional differential equations involving infinite delays through developing some Halanay inequalities. We emphasize that it is crucial to explore generalized exponential stability because it provides a new view for dynamic systems’ stability.

The primary objective of this research is to determine adequate conditions that ensure generalized mean square exponential stability for a class of stochastic functional differential equations, including neutral stochastic differential functional equations. The methodology used involves a comparative approach along with a contradiction-based proof, ensuring the practicality of the proposed conditions. These findings not only enhance the understanding of stability in such systems but also improve upon previously established findings.

The remainder of this research is arranged as outlined below. In Section 2, this research introduces the essential symbols and preliminary concepts. In Section 3, we consider the generalized exponential stability for stochastic functional differential equations. In Section 4, we provide conditions for the mean square generalized exponential stability of neutral stochastic functional differential equations. In Section 5, we provide comparisons to recent findings and offer examples to demonstrate the benefits of our findings.

2. Preliminaries

Let ${\mathbb{N}}_n=\left\{1,2,\cdots ,n\right\}$. $\left(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},P\right)$ represents a fully defined probability space, where ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ is a right-continuous normal filtration, and ${\mathcal{F}}_0$ includes all $\mathit{P}$-null sets. Let $\mathcal{C}:=\mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$ denote the Banach space of all continuous functions defined on the interval $\left[-h,0\right]$, equipped with the norm ${∥\phi ∥}_C={max}_{s\in [-h,0]}\left|\phi \left(s\right)\right|$, where $\parallel ·\parallel$ denotes the Euclidean norm on ${\mathbb{R}}^n$, and ${\parallel ·\parallel }_{\mathit{F}}$ denotes the Frobenius norm for matrices. Let $\mathit{B}\left(t\right)={\left(B_1\left(t\right),\cdots ,B_m\left(t\right)\right)}^{\mathit{T}}$ donate an m-dimensional Brownian motion defined on $\left(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0},P\right)$. Let ${\mathit{C}}_{\mathcal{F}}^{\mathit{b}}{\mathrm{t}}_0\left(\left[-h,0\right];{\mathbb{R}}^n\right)$ represent the set of all ${\mathcal{F}}_0$-measurable, bounded, $\mathit{C}\left([-h,0];{\mathbb{R}}^n\right)$-valued random variables. $\mathbb{E}$ is an expectation operator.

Consider the stochastic functional differential equation

$$\left\{\begin{array}{cc}& dx\left(t\right)=f\left(x_t,t\right)dt+g\left(x_t,t\right)dB\left(t\right),t\ge t_0\ge 0,\hfill \\ & x_{t_0}=\xi \in C_{{\mathcal{F}}_{t_0}}^b\left(\left[-h,0\right];{\mathbb{R}}^n\right),\hfill \end{array}\right.$$

(1)

where $f:{\mathbb{R}}_{+}\times \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)\to {\mathbb{R}}^n,g:{\mathbb{R}}_{+}\times \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)\to {\mathbb{R}}^{n\times m}$. Furthermore, $x_t=\left\{x\left(t+\theta \right):-h\le \theta \le 0\right\}$, which is treated as a $\mathit{C}\left([-h,0];{\mathbb{R}}^n\right)$-valued random process. The functions f and g fulfill the local Lipschitz criterion with Lipschitz constant ${\mathit{L}}_n>0$ and adhere to the linear growth requirement. These conditions are outlined in [36].

Definition 1.

A solution of $\left(1\right)$ is a continuous ${\mathcal{F}}_t$-adapted process $x\left(t\right),t\ge t_0-h$, if it meets the requirement stated above, such that

$$x\left(t\right)=\xi \left(0\right)+{\int }_{t_0}^{t}f\left(x_s,s\right)ds+{\int }_{t_0}^{t}g\left(x_s,s\right)dB\left(s\right),$$

(2)

for each $t\ge t_0$. As is known, for a specified $\xi \in {\mathit{C}}_{\mathcal{F}}^{\mathit{b}}{\mathrm{t}}_0\left([-h,0];{\mathbb{R}}^n\right)$, Equation $\left(1\right)$ consistently possesses a unique continuous solution $x\left(t,t_0,\xi \right)$. In addition, this solution satisfies the property

$$\mathbb{E}\left[\underset{t_0-h\le t\le T}{sup}{∥x\left(t,t_0,\xi \right)∥}^p\right]<\infty ,$$

(3)

for $T\ge t_0$ and any $p>0$. Suppose that $f\left(0,t\right)=0$ and $g\left(0,t\right)=0$ for any $t>0$; Equation $\left(1\right)$ has the zero solution $x\left(t\right)=0$ with the zero initial data at $t_0$.

Definition 2.

The trivial solution of Equation $\left(1\right)$ is mean square generalized exponentially stable, if for every $t_0\ge 0$ and each $\xi \in {\mathit{C}}_{\mathcal{F}}^{\mathit{b}}{\mathrm{t}}_0\left([-h,0];{\mathbb{R}}^n\right)$, there exist a constant K and an integrable function $c_1\left(u\right)\ge 0$ such that the following condition is satisfied,

$$\underset{t\to \infty }{lim}{\int }_{t_0}^t{c}_1\left(u\right)du\to +\infty$$

and

$$\mathbb{E}{∥x\left(t,t_0,\xi \right)∥}^2\le K{e}^{-{\int }_{t_0}^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t\ge t_0,$$

(4)

where $-{\int }_{t_0}^t{c}_1\left(u\right)du$ represents the typical rate of decline.

Remark 1.

If we use $\beta \left(t-t_0\right)$, $\beta ln \left(t-t_0+1\right)$ and $\beta ln \left(ln \left(t-t_0+e\right)\right)$ instead of ${\int }_{t_0}^t{c}_1\left(u\right)du$, respectively, then Equation $\left(1\right)$ is stable in the mean square, exhibiting exponential, polynomial, and logarithmic stability.

In order to compare this with some other forms of stability, we consider the definition of the h-type function and the h-stability.

Definition 3.

The function $h:{\mathbb{R}}_{+}\to (0,+\infty )$ is referred to as an h-type function as long as it meets the criteria listed below.

(i) 

It is non-decreasing and continuously differentiable in ${\mathbb{R}}_{+}$.

(ii) 

$h\left(0\right)=1$, ${lim}_{t\to \infty }h\left(t\right)=\infty$ and $J={sup}_{t>0}\left|{\displaystyle \frac{h^{\prime }\left(t\right)}{h\left(t\right)}}\right|<\infty$.

(iii) 

For any $t\ge 0,s\ge 0$, $h(t+s)\le h\left(t\right)h\left(s\right)$.

Definition 4.

The solution $x\left(t,t_0,\xi \right)$ of Equation $\left(1\right)$ with the starting value $\xi \in \mathcal{C}$ is considered

(i) 

mean square h-stable if ${lim\; sup}_{t\to \infty }{\displaystyle \frac{\log \mathbb{E}\left[\right|x\left(t\right){|}^2]}{\log h\left(t\right)}}\le -\kappa$ if there exists a positive constant κ;

(ii) 

almost surely h-stable if ${lim\; sup}_{t\to \infty }{\displaystyle \frac{\log \mathbb{E}\left|x\right(t\left)\right|}{\log h\left(t\right)}}\le -\kappa$ if there exists a positive constant κ.

We notice that generalized exponential stability is broader in scope compared to h-stability. In fact, if the solution of $\left(1\right)$ has h-stability, then the solution of $\left(1\right)$ has generalized exponential stability with $\rho \left(t\right)={\displaystyle \frac{1}{h\left(t\right)}}{h}^{\prime }\left(t\right)$.

Conversely, we remark that generalized exponential stability does not imply h-stability. Suppose that the solution of Equation $\left(1\right)$ possesses generalized exponential stability with $\rho \left(t\right)$. Denote $h\left(t\right)=e^{{\int }_0^t\rho \left(s\right)ds}$. Then,

$$\begin{array}{cc}\hfill h(t+s)=& e^{{\int }_0^{t+s}\rho \left(s\right)ds}=e^{{\int }_0^t\rho \left(s\right)ds+{\int }_t^{t+s}\rho \left(s\right)ds}\hfill \\ \hfill =& h\left(t\right)e^{{\int }_t^{t+s}\rho \left(s\right)ds}=h\left(t\right)e^{{\int }_0^s\rho (u+t)du}.\hfill \end{array}$$

(5)

Obviously, if $\rho (·)$ is strictly monotonically increasing on $[0,+\infty )$, then $e^{{\int }_0^s\rho (u+t)du}>e^{{\int }_0^s\rho \left(u\right)du}$ for $t>0$, which means that $h\left(s\right)h\left(t\right)<h\left(t+s\right)$ for any $t>0$ and $s\ge 0$. In this case, it can be concluded that the solution to Equation $\left(1\right)$ does not exhibit h-stability.

3. Exponential Stability for Stochastic Functional Equations

We consider the generalized exponential stability for Equation $\left(1\right)$ in this segment. To present the primary outcome of this section, we first introduce a few functions. Let ${\eta }_i\left(t,\theta \right):{\mathbb{R}}_{+}\times \left[-h,0\right]\to \mathbb{R},\left(i=1,2\right)$ be non-decreasing for any $t\in {\mathbb{R}}_{+}$. Moreover, ${\eta }_i\left(t,\theta \right)$ is left-continuous in $-h\le \theta \le 0$. Suppose that

$$L_i\left(t,\varphi \right):={\int }_{-h}^0\varphi \left(\theta \right)d\left[{\eta }_i\left(t,\theta \right)\right],t\in {\mathbb{R}}_{+},i=1,2,$$

(6)

is a Borel-measurable function that is locally bounded for every $\varphi \in C\left(\left[-h,0\right];\mathbb{R}\right)$. In $\left(6\right)$, the integral is Riemann–Stieltjes integral.

Theorem 1.

Let $\gamma \left(·\right):{\mathbb{R}}_{+}\to \mathbb{R}$ be a Borel-measurable function that is locally bounded. Suppose that the following assumptions hold for each $t\in {\mathbb{R}}_{+},\phi \in \mathcal{C}$:

$$\phi {\left(0\right)}^{T}f\left(\phi ,t\right)\le \gamma \left(t\right){∥\phi \left(0\right)∥}^2+{\int }_{-h}^0{∥\phi \left(\theta \right)∥}^{2}d\left[{\eta }_1\left(t,\theta \right)\right]$$

(7)

and

$${∥g\left(\phi ,t\right)∥}_F^2\le {\int }_{-h}^0{∥\phi \left(\theta \right)∥}^{2}d\left[{\eta }_2\left(t,\theta \right)\right].$$

(8)

If $t\in {\mathbb{R}}_{+}$ and there exists $c_1\left(u\right)>0$ such that, for $t\in {\mathbb{R}}_{+}$,

$$2\gamma \left(t\right)+2{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1\left(t,\theta \right)\right]+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2\left(t,\theta \right)\right]\le -c_1\left(u\right),$$

(9)

then the zero solution of Equation $\left(1\right)$ is mean square generalized exponentially stable, with the rate of decline given by ${\int }_{t_0}^t{c}_1\left(u\right)du$ for any $\xi \in \mathcal{C}$.

Proof. 

Let $\xi \in \mathcal{C}$ such that $\mathbb{E}{∥\xi ∥}_C^2>0$. Define the following functions:

$$x\left(t\right):=x\left(t,t_0,\xi \right),t\ge t_0-h,$$

$$X\left(t\right):=\mathbb{E}{∥x\left(t,t_0,\xi \right)∥}^2,t\ge t_0-h$$

and

$$Z\left(t\right):=K{e}^{-{\int }_{t_0}^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t\ge t_0-h.$$

Fix $K>1$. From $\left(1\right)$, and with K being large enough, we conclude that

$$\mathbb{E}{∥x\left(t,t_0,\xi \right)∥}^2<Z\left(t\right),t_0-h\le t\le t_0.$$

We demonstrate that, when any $t>t_0$,

$$X\left(t\right)\le Z\left(t\right).$$

(10)

Suppose that one can find $t_1>t_0$ such that

$$X\left(t_1\right)>Z\left(t_1\right).$$

(11)

Define $t_{*}:=inf\left\{t>t_0:X\left(t\right)>Z\left(t\right)\right\}$. Obviously, because $X\left(t\right)$ and $Z\left(t\right)$ are continuous functions,

$$X\left(t\right)\le Z\left(t\right),t_0\le t\le t_{*},$$

(12)

$$X\left(t_{*}\right)=K{e}^{-{\int }_{t_0}^{t_{*}}{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2$$

(13)

and

$$\mathbb{E}{∥x\left(t_n\right)∥}^2>K{e}^{-{\int }_{t_0}^{t_n}{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t_n\in \left(t_{*},t_{*}+{\displaystyle \frac{1}{n}}\right),n\in \mathbb{N}.$$

(14)

Consider the following real-valued function for any $t\in \mathbb{R}$, $x\in {\mathbb{R}}^n$:

$$V\left(x,t\right)=e^{{\int }_{t_0}^t{c}_2\left(u\right)du}{∥x∥}^2.$$

(15)

Utilizing Itô’s formula in Equation $\left(15\right)$ [3], we obtain

$$\begin{array}{cc}\hfill e^{{\int }_{t_0}^t{c}_2\left(u\right)du}{\parallel x\left(t\right)\parallel }^2=& {∥x\left(t_0\right)∥}^2+{\int }_{t_0}^t{c}_2\left(s\right)e^{{\int }_{t_0}^s{c}_2\left(u\right)du}{\parallel x\left(s\right)\parallel }^{2}ds\hfill \\ \hfill & +2{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}{x}^T\left(s\right)f\left(x_s,s\right)ds+{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}\sum _{k=1}^m{∥g_k\left(x_s,s\right)∥}^{2}ds\hfill \\ \hfill & +{\int }_{t_0}^t{V}_x(x\left(s\right),s)g\left(x_s,s\right)dB\left(s\right).\hfill \end{array}$$

From $\left(7\right)$ and $\left(8\right)$, we obtain

$$\begin{array}{cc}\hfill e^{{\int }_{t_0}^t{c}_2\left(u\right)du}{\parallel x\left(t\right)\parallel }^2\le & {∥x\left(t_0\right)∥}^2+{\int }_{t_0}^t{c}_2\left(s\right)e^{{\int }_{t_0}^s{c}_2\left(u\right)du}{\parallel x\left(s\right)\parallel }^{2}ds\hfill \\ \hfill & +2{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}\left({\gamma \left(s\right)\parallel x\left(s\right)\parallel }^2+{\int }_{-h}^0{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_1(s,\theta )\right]\right)ds\hfill \\ \hfill & +{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}\left({\int }_{-h}^0{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_2(s,\theta )\right]\right)ds\hfill \\ \hfill & +{\int }_{t_0}^t{V}_x(x\left(s\right),s)g\left(x_s,s\right)dB\left(s\right).\hfill \\ \end{array}$$

Based on $\left(3\right)$ and the zero mean property of Brownian motion, the above equation can be rewritten as

$$\begin{array}{cc}\hfill e^{{\int }_{t_0}^t{c}_2\left(u\right)du}\mathbb{E}{\parallel x\left(t\right)\parallel }^2\le & \mathbb{E}{∥x\left(t_0\right)∥}^2+{\int }_{t_0}^t{c}_2\left(s\right)e^{{\int }_{t_0}^s{c}_2\left(u\right)du}\mathbb{E}{\parallel x\left(s\right)\parallel }^{2}ds\hfill \\ \hfill & +2{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}\mathbb{E}\left({\gamma \left(s\right)\parallel x\left(s\right)\parallel }^2+{\int }_{-h}^0{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_1(s,\theta )\right]\right)ds\hfill \\ \hfill & +{\int }_{t_0}^t{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}\mathbb{E}\left({\int }_{-h}^0{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_2(s,\theta )\right]\right)ds,t\ge t_0.\hfill \\ \end{array}$$

From ${\eta }_1\left(s,\theta \right)$ and ${\eta }_2\left(s,\theta \right)$, which are non-decreasing for any $s\le t_{*}$, $\theta \in \left[-h,0\right]$, along with $\left(12\right)$, we can find

$${\int }_{-h}^0\mathbb{E}{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_1(s,\theta )\right]\le K{e}^{-{\int }_{t_0}^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1(s,\theta )\right]\mathbb{E}{∥\xi ∥}_C^2$$

(16)

and

$${\int }_{-h}^0\mathbb{E}{\parallel x(s+\theta )\parallel }^{2}d\left[{\eta }_2(s,\theta )\right]\le K{e}^{-{\int }_{t_0}^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2(s,\theta )\right]\mathbb{E}{∥\xi ∥}_C^2.$$

(17)

From $\left(16\right)$ and $\left(17\right)$, we have

$$\begin{array}{cc}\hfill e^{{\int }_{t_0}^{t_{*}}{c}_2\left(u\right)du}\mathbb{E}{∥x\left(t_{*}\right)∥}^2\le & \mathbb{E}{∥x\left(t_0\right)∥}^2+K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}{e}^{-{\int }_{t_0}^s{c}_1\left(u\right)du}\left(c_2\left(s\right)+2\gamma \left(s\right)\right)ds\hfill \\ \hfill & +2K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}{e}^{-{\int }_{t_0}^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1(s,\theta )\right]ds\hfill \\ \hfill & +K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)du}{e}^{-{\int }_{t_0}^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2(s,\theta )\right]ds\hfill \\ \hfill & =\mathbb{E}{∥x\left(t_0\right)∥}^2+K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)-c_1\left(u\right)du}{c}_2\left(s\right)ds\hfill \\ \hfill & +K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)-c_1\left(u\right)du}\left[2\gamma \left(s\right)+2{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1(s,\theta )\right]\right.\hfill \\ \hfill & \left.+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2(s,\theta )\right]\right]ds.\hfill \\ \end{array}$$

From condition $\left(9\right)$, we obtain

$$\begin{array}{cc}\hfill e^{{\int }_{t_0}^{t_{*}}{c}_2\left(u\right)du}\mathbb{E}{∥x\left(t_{*}\right)∥}^2& \le \mathbb{E}{∥x\left(t_0\right)∥}^2+K\mathbb{E}{∥\xi ∥}_C^2{\int }_{t_0}^{t_{*}}{e}^{{\int }_{t_0}^s{c}_2\left(u\right)-c_1\left(u\right)du}\left(c_2\left(s\right)-c_1\left(s\right)\right)ds\hfill \\ \hfill & =\mathbb{E}{∥x\left(t_0\right)∥}^2+K\mathbb{E}{∥\xi ∥}_C^2{e}^{{\int }_{t_0}^{t_{*}}{c}_2\left(u\right)-c_1\left(u\right)du}-K\mathbb{E}{∥\xi ∥}_C^2\hfill \\ \hfill & =\left(\mathbb{E}{∥x\left(t_0\right)∥}^2-K\mathbb{E}{\parallel \xi \parallel }_C^2\right)+K\mathbb{E}{\parallel \xi \parallel }_C^2{e}^{{\int }_{t_0}^{t_{*}}{c}_2\left(u\right)-c_1\left(u\right)du}.\hfill \\ \end{array}$$

We obtain from Equation $\left(12\right)$, $X\left(t_0\right)<Z\left(t_0\right)$,

$$\mathbb{E}{∥x\left(t_{*}\right)∥}^2<K{e}^{-{\int }_{t_0}^{t_{*}}{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,$$

which is inconsistent with $\left(12\right)$. This implies that

$$\mathbb{E}{∥x\left(t\right)∥}^2\le K{e}^{-{\int }_{t_0}^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t\ge t_0.$$

This demonstration has been completed. □

Remark 2.

It is worth noting that Lu et al. [34] explored the global generalized exponential stability for a class of non-autonomous cellular neural networks by using generalized Halanay inequalities. However, our condition $\left(9\right)$ is a generalization of $\left(3\right)$ in [34]. Our findings are novel and highly beneficial for applications in “mixed” delay SDEs, encompassing point time delays, varying time delays, and distributed time delays.

Corollary 1.

Let ${\Theta }_1\left(·,·\right),{\Theta }_2\left(·,·\right):{\mathbb{R}}_{+}\times \left[-h,0\right]\to {\mathbb{R}}_{+}.{\gamma }_i\left(·\right),\zeta \left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i=0,1,2,\cdots ,n$ and $h_i\left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i=0,1,2,\cdots ,n$, be Borel-measurable functions that are locally bounded, with $t\in {\mathbb{R}}_{+}$, $0:=h_0\left(t\right)\le h_1\left(t\right)\le h_2\left(t\right)\le \cdots \le h_n\left(t\right)\le h$. Suppose that the following assumptions hold for each $t\in {\mathbb{R}}_{+},\phi \in \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$,

$$\phi {\left(0\right)}^{T}f\left(\phi ,t\right)\le \sum _{i=0}^n{\gamma }_i\left(t\right){∥\phi \left(-h_i\left(t\right)\right)∥}^2+{\int }_{-h}^0{\Theta }_1\left(t,s\right){∥\phi \left(s\right)∥}^{2}ds$$

(18)

and

$${\left({∥g\left(\phi ,t\right)∥}_F\right)}^2\le \sum _{i=0}^n{\zeta }_i\left(t\right){∥\phi \left(-h_i\left(t\right)\right)∥}^2+{\int }_{-h}^0{\Theta }_2\left(t,s\right){∥\phi \left(s\right)∥}^{2}ds.$$

(19)

If, for $t\in {\mathbb{R}}_{+},\phi \in \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$, and one can find $c_1\left(u\right)>0$ such that

$$\begin{array}{cc}\hfill -c_1\left(u\right)\ge & 2\left(\sum _{i=0}^n{e}^{-{\int }_{\theta }^{\theta -h_i\left(t\right)}{c}_1\left(u\right)du}{\gamma }_i\left(t\right)+{\int }_{-h}^0{e}^{-{\int }_{\theta }^{\theta +s}{c}_1\left(u\right)du}{\Theta }_1\left(t,s\right)ds\right)\hfill \\ \hfill & +\sum _{i=0}^n{e}^{-{\int }_{\theta }^{\theta -h_i\left(t\right)}{c}_1\left(u\right)du}{\zeta }_i\left(t\right)+{\int }_{-h}^0{e}^{-{\int }_{\theta }^{\theta +s}{c}_1\left(u\right)du}{\Theta }_2\left(t,s\right)ds,\hfill \end{array}$$

(20)

then the trivial solution of Equation $\left(1\right)$ is mean square generalized exponentially stable, with the rate of decline given by ${\int }_{t_0}^t{c}_1\left(u\right)du$ for any $\xi \in {\mathit{C}}_{\mathcal{F}}^{\mathit{b}}{\mathrm{t}}_0\left(\left[-h,0\right];{\mathbb{R}}^n\right)$.

Proof. 

Consider the following real-valued functions for $t\ge 0,-h\le s\le 0$

$$u_i\left(t,s\right):=\left\{\begin{array}{cc}0\hfill & \mathrm{if}s\in \left[-h,-h_i\left(t\right)\right],\hfill \\ {\gamma }_i\left(t\right)\hfill & \mathrm{if}s\in \left(-h_i\left(t\right),0\right],\hfill \end{array}\right.$$

$${\eta }_1\left(t,s\right):=\sum _{i=1}^n{u}_i\left(t,s\right)+{\int }_{-h}^s{\Theta }_1\left(t,r\right)dr,$$

(21)

and

$$v_i\left(t,s\right):=\left\{\begin{array}{cc}0\hfill & \mathrm{if}s\in \left[-h,-h_i\left(t\right)\right],\hfill \\ {\zeta }_i\left(t\right)\hfill & \mathrm{if}s\in \left(-h_i\left(t\right),0\right],\hfill \end{array}\right.$$

$${\eta }_2\left(t,s\right):=\sum _{i=0}^n{v}_i\left(t,s\right)+{\int }_{-h}^s{\Theta }_2\left(t,r\right)dr.$$

(22)

We can derive the following by applying the properties of Riemann–Stieltjes integrals [37],

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\int }_{-h}^s{\Theta }_i\left(t,r\right)dr\right]={\int }_{-h}^0\varphi \left(s\right){\Theta }_i\left(t,s\right)ds,$$

(23)

for any $\varphi \left(·\right)\in \mathcal{C}$, $t\in {\mathbb{R}}_{+}$, $i=1,2.$ From $\left(21\right)$, $\left(22\right)$ and $\left(23\right)$, we have

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\eta }_1\left(t,s\right)\right]=\sum _{i=1}^n{\gamma }_i\left(t\right)\varphi \left(-h_i\left(t\right)\right)+{\int }_{-h}^0\varphi \left(s\right){\Theta }_1\left(t,s\right)ds$$

(24)

and

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\eta }_2\left(t,s\right)\right]=\sum _{i=1}^n{\zeta }_i\left(t\right)\varphi \left(-h_i\left(t\right)\right)+{\int }_{-h}^0\varphi \left(s\right){\Theta }_2\left(t,s\right)ds,$$

(25)

for any $\varphi \left(·\right)\in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$, $t\in {\mathbb{R}}_{+}$. Thus, $\left(18\right)$ and $\left(24\right)$ mean that $\left(7\right)$ holds; $\left(19\right)$ and $\left(25\right)$ mean that $\left(8\right)$ holds; and $\left(20\right)$, $\left(24\right)$ and $\left(25\right)$ mean that $\left(9\right)$ holds. Thus, this proof is simplified to Theorem 1. The deduction comes from Theorem 1. □

Corollary 2.

Let ${\eta }_i:=\left[-h,0\right]\to {\mathbb{R}}_{+}$ be non-decreasing functions and one can find a constant γ. Suppose that the following assumptions hold for any $t\in {\mathbb{R}}_{+},\phi \in \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$

$$\phi {\left(0\right)}^{T}f\left(\phi ,t\right)\le \gamma {∥\phi \left(0\right)∥}^2+{\int }_{-h}^0{∥\phi \left(s\right)∥}^{2}d\left[{\eta }_1\left(s\right)\right],$$

(26)

$${\left({∥g\left(\phi ,t\right)∥}_F\right)}^2\le {\int }_{-h}^0{∥\phi \left(s\right)∥}^{2}d\left[{\eta }_2\left(s\right)\right]$$

(27)

and

$$2\gamma +2\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]<0.$$

(28)

Thus, the trivial solution of Equation $\left(1\right)$ is mean square generalized exponentially stable.

Proof. 

Obviously, due to continuity, Equation $\left(28\right)$ can be rewritten as

$$2\gamma +2e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]<-c_1\left(u\right),$$

for $c_1\left(u\right)>0$ and a small enough value. With ${\eta }_i\left(·\right)$ being non-decreasing functions, we obtain

$$\begin{array}{cc}\hfill & 2\gamma +2{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1\left(\theta \right)\right]+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2\left(\theta \right)\right]\hfill \\ \hfill & \le 2\gamma +2e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]\hfill \\ \hfill & <-c_1\left(u\right).\hfill \end{array}$$

Hence, $\left(28\right)$ ensures that $\left(9\right)$ holds. Thus, this proof is deduced to Theorem 1. This demonstration has been completed. □

Based on the above inference, we directly derive the following Corollary 3.

Corollary 3.

Let $h_i\left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i=0,1,2,3,\cdots ,n$, be Borel-measurable functions that are locally bounded, with $t\in {\mathbb{R}}_{+}$, $0:=h_0\left(t\right)\le h_1\left(t\right)\le h_2\left(t\right)\le \cdots \le h_n\left(t\right)\le h$. Assume that one can find constants ${\gamma }_i$, ${\zeta }_i$, $i=0,1,2,3,\cdots ,n$, and ${\theta }_i:\left[-h,0\right]\to {\mathbb{R}}_{+}$ are Borel-measurable functions for $i=1,2$, such that, for each $t\in {\mathbb{R}}_{+}$, $\phi \in \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$,

$$\phi {\left(0\right)}^{T}f\left(\phi ,t\right)\le \sum _{i=0}^n{\gamma }_i{∥\phi \left(-h_i\left(t\right)\right)∥}^2+{\int }_{-h}^0{\theta }_1\left(s\right){∥\phi \left(s\right)∥}^{2}ds,$$

(29)

$${\left({∥g\left(\phi ,t\right)∥}_F\right)}^2\le \sum _{i=0}^n{\zeta }_i{∥\phi \left(-h_i\left(t\right)\right)∥}^2+{\int }_{-h}^0{\theta }_2\left(s\right){∥\phi \left(s\right)∥}^{2}ds$$

(30)

and

$$2\left(\sum _{i=0}^n{\gamma }_i+{\int }_{-h}^0{\theta }_1\left(s\right)ds\right)+\sum _{i=0}^n{\zeta }_i+{\int }_{-h}^0{\theta }_2\left(s\right)ds<0.$$

(31)

Thus, the trivial solution of Equation $\left(1\right)$ is mean square generalized exponentially stable.

4. Exponential Stability for Neutral Stochastic Functional Equations

In this section, we consider the following neutral stochastic functional differential equation,

$$\left\{\begin{array}{cc}& d\left[x\left(t\right)-G\left(x_t\right)\right]=f\left(t,x_t\right)dt+g\left(t,x_t\right)dw\left(t\right),t\ge 0,\hfill \\ & x_0=\xi ,\hfill \end{array}\right.$$

(32)

where $x_t=\left\{x\left(t+\theta \right):\theta \in \left[-h,0\right]\right\}$, which is treated as a $\mathit{C}\left([-h,0];{\mathbb{R}}^n\right)$-valued random process and $\xi =\left\{\xi \left(s\right):-h\le s\le 0\right\}\in C_{{\mathcal{F}}_0}^b\left(\left[-h,0\right],{\mathbb{R}}^n\right)$ and $G:C\left(\left[-h,0\right];{\mathbb{R}}^n\right)\to {\mathbb{R}}^n$, $f:\mathcal{C}\times {\mathbb{R}}_{+}\to {\mathbb{R}}^n$, $g:\mathcal{C}\times {\mathbb{R}}_{+}\to {\mathbb{R}}^{n\times m}$. We can refer to [38] for details about the existence and uniqueness of the solution of $\left(32\right)$. The functions f, g and G fulfill the local Lipschitz criterion with Lipschitz constant ${\mathit{L}}_n>0$ and adhere to the linear growth requirement. Further, the functions G fulfill Lipschitz constant $L_n<1$. We can find the only solution of Equation $\left(32\right)$, as well as the solution remaining finite at all moments.

Definition 5.

A solution of $\left(32\right)$ is a continuous ${\mathcal{F}}_t$-adapted process $x\left(t\right)$, $t\ge -h$, if it meets the requirement stated above, such that

$$x\left(t\right)-G\left(x_t\right)=\xi \left(0\right)-G\left(x_0\right)+{\int }_0^{t}f\left(s,x_s\right)ds+{\int }_0^{t}g\left(s,x_s\right)dw\left(s\right),$$

(33)

for any $t\ge 0$. Suppose that $G\left(0\right)\equiv 0$, $f\left(t,0\right)\equiv 0$, $g\left(t,0\right)\equiv 0$ for any $t>0$; Equation $\left(32\right)$ has a zero solution $x\left(t\right)=0$.

Definition 6.

The zero solution of Equation $\left(32\right)$ is mean square generalized exponentially stable if, for every $t\ge 0$ and each $\xi \in {\mathit{C}}_{\mathcal{F}}^{\mathit{b}}{\mathrm{t}}_0\left([-h,0];{\mathbb{R}}^n\right)$, one can find a constant K and an integrable function $c_1\left(u\right)\ge 0$ such that the following condition is satisfied:

$$\underset{t\to \infty }{lim}{\int }_0^t{c}_1\left(u\right)du\to +\infty$$

(34)

and

$$\mathbb{E}∥x\left(t,\xi \right)∥\le K{e}^{-{\int }_0^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,$$

(35)

where $-{\int }_0^t{c}_1\left(u\right)du$ is the common decay rate.

Moreover, suppose that one can find a constant $0<k<1$ such that, for each $\phi \in L_{\mathcal{F}}^2\left(\left[-h,0\right];{\mathbb{R}}^n\right)$,

$${\left|G\left(\phi \right)\right|}^2\le k\underset{-h\le \theta \le 0}{sup}{\left|\phi \left(\theta \right)\right|}^2.$$

(36)

Lemma 1.

Assume that Equation $\left(36\right)$ holds with $k\in \left(0,1\right)$, $\lambda \in \left[0,\infty \right)$, $K\in \left(1,\infty \right)$ and an integrable function $c\left(u\right)\ge 0$,

$$\underset{t\to \infty }{lim}{\int }_0^{t}c\left(u\right)du\to +\infty .$$

If

$$e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2\le K\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2$$

(37)

for each $t\in \left[0,\lambda \right]$,

$$e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2\le {\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2.$$

Proof. 

Assume $\epsilon \in \left(k,1\right)$. For each $t\in \left[0,\lambda \right]$, through Young‘s Inequality, we derive

$$\begin{array}{cc}\hfill \mathbb{E}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2& \ge \left(1-\epsilon \right)\mathbb{E}{\left|x\left(t\right)\right|}^2-\left({\epsilon }^{-1}-1\right)\mathbb{E}{\left|G\left(x_t\right)\right|}^2.\hfill \end{array}$$

Then, by $\left(36\right)$, we obtain

$$\mathbb{E}{\left|x\left(t\right)\right|}^2\le {\displaystyle \frac{1}{1-\epsilon }}\mathbb{E}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2+{\displaystyle \frac{k}{\epsilon }}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(t+\theta \right)\right|}^2.$$

From $\left(37\right)$, we have

$$\begin{array}{cc}\hfill e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2& \le {\displaystyle \frac{K}{1-\epsilon }}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2+{\displaystyle \frac{k}{\epsilon }}\underset{-h\le \theta \le 0}{sup}\left[e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t+\theta \right)\right|}^2\right]\hfill \\ \hfill & \le {\displaystyle \frac{K}{1-\epsilon }}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2+{\displaystyle \frac{k}{\epsilon }}\underset{-h\le t\le \lambda }{sup}\left[e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2\right],\hfill \end{array}$$

for all $t\in \left[0,\lambda \right]$. Additionally, for $t\in \left[-h,0\right]$, this is also true. Then,

$$\underset{-h\le t\le \lambda }{sup}\left[e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2\right]\le {\displaystyle \frac{K}{1-\epsilon }}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2+{\displaystyle \frac{k}{\epsilon }}\underset{-h\le t\le \lambda }{sup}\left[e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2\right].$$

Since $k<\epsilon <1$, we obtain

$$\underset{-h\le t\le \lambda }{sup}\left[e^{{\int }_0^{t}c\left(u\right)du}\mathbb{E}{\left|x\left(t\right)\right|}^2\right]\le {\displaystyle \frac{K\epsilon }{\left(1-\epsilon \right)\left(\epsilon -k\right)}}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2.$$

Lastly, by setting $\epsilon =\sqrt{k}$, we arrive at the expected result. This demonstration has been completed. □

Theorem 2.

Let $\gamma \left(·\right):{\mathbb{R}}_{+}\to \mathbb{R}$ be a Borel-measurable function that is locally bounded. Suppose that $\left(36\right)$ holds and $k\in \left(0,1\right)$, and the following assumptions hold for each $t\in {\mathbb{R}}_{+},\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$:

$$\left(2{\left(\phi \left(0\right)-G\left(\phi \right)\right)}^{T}f\left(\phi ,t\right)\right)\le \gamma \left(t\right){\left|\phi \left(0\right)\right|}^2+{\int }_{-h}^0{\left|\phi \left(\theta \right)\right|}^{2}d\left[{\eta }_1\left(t,\theta \right)\right]$$

(38)

and

$$\left(trace\left[g^T\left(t,\phi \right)g\left(t,\phi \right)\right]\right)\le {\int }_{-h}^0{\left|\phi \left(\theta \right)\right|}^{2}d\left[{\eta }_2\left(t,\theta \right)\right].$$

(39)

If $t\in {\mathbb{R}}_{+}$ and one can find $c_1\left(u\right)>0$ such that

$$\gamma \left(t\right)+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1\left(t,\theta \right)\right]+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2\left(t,\theta \right)\right]\le -{\left(1-\sqrt{k}\right)}^2{c}_1\left(u\right),$$

(40)

then the zero solution of Equation $\left(32\right)$ is mean square generalized exponentially stable. In particular, the decay rate is ${\int }_0^t{c}_1\left(u\right)du$ for each $\xi \in C_{{\mathcal{F}}_0}^b\left(\left[-h,0\right];{\mathbb{R}}^n\right)$.

Proof. 

Suppose that $\xi \in C_{{\mathcal{F}}_0}^b\left(\left[-h,0\right];{\mathbb{R}}^n\right)$ such that $\mathbb{E}{∥\xi ∥}_C^2>0$. Equation $\left(32\right)$ has a solution $x\left(t\right),-h\le t$. Let $X\left(t\right):=\mathbb{E}{\left|x\left(t\right)-G\left(t,x_t\right)\right|}^2$ and $Z\left(t\right):=K{e}^{-{\int }_0^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2$, $t\ge -h$. Thus, ensuring that $K>1$ and is sufficiently large, we derive

$$X\left(t\right)\le Z\left(t\right),t\in \left[-h,0\right].$$

We demonstrate that, when any $t\ge 0$,

$$X\left(t\right)\le Z\left(t\right).$$

Assume that one can find $t_1>0$ such that

$$X\left(t_1\right)>Z\left(t_1\right).$$

(41)

Define $t_{*}:=inf\left\{t>0:X\left(t\right)>Z\left(t\right)\right\}$. Obviously, because $X\left(t\right)$ and $Z\left(t\right)$ are continuous functions,

$$X\left(t\right)\le Z\left(t\right),t\in \left[0,t_{*}\right],X\left(t_{*}\right)=K{e}^{-{\int }_0^{t_{*}}{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2$$

(42)

and

$$\mathbb{E}{\left|x\left(t_n\right)-G\left(x_{t_n}\right)\right|}^2>K{e}^{-{\int }_0^{t_n}{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t_n\in \left(t_{*},t_{*}+{\displaystyle \frac{1}{n}}\right),n\in \mathbb{N}.$$

(43)

Consider the following real-valued function for $x\in {\mathbb{R}}^n$,

$$V\left(x,t\right)=e^{{\int }_0^t{c}_2\left(u\right)du}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2.$$

(44)

Utilizing Itô’s formula in $\left(44\right)$, we have

$$\begin{array}{cc}\hfill e^{{\int }_0^t{c}_2\left(u\right)du}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2=& {\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2+{\int }_0^t{c}_2\left(s\right)e^{{\int }_0^s{c}_2\left(u\right)du}{\left|x\left(s\right)-G\left(x_s\right)\right|}^{2}ds\hfill \\ \hfill & +2{\int }_0^t{e}^{{\int }_0^s{c}_2\left(u\right)du}{\left(x\left(s\right)-G\left(x_s\right)\right)}^{T}f\left(x_s,s\right)ds\hfill \\ \hfill & +{\int }_0^t{e}^{{\int }_0^s{c}_2\left(u\right)du}trace\left[g^T\left(x_s,s\right)g\left(x_s,s\right)\right]ds\hfill \\ \hfill & +{\int }_0^t{V}_x(x\left(s\right),s)g\left(x_s,s\right)dB\left(s\right).\hfill \end{array}$$

From $\left(38\right)$, $\left(39\right)$ and the property of Brownian motion, we obtain

$$\begin{array}{cc}\hfill \mathbb{E}\left(e^{{\int }_0^t{c}_2\left(u\right)du}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2\right)\le & \mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2+\mathbb{E}{\int }_0^t{c}_2\left(s\right)e^{{\int }_0^s{c}_2\left(u\right)du}{\left|x\left(s\right)-G\left(x_s\right)\right|}^{2}ds\hfill \\ \hfill & +{\int }_0^t\gamma \left(s\right)e^{{\int }_0^s{c}_2\left(u\right)du}\mathbb{E}{\left|x\left(s\right)\right|}^{2}ds\hfill \\ \hfill & +{\int }_0^t{e}^{{\int }_0^s{c}_2\left(u\right)du}\left({\int }_{-h}^0\mathbb{E}{\left|x\left(s+\theta \right)\right|}^{2}d\left[{\eta }_1\left(s,\theta \right)\right]\right)ds\hfill \\ \hfill & +{\int }_0^t{e}^{{\int }_0^s{c}_2\left(u\right)du}\left({\int }_{-h}^0\mathbb{E}{\left|x\left(s+\theta \right)\right|}^{2}d\left[{\eta }_2\left(s,\theta \right)\right]\right)ds.\hfill \end{array}$$

Since ${\eta }_1\left(s,\theta \right)$ and ${\eta }_2\left(s,\theta \right)$ are non-decreasing for any $s\le t_{*}$, $\theta \in \left[-h,0\right]$, along with Equations $\left(42\right)$ and $\left(37\right)$, we derive

$${\int }_{-h}^0\mathbb{E}{\left|x(s+\theta )\right|}^{2}d\left[{\eta }_1(s,\theta )\right]\le {\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\mathbb{E}{∥\xi ∥}_C^2{e}^{-{\int }_0^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1(s,\theta )\right]$$

(45)

and

$${\int }_{-h}^0\mathbb{E}{\left|x(s+\theta )\right|}^{2}d\left[{\eta }_2(s,\theta )\right]\le {\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\mathbb{E}{∥\xi ∥}_C^2{e}^{-{\int }_0^s{c}_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2(s,\theta )\right].$$

(46)

From $\left(45\right)$ and $\left(46\right)$, we have

$$\begin{array}{cc}\hfill e^{{\int }_0^{t_{*}}{c}_2\left(u\right)du}& \mathbb{E}{\left|x\left(t_{*}\right)-G\left(x_{t_{*}}\right)\right|}^2\le \mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2\hfill \\ \hfill & +K\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)du}{e}^{-{\int }_0^s{c}_1\left(u\right)du}\left(c_2\left(s\right)+{\displaystyle \frac{\gamma \left(s\right)}{{\left(1-\sqrt{k}\right)}^2}}\right)ds\hfill \\ \hfill & +{\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)-c_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1(s,\theta )\right]ds\hfill \\ \hfill & +{\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)-c_1\left(u\right)du}{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2(s,\theta )\right]ds\hfill \\ \hfill & =\mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2+K\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)-c_1\left(u\right)du}{c}_2\left(s\right)ds\hfill \\ \hfill & +{\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)-c_1\left(u\right)du}·\hfill \\ \hfill & \left[\gamma \left(s\right)+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1\left(s,\theta \right)\right]+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2\left(s,\theta \right)\right]\right]ds.\hfill \\ \end{array}$$

From condition $\left(40\right)$, we have

$$\begin{array}{cc}\hfill e^{{\int }_0^{t_{*}}{c}_2\left(u\right)du}\mathbb{E}{\left|x\left(t_{*}\right)-G\left(x_{t_{*}}\right)\right|}^2\le & \mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2\hfill \\ \hfill & +K\mathbb{E}{∥\xi ∥}_C^2{\int }_0^{t_{*}}{e}^{{\int }_0^s{c}_2\left(u\right)-c_1\left(u\right)du}\left(c_2\left(s\right)-c_1\left(s\right)\right)ds\hfill \\ \hfill & =\mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2+K\mathbb{E}{∥\xi ∥}_C^2\left(e^{{\int }_0^{t_{*}}{c}_2\left(u\right)-c_1\left(u\right)}-1\right)\hfill \\ \hfill & =\mathbb{E}{\left|\xi \left(0\right)-G\left(\xi \right)\right|}^2-K\mathbb{E}{∥\xi ∥}_C^2\hfill \\ \hfill & +K\mathbb{E}{∥\xi ∥}_C^2{e}^{{\int }_0^{t_{*}}{c}_2\left(u\right)du}{e}^{-{\int }_0^{t_{*}}{c}_1\left(u\right)du}\hfill \\ \hfill & <K\mathbb{E}{∥\xi ∥}_C^2{e}^{{\int }_0^{t_{*}}{c}_2\left(u\right)du}{e}^{-{\int }_0^{t_{*}}{c}_1\left(u\right)du},\hfill \end{array}$$

which is a contradiction of $\left(42\right)$. This means that, for $t\ge 0$,

$$\mathbb{E}\left|X\left(t\right)\right|\le K{e}^{-{\int }_0^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,$$

(47)

i.e.,

$$\mathbb{E}{\left|x\left(t\right)\right|}^2\le {\displaystyle \frac{K}{{\left(1-\sqrt{k}\right)}^2}}{e}^{-{\int }_0^t{c}_1\left(u\right)du}\mathbb{E}{∥\xi ∥}_C^2,t\ge 0.$$

(48)

This demonstration has been completed. □

Corollary 4.

Let ${\Theta }_1\left(·,·\right),{\Theta }_2\left(·,·\right):{\mathbb{R}}_{+}\times \left[-h,0\right]\to {\mathbb{R}}_{+}.{\gamma }_i\left(·\right),\zeta \left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i=0,1,2,\cdots ,n$ and $h_i\left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i=0,1,2,\cdots ,n$, be Borel-measurable functions that are locally bounded, with $0:=h_0\left(t\right)\le h_1\left(t\right)\le h_2\left(t\right)\le \cdots \le h_n\left(t\right)\le h$, $t\in {\mathbb{R}}_{+}$. Suppose that the following conditions are satisfied for each $t\in {\mathbb{R}}_{+}$, $\phi \in \mathit{C}\left(\left[-h,0\right];{\mathbb{R}}^n\right)$:

$$\mathbb{E}\left(2{\left(\phi \left(0\right)-G\left(\phi \right)\right)}^{T}f\left(\phi ,t\right)\right)\le \sum _{i=0}^n{\gamma }_i\left(t\right)\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\Theta }_1\left(t,s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds$$

(49)

and

$$\mathbb{E}\left(trace\left[g^T\left(\phi ,t\right)g\left(\phi ,t\right)\right]\right)\le \sum _{i=0}^n{\zeta }_i\left(t\right)\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\Theta }_2\left(t,s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds.$$

(50)

If one can find $c_1\left(u\right)>0$ such that, for $t\in {\mathbb{R}}_{+}$,

$$\begin{array}{cc}\hfill -{\left(1-\sqrt{k}\right)}^2{c}_1\left(u\right)\ge & \sum _{i=0}^n{e}^{-{\int }_{\theta }^{\theta -h_i\left(t\right)}{c}_1\left(u\right)du}{\gamma }_i\left(t\right)+{\int }_{-h}^0{e}^{-{\int }_{\theta }^{\theta +s}{c}_1\left(u\right)du}{\Theta }_1\left(t,s\right)ds\hfill \\ \hfill & +\sum _{i=0}^n{e}^{-{\int }_{\theta }^{\theta -h_i\left(t\right)}{c}_1\left(u\right)du}{\zeta }_i\left(t\right)+{\int }_{-h}^0{e}^{-{\int }_{\theta }^{\theta +s}{c}_1\left(u\right)du}{\Theta }_2\left(t,s\right)ds,\hfill \end{array}$$

(51)

then the trivial solution of Equation $\left(32\right)$ is mean square generalized exponentially stable, and the rate of decline is given by ${\int }_{t_0}^t{c}_1\left(u\right)du$ for any $\xi \in C_{{\mathcal{F}}_0}^b\left(\left[-h,0\right];{\mathbb{R}}^n\right)$.

Proof. 

Consider the following real-valued functions for $t\ge 0$, $-h\le s\le 0$:

$$u_i\left(t,s\right):=\left\{\begin{array}{cc}0\hfill & \mathrm{if}s\in \left[-h,-h_i\left(t\right)\right],\hfill \\ {\gamma }_i\left(t\right)\hfill & \mathrm{if}s\in \left(-h_i\left(t\right),0\right],\hfill \end{array}\right.$$

$${\eta }_1\left(t,s\right):=\sum _{i=1}^n{u}_i\left(t,s\right)+{\int }_{-h}^s{\Theta }_1\left(t,r\right)dr,$$

(52)

and

$$v_i\left(t,s\right):=\left\{\begin{array}{cc}0\hfill & \mathrm{if}s\in \left[-h,-h_i\left(t\right)\right],\hfill \\ {\zeta }_i\left(t\right)\hfill & \mathrm{if}s\in \left(-h_i\left(t\right),0\right],\hfill \end{array}\right.$$

$${\eta }_2\left(t,s\right):=\sum _{i=0}^n{v}_i\left(t,s\right)+{\int }_{-h}^s{\Theta }_2\left(t,r\right)dr.$$

(53)

By applying the properties of the Riemann–Stiemann integrals [37], we have

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\int }_{-h}^s{\Theta }_i\left(t,r\right)dr\right]={\int }_{-h}^0\varphi \left(s\right){\Theta }_i\left(t,s\right)ds,$$

(54)

for each $i=1,2$ and any $\varphi \left(·\right)\in \mathcal{C}$, $t\in {\mathbb{R}}_{+}$. Thus, from $\left(52\right)$, $\left(53\right)$ and $\left(54\right)$, we have

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\eta }_1\left(t,s\right)\right]=\sum _{i=1}^n{\gamma }_i\left(t\right)\varphi \left(-h_i\left(t\right)\right)+{\int }_{-h}^0\varphi \left(s\right){\Theta }_1\left(t,s\right)ds$$

(55)

and

$${\int }_{-h}^0\varphi \left(s\right)d\left[{\eta }_2\left(t,s\right)\right]=\sum _{i=0}^n{\zeta }_i\left(t\right)\varphi \left(-h_i\left(t\right)\right)+{\int }_{-h}^0\varphi \left(s\right){\Theta }_2\left(t,s\right)ds.$$

(56)

Therefore, $\left(49\right)$ and $\left(55\right)$ mean that $\left(38\right)$ holds; $\left(50\right)$ and $\left(56\right)$ mean that $\left(39\right)$ holds; and $\left(51\right)$, $\left(55\right)$ and $\left(56\right)$ mean that $\left(40\right)$ holds. Thus, this proof is simplified to Theorem 2. The deduction comes from Theorem 2. □

Corollary 5.

Suppose that $\left(36\right)$ holds with $k\in \left(0,1\right)$. Let ${\eta }_i:=\left[-h,0\right]\to {\mathbb{R}}_{+}$ be non-decreasing functions and one can find a constant γ. Suppose that the following assumptions hold for any $t\in {\mathbb{R}}_{+},\phi \in \mathcal{C}$:

$$\mathbb{E}{\left(2\phi \left(0\right)-G\left(\phi \right)\right)}^{T}f\left(\phi ,t\right)\le \gamma {\left|\phi \left(0\right)\right|}^2+{\int }_{-h}^0{\left|\phi \left(s\right)\right|}^{2}d\left[{\eta }_1\left(s\right)\right]$$

(57)

and

$$\mathbb{E}\left(trace\left[g^T\left(\phi ,t\right)g\left(\phi ,t\right)\right]\right)\le {\int }_{-h}^0{\left|\phi \left(s\right)\right|}^{2}d\left[{\eta }_2\left(s\right)\right],$$

(58)

$$\gamma +\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]<0.$$

(59)

Thus, the zero solution of Equation $\left(32\right)$ is generalized exponentially stable in the mean square.

Proof. 

Obviously, due to continuity, Equation $\left(59\right)$ can be rewritten as

$$\gamma +e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]<-{\left(1-\sqrt{k}\right)}^2{c}_1\left(u\right),$$

for $c_1\left(u\right)>0$ and a small enough value. With ${\eta }_i\left(·\right)$ being non-decreasing functions, we obtain

$$\begin{array}{cc}\hfill & \gamma +{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_1\left(\theta \right)\right]+{\int }_{-h}^0{e}^{-{\int }_s^{s+\theta }{c}_1\left(u\right)du}d\left[{\eta }_2\left(\theta \right)\right]\hfill \\ \hfill & \le \gamma +e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_1\left(0\right)-{\eta }_1\left(-h\right)\right]+e^{{\int }_{s-h}^s{c}_1\left(u\right)du}\left[{\eta }_2\left(0\right)-{\eta }_2\left(-h\right)\right]\hfill \\ \\ \hfill & <-{\left(1-\sqrt{k}\right)}^2{c}_1\left(u\right).\hfill \end{array}$$

Hence, $\left(59\right)$ ensures that $\left(40\right)$ holds. The proof is deduced to Theorem 2. The demonstration has been completed. □

Based on the above inference, we directly derive the following Corollary 6.

Corollary 6.

Suppose that $\left(36\right)$ holds with $k\in \left(0,1\right)$. Let $h_i\left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i\in {\underline{n}}_0$, be Borel-measurable functions that are locally bounded, with $t\in {\mathbb{R}}_{+}$, $0:=h_0\left(t\right)\le h_1\left(t\right)\le h_2\left(t\right)\le \cdots \le h_n\left(t\right)\le h$. Assume that one can find constants ${\gamma }_i$, ${\zeta }_i$, $i\in {\underline{n}}_0$, and ${\theta }_i:\left[-h,0\right]\to {\mathbb{R}}_{+}$ are Borel-measurable functions for $i=1,2$, such that

$$\mathbb{E}{\left(2\phi \left(0\right)-G\left(\phi \right)\right)}^{T}f\left(\phi ,t\right)\le \sum _{i=0}^n{\gamma }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_1\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds,$$

(60)

$$\mathbb{E}\left(trace\left[g^T\left(\phi ,t\right)g\left(\phi ,t\right)\right]\right)\le \sum _{i=0}^n{\zeta }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_2\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds,$$

(61)

for each $t\in {\mathbb{R}}_{+}$, $\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$, and

$$\sum _{i=0}^n{\gamma }_i+{\int }_{-h}^0{\theta }_1\left(s\right)ds+\sum _{i=0}^n{\zeta }_i+{\int }_{-h}^0{\theta }_2\left(s\right)ds<0.$$

(62)

Thus, the zero solution of Equation $\left(32\right)$ is generalized exponentially stable in the mean square.

Corollary 7.

Assume that $\left(36\right)$ holds with $0<k<1$. Let $h_i\left(·\right):{\mathbb{R}}_{+}\to \mathbb{R},i\in {\underline{n}}_0$, be Borel-measurable functions that are locally bounded, with $t\in {\mathbb{R}}_{+}$, $0:=h_0\left(t\right)\le h_1\left(t\right)\le h_2\left(t\right)\le \cdots \le h_n\left(t\right)\le h$. Assume that one can find constants ${\gamma }_i$, ${\zeta }_i$, ${\rho }_i$, $i\in {\underline{n}}_0$, and ${\theta }_i:\left[-h,0\right]\to {\mathbb{R}}_{+}$ are Borel-measurable functions for $i=1,2,3$, such that

$$\mathbb{E}{\left(2\phi \left(0\right)\right)}^{T}f\left(\phi ,t\right)\le \sum _{i=0}^n{\gamma }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_1\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds,$$

(63)

$$\mathbb{E}{\left|f\left(\phi ,t\right)\right|}^2\le \sum _{i=0}^n{\rho }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_2\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds,$$

(64)

$$\mathbb{E}\left(trace\left[g^T\left(\phi ,t\right)g\left(\phi ,t\right)\right]\right)\le \sum _{i=0}^n{\zeta }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_3\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds,$$

(65)

for any $t\in {\mathbb{R}}_{+}$, $\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$, and

$$\sum _{i=0}^n{\gamma }_i+{\int }_{-h}^0{\theta }_1\left(s\right)ds+\sum _{i=0}^n{\zeta }_i+{\int }_{-h}^0{\theta }_3\left(s\right)ds+\sqrt{k}\sum _{i=0}^n{\rho }_i+\sqrt{k}{\int }_{-h}^0{\theta }_2\left(s\right)ds+\sqrt{k}<0.$$

(66)

Therefore, the trivial solution of Equation $\left(32\right)$ is generalized exponentially stable in the mean square.

Proof. 

From $\left(36\right)$, $\left(63\right)$ and $\left(64\right)$, we have

$$\begin{array}{cc}\hfill \mathbb{E}\left({\left(2\phi \left(0\right)-G\left(\phi \right)\right)}^{T}f\left(\phi ,t\right)\right)& \le 2\mathbb{E}\left({\phi }^T\left(0\right)f\left(\phi ,t\right)+\left|G\left(\phi \right)\right|\left|f\left(\phi ,t\right)\right|\right)\hfill \\ \hfill & \le \sum _{i=0}^n{\gamma }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+{\int }_{-h}^0{\theta }_1\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\sqrt{k}\sum _{i=0}^n{\rho }_i\mathbb{E}{\left|\phi \left(-h_i\left(t\right)\right)\right|}^2+\sqrt{k}{\int }_{-h}^0{\theta }_2\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds\hfill \\ \hfill & +\sqrt{k}\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|\phi \left(\theta \right)\right|}^2.\hfill \end{array}$$

Therefore, $\left(63\right)$, $\left(64\right)$, $\left(65\right)$ and $\left(66\right)$ guarantee that $\left(60\right)$, $\left(61\right)$ and $\left(62\right)$ hold. From Corollary 6, we can achieve the results that we seek. □

5. Comparison to Known Results

In this part, we compare the existing results with those given in this paper. At the same time, we provide three examples to demonstrate the effectiveness of the results that we have obtained.

We compare the results of Theorem 2 with those in [39]. In [39], the author finds that the zero solution of Equation $\left(32\right)$ is mean square exponentially stable if one can find some constants $K>1$, $\delta >0$ and $k\in \left(0,1\right)$, such that

$$e^{\delta t}\mathbb{E}{\left|x\left(t\right)-G\left(x_t\right)\right|}^2\le K\underset{-h\le \theta \le 0}{sup}\mathbb{E}{\left|x\left(\theta \right)\right|}^2.$$

(67)

Moreover, one can find $\beta >0$ such that

$$\gamma \left(t\right)+{\int }_{-h}^0{e}^{-\beta \theta }d\left[{\eta }_1\left(t,\theta \right)\right]+{\int }_{-h}^0{e}^{-\beta \theta }d\left[{\eta }_2\left(t,\theta \right)\right]\le -{\left(1-\sqrt{k}\right)}^2\beta ,$$

(68)

for $t\in {\mathbb{R}}_{+}$ and $\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^d\right)$. In this paper, assuming that Theorem 2 holds and

$$c_1\left(u\right)\equiv \beta ,{\int }_0^t\beta du=\beta t,$$

(69)

we can obtain the same result. This means that our hypothesis is weaker. Our conditions are more suitable than $\left(67\right)$ and $\left(68\right)$. This paper extends the results of [39].

Moreover, the exponential stability of SDEs is researched in [36]. In [36], the author has shown that the trivial solution of Equation $\left(1\right)$ is mean square exponentially stable if one can find $\beta >0$ such that

$$2\gamma \left(t\right)+2{\int }_{-h}^0{e}^{-\beta \theta }d\left[{\eta }_1\left(t,\theta \right)\right]+{\int }_{-h}^0{e}^{-\beta \theta }d\left[{\eta }_2\left(t,\theta \right)\right]\le -\beta .$$

(70)

for any $t\in {\mathbb{R}}_{+}$. In this paper, if $c_1\left(u\right)=\beta$, we can obtain the same result via our Theorem 1. This indicates that the findings of this work enhance and generalize the findings in [36].

Remark 3.

When we use $\beta t$, $\beta ln \left(t+1\right)$ and $\beta ln \left(ln \left(t+e\right)\right)$ instead of ${\int }_0^t{c}_1\left(u\right)du$, respectively, then Equation $\left(32\right)$ is stable in the mean square, exhibiting exponential, polynomial, and logarithmic stability.

We now present three examples to demonstrate the effectiveness of the findings of this work.

Example 1.

Consider the neutral SDE

$$d\left[x\left(t\right)-G\left(x_t\right)\right]=\left(f_0\left(x_t,t\right)+f_1\left(x_t,t\right)\right)dt+g\left(x_t,t\right)dB\left(t\right)$$

(71)

for $t\ge 0$ with initial value $x_0=\xi \in C\left(\left[-h,0;{\mathbb{R}}^n\right]\right)$, $k\in \left(0,1\right)$, where $f_0:{\mathbb{R}}_{+}\times {\mathbb{R}}^n\to {\mathbb{R}}^n,f_1:{\mathbb{R}}_{+}\times C\left(\left[-h,0;{\mathbb{R}}^n\right]\right)\to {\mathbb{R}}^n$ and $g:{\mathbb{R}}_{+}\times C\left(\left[-h,0;{\mathbb{R}}^n\right]\right)\to {\mathbb{R}}^{m\times n}$, $w\left(t\right)$ is an m-dimension Brownian motion. The functions $f_0$, g, $f_1$ and G fulfill the local Lipschitz criterion with Lipschitz constant ${\mathit{L}}_n>0$ and adhere to the linear growth requirement. Further, the functions G fulfill Lipschitz constant $L_n<1$, and $\left(36\right)$ holds. Suppose that one can find a constant $\alpha \in \mathbb{R}$ and two non-decreasing left-continuous functions ${\eta }_1\left(t\right),{\eta }_2\left(t\right):\left[-h,0\right]\to {\mathbb{R}}_{+}$ such that, for $t\in {\mathbb{R}}_{+},\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$,

$$\mathbb{E}\left({\phi }^T\left(0\right)f_0\left(\phi \left(0\right),t\right)\right)\le \alpha \mathbb{E}{\left|\phi \left(0\right)\right|}^2,$$

(72)

$$\mathbb{E}\left|f_1\left(\phi ,t\right)\right|\le {\int }_{-h}^0{\eta }_1\left(s\right)\mathbb{E}\left|\phi \left(s\right)\right|ds$$

(73)

and

$$\mathbb{E}\left(trace\left[g^T\left(\phi ,t\right)g\left(\phi ,t\right)\right]\right)\le {\int }_{-h}^0{\eta }_2\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds.$$

(74)

Let $f\left(x,t\right)=f_0\left(\phi \left(0\right),t\right)+f_1\left(\phi ,t\right)$. On the basis of $\left(72\right)$ and $\left(73\right)$, they are merged into the following formula:

$$\mathbb{E}\left(2{\phi }^T\left(0\right)f\left(x,\phi \right)\right)\le \left(2\alpha +{\int }_{-h}^0{\eta }_1\left(s\right)ds\right)\mathbb{E}{\left|\phi \left(0\right)\right|}^2+{\int }_{-h}^0{\eta }_1\left(s\right)\mathbb{E}{\left|\phi \left(s\right)\right|}^{2}ds.$$

(75)

Moreover, let us suppose that, for $t\in {\mathbb{R}}_{+},\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$,

$$\mathbb{E}{∥f_0\left(\phi \left(0\right),t\right)∥}^2\le \lambda \mathbb{E}{\left|\phi \left(0\right)\right|}^2.$$

(76)

By Corollaries 6 and 7, we conclude that the trivial solution of Equation $\left(71\right)$ is mean square generalized exponentially stable if

$$2\alpha +2{\int }_{-h}^0{\eta }_1\left(s\right)ds+\sqrt{k}+\sqrt{k}\lambda +\sqrt{k}h{\int }_{-h}^0{\eta }_1^2\left(s\right)ds+{\int }_{-h}^0{\eta }_2\left(s\right)ds<0.$$

(77)

Example 2.

Consider the scalar stochastic differential with distributed delays

$$\begin{array}{c}\hfill dx\left(t\right)=\left(a_{1}x\left(t\right)+a_{2}x\left(t-h_1\right)+a_3{\int }_{-h}^{0}x\left(t+s\right)ds\right)ds+\\ \hfill \left(b_{1}x\left(t-h_2\right)+b_2{\int }_{-h}^{0}x\left(t+s\right)ds\right)dB\left(t\right),\end{array}$$

(78)

for $t\ge 0$, where $B\left(t\right)$ represents a one-dimensional Brownian motion and $a_1$, $a_2$, $a_3$, $b_1$, $b_2$$\in \mathbb{R}$.

Let

$$f\left(t,\phi \right):=a_1\phi \left(0\right)+a_2\phi \left(-h_1\right)+a_3{\int }_{-h}^0\phi \left(s\right)ds,$$

(79)

and

$$g\left(t,\phi \right):=b_1\phi \left(-h_2\right)+b_2{\int }_{-h}^0\phi \left(s\right)ds,$$

(80)

for $t\in {\mathbb{R}}_{+}$, $\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$. Thus, based on Equations $\left(79\right)$ and $\left(80\right)$, we derive the following equations:

$$\begin{array}{cc}\hfill \phi {\left(0\right)}^{T}f\left(\phi ,t\right)& =a_1\phi {\left(0\right)}^2+a_2\phi \left(0\right)\phi \left(-h_1\right)+a_3\phi \left(0\right){\int }_{-h}^0\phi \left(s\right)ds\hfill \\ \hfill & \le \left(a_1+{\displaystyle \frac{a_2}{2}}+{\displaystyle \frac{a_3}{2}}\right)\phi {\left(0\right)}^2+{\displaystyle \frac{a_2}{2}}\phi {\left(-h_1\right)}^2+{\displaystyle \frac{a_3}{2}}{\int }_{-h}^0\phi {\left(s\right)}^{2}ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill g^2\left(\phi ,t\right)& ={\left(b_1\phi \left(-h_2\right)+b_2{\int }_{-h}^0\phi \left(s\right)ds\right)}^2\hfill \\ \hfill & \le \left(b_1^2+b_1{b}_2\right)\phi {\left(-h_2\right)}^2+\left(b_2^2+b_1{b}_2\right){\int }_{-h}^0\phi {\left(s\right)}^{2}ds.\hfill \end{array}$$

Then, according to Corollary 3, we conclude that

$$2\left(a_1+a_2+a_3\right)+b_1^2+2b_1{b}_2+b_2^2<0.$$

(81)

Thus, the zero solution of Equation $\left(78\right)$ is mean square generalized exponentially stable, with the rate of decline expressed as ${\int }_{t_0}^t{c}_1\left(u\right)du$ for any $\xi \in \mathcal{C}$.

Example 3.

Consider the scalar linear time-varying SDE with delay

$$d\left[x\left(t\right)-G\left(x_t\right)\right]=\left(-a\left(t\right)x\left(t\right)+b\left(t\right)x\left(t-h_1\left(t\right)\right)\right)dt+c\left(t\right)x\left(t-h_2\left(t\right)\right)dB\left(t\right),$$

(82)

for $t\ge t_0\ge 0$, where $B\left(t\right)$ represents a one-dimensional Brownian motion and $a\left(t\right)$, $b\left(t\right),c\left(t\right),h_1\left(t\right),$$h_2\left(t\right):{\mathbb{R}}_{+}\to \mathbb{R}$ are continuous functions, and $h_1\left(t\right),h_2\left(t\right)\in \left[0,h\right]$ for $h>0$. Assume that $\left(36\right)$ holds and $k\in \left(0,1\right)$.

Define some functions:

$$f\left(t,\phi \right):=-a\left(t\right)\phi \left(0\right)+b\left(t\right)\phi \left(-h_1\left(t\right)\right),$$

(83)

and

$$g\left(t,\phi \right):=c\left(t\right)\phi \left(-h_2\left(t\right)\right),$$

(84)

for $t\in {\mathbb{R}}_{+}$, $\phi \in C\left(\left[-h,0\right];{\mathbb{R}}^n\right)$. Then, through Young‘s inequality, we derive

$$\begin{array}{c}\hfill 2\phi \left(0\right)f\left(t,\phi \right)\le \left|b\left(t\right)\right|\left({\phi }^2\left(0\right)+{\phi }^2\left(-h_1\left(t\right)\right)\right)-2a\left(t\right){\left|\phi \left(0\right)\right|}^2.\end{array}$$

(85)

Thus, through Young‘s inequality and Equation $\left(36\right)$, we derive

$$\begin{array}{c}\hfill 2G\left(\phi \right)f\left(t,\phi \right)\le 2\sqrt{k}\left[a^2\left(t\right){\left|\phi \left(0\right)\right|}^2+b^2\left(t\right){\phi }^2\left(-h_1\left(t\right)\right)\right]+\sqrt{k}\underset{-\tau \le s\le 0}{sup}{\left|\phi \left(s\right)\right|}^2\end{array}$$

(86)

and

$$g^2\left(t,\phi \right)=c^2\left(t\right){\phi }^2\left(-h_2\left(t\right)\right).$$

(87)

From Corollary 4 and $\left(69\right)$, we can obtain the following results:

$$2\left|b\left(t\right)\right|+2\sqrt{k}{b}^2\left(t\right)+\sqrt{k}+2\sqrt{k}{a}^2\left(t\right)+c^2\left(t\right)-2a\left(t\right)<0.$$

(88)

Our findings are consistent with the results in [39]. Then, the trivial solution of $\left(82\right)$ is generalized exponentially stable in the mean square if $\left(88\right)$ holds.

6. Conclusions

This paper has discussed the mean square generalized exponential stability of neutral stochastic functional differential equations and SDEs by employing comparison principles and contradiction-based proofs. We extend and refine several existing results, demonstrating the applicability of our criteria across a wider range of systems.

Compared to traditional exponential stability, generalized exponential stability encompasses common stability concepts and offers new insights into polynomial and logarithmic stability. Our findings offer a more versatile framework for the analysis of system stability in complex noisy environments, offering significant theoretical and practical value, particularly in biological systems, neural networks, and delay systems. Future research can build upon these results, extending them to higher-dimensional and more complex stochastic systems.