Stability in mean and almost sure stability for uncertain pantograph differential equations

Abstract

Uncertain pantograph differential equations are an important class of pantograph differential equations driven by uncertain process. This paper investigates two types of stability, namely stability in mean and almost sure stability, for uncertain pantograph differential equations. In detail, the concepts of stability in mean and almost sure stability for uncertain pantograph differential equations are presented. Moreover, we reveal the sufficient conditions for uncertain pantograph differential equations being stable in mean and stable almost surely. Finally, this paper attempts to explore the relationships among stability in mean, almost sure stability as well as stability in measure.

Keywords

Uncertain pantograph differential equation stability in mean almost sure stability uncertainty theory

1 Introduction

As an important delay differential equation, pantograph differential equation is concerned as form [1] $d X_{t} = f (t, X_{t}, X_{qt}) d t$ with q ∈ (0, 1), and plays a significant role in many applications, such as electrodynamics, astrophysics, nonlinear dynamical systems and cell growth [12 , 27]. In addition, a simple numerical discretization of the pantograph equation was analyzed in [23]. In order to obtain the approximate solution of generalized pantograph equations with variable coefficients, a numerical method based on polynomial approximation using Hermite polynomial basis was presented in [30]. And Koto [19] discussed the stability properties of Runge-Kutta methods applied to pantograph equations. Recently, the Chebyshev polynomials and a collocation method were applied to the solution of pantograph equations in [31].

However, nondeterminacy can be seen everywhere in the real world, many authors began to explore new type of pantograph differential equations with non-deterministic input. Within the framework of probability theory, stochastic pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d W_{t}$ with a Wiener process W_t was put forward [4]. A large number of theoretical researches about stochastic pantograph differential equations begun. Fan et al. [7] introduced the sufficient conditions that guarantee the existence and uniqueness of a strong solution to the nonlinear stochastic pantograph differential equations. In addition, the stability conditions of the analytical solutions of the nonlinear stochastic pantograph differential equations was discussed in [8]. Mao et al. [24] proved the existence and uniqueness of solutions to stochastic pantograph equations with diffusion and Lévy jumps. More researches about stochastic pantograph differential equations can be referred to [13, 14].

Fuzzy set theory is another powerful tool to describe nondeterminacy and to process vague or subjective information in mathematical models. Within the framework of fuzzy set theory, fuzzy number [2], fuzzy differential equation [6] and its extended forms, such as fuzzy delay differential equation [18], fuzzy fractional differential equation [3, 25], have become the current research hotspots. Unfortunately, there are few articles relating to fuzzy pantograph differential equations.

When indeterminacy behaves neither randomness nor fuzziness, uncertainty theory [20] established based on uncertain measure, can be applied. Moreover, uncertain variable and uncertain process was defined to represent the quantity with uncertainty, and to describe the evolution of an uncertain phenomenon, respectively [21]. After that, uncertain differential equation was proposed to further deal with uncertain dynamic systems. Much attention has been drawn to this topic. As a result, a lot of work in theory about uncertain differential equations sprang up. For example, the concept of stability in measure for uncertain differential equation was presented in [22], and some stability theorems were proved by Yao et al. [32] and Sheng and Wang [26].

Within the framework of uncertainty theory, uncertain delay differential equation [10] $d X_{t} = f (t, X_{t}, X_{t - τ}) d t + g (t, X_{t}, X_{t - τ}) d C_{t}$ with time delay τ > 0 and a Liu process C_t was proposed to describe such system, which depends on the current state and the previous states. Ge and Zhu [10] took the lead in exploring the existence and uniqueness of solutions for uncertain delay differential equations. Almost sure stability for uncertain delay differential equations was discussed in [28]. Recently, another type of stability, called stability in distribution, for uncertain delay differential equations was proposed in [17].

In this paper, a very special unbounded uncertain delay differential equations called uncertain pantograph differential equations [29] $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with q ∈ (0, 1) and a Liu process C_t is discussed. Such type of pantograph differential equations driven by uncertain process is widely used in engineering, population dynamics and industrial robotics under an environment with uncertain disturbance. Stability analysis means sensitivity of the state when a system has small changes in the initial status. Up to now, stability in measure for uncertain pantograph differential equations was discussed in [29]. As we all known, both stability in mean and almost sure stability play a vital role in various types of differential equations [9 , 33].

However, none of previous studies have attempted to investigate stability in mean and almost sure stability for uncertain pantograph differential equations. Consequently, this paper attempts to establish the theoretical research about stability in mean and almost sure stability for such differential equations. The main contribution of this paper includes three aspects.

(1) Give the definitions of stability in mean and almost sure stability for uncertain pantograph differential equations.

(2) Deduce some stability theorems.

(3) Analyze the relationship among stability in mean, almost sure stability and stability in measure.

The structure of this paper is organized as follows. Section 2 recalls some basic concepts and theorems in uncertainty theory, including uncertain variable, uncertain process and uncertain pantograph differential equation. In Sections 3 and 4, stability in mean and almost sure stability for uncertain pantograph differential equation are explored, respectively. Meanwhile, we attempt to analyze the relationships among stability in mean, almost sure stability as well as stability in measure. Finally, we make a brief conclusion in Section 5.

2 Preliminaries

In this section, we introduce some definitions and theorems in uncertainty theory, including uncertain variable, uncertain process and uncertain pantograph differential equation.

2.1 Uncertain variable

To provide a quantitative measurement that an uncertain phenomenon will occur, uncertain measure was proposed and defined below.

Definition 2.1. (see [20, 22]) Let ℒ be a σ-algebra on a nonempty set Γ. A set function ℳ is called an uncertain measure if it satisfies four axioms:

(1) ℳ {Γ} =1 for the universal set Γ.

(2) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ, where Λ^c is the complementary set of Λ.

(3) For every countable sequence of {Λ_k} ∈ ℒ, we have $ℳ {⋃_{k = 1}^{\infty} Λ_{k}} \leq \sum_{k = 1}^{\infty} ℳ {Λ_{k}} .$

(4) Let (Γ_k, ℒ_k, _ℳk) be uncertainty spaces for k = 1, 2, ⋯ The product uncertain measure ℳ is an uncertain measure satisfying $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}}$ where Λ_k ∈ ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.2. (see [20]) An uncertain variable is a function ξ from an uncertainty space to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B.

Definition 2.3. (see [22]) Let ξ be an uncertain variable. The expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} ℳ {ξ \geq t} d t - \int_{- \infty}^{0} ℳ {ξ \leq t} d t$ provided that at least one of the above two integrals is finite, where ℳ is an uncertain measure.

Theorem 2.1. (see [20]) Let ξ be an uncertain variable. Then for any x > 0, we have $ℳ {| ξ | \geq x} \leq \frac{E [| ξ |]}{x}$ where ℳ is an uncertain measure and E [·] denotes expected value.

2.2 Uncertain process

Uncertain process was defined as a sequence of uncertain variables driven by time (see [21]). As an important uncertain process, a Liu process was defined below.

Definition 2.4. (see [22]) An uncertain process C_t with t ≥ 0 is called a Liu process if

(1) C₀ = 0 and almost all sample paths are Lipschitz continuous.

(2) C_t is a stationary independent increment process.

(3) every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t².

Theorem 2.2. (see [5]) Suppose that C_t is a Liu process on uncertainty space (Γ, ℒ, ℳ), and X_t is an integrable uncertain process on [a, b] with respect to t. Then we have $| \int_{a}^{b} X_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a}^{b} | X_{t} (γ) | d t$ where K (γ) is the Lipschitz constant of C_t (γ) for any γ ∈ Γ.

Theorem 2.3. (see[32]) Suppose that C_t is a Liu process on uncertainty space (Γ, ℒ, ℳ). Then there exists an uncertain variable K such that K (γ) is a Lipschitz constant of the sample path C_t (γ) for each γ ∈ Γ, $lim_{x \to + \infty} ℳ {γ \in Γ | K (γ) \leq x} = 1$ and $ℳ {γ \in Γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$

2.3 Uncertain pantograph differential equation

Definition 2.5. (see [29]) Suppose that C_t is a Liu process, and f and g are two real-valued functions. $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ (1) with 0 ≤ t ≤ T is called an uncertain pantograph differential equation, where 0 < q < 1, f is usually called the drift function and g is called the diffusion function.

Definition 2.6. (see [29]) Uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 is said to be stable in measure if for two different solutions X_t and Y_t, we have $lim_{| X_{0} - Y_{0} | \to 0} ℳ {sup_{t \in [0, T]} | X_{t} - Y_{t} | > ɛ} = 0$ for any ɛ > 0.

3 Stability in mean

In this section, we mainly establish stability in mean for uncertain pantograph differential equation. Firstly, the definition of stability in mean is given as follows.

Definition 3.1. Uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 is said to be stable in mean if for any two solutions X_t and Y_t, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] = 0 .$

The integral form of uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with the initial value x₀ (X₀ = x₀) is expressed as $\begin{matrix} X_{t} = X_{0} + \int_{0}^{t} f (s, X_{s}, X_{qs}) d s \\ + \int_{0}^{t} g (s, X_{s}, X_{qs}) d C_{s} . \end{matrix}$

Theorem 3.1. Suppose that uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 has a unique solution for each given initial value. Then it is stable in mean if the coefficients f (t, x, y) and g (t, x, y) satisfy ${\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ \leq F_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \\ | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq G_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ (2) for $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T], where F_t and G_t are two bounded functions satisfying $\int_{0}^{T} F_{t} d t < + \infty$ and $\int_{0}^{T} G_{t} d t < \frac{π}{2 \sqrt{3}},$ respectively.

Proof. Let X_t and Y_t be two solutions of uncertain pantograph differential Equation (1) with different initial values x₀ (X₀ = x₀) and y₀ (Y₀ = y₀), respectively. Then for a Lipschitz continuous sample C_t (γ), we have $\begin{matrix} X_{t} (γ) = X_{0} + \int_{0}^{t} f (s, X_{s} (γ), X_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{qs} (γ)) d C_{s} (γ) \end{matrix}$ and $\begin{matrix} Y_{t} (γ) = Y_{0} + \int_{0}^{t} f (s, Y_{s} (γ), Y_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, Y_{s} (γ), Y_{qs} (γ)) d C_{s} (γ) . \end{matrix}$

By using Condition (2) and Theorem 2.2, we have

$\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s \\ + \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s | \\ + | \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} F_{s} (| X_{s} (γ) - Y_{s} (γ) | | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ + K (γ) \int_{0}^{t} G_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ = | X_{0} - Y_{0} | \\ + \int_{0}^{t} (F_{s} + K (γ) G_{s}) \\ \cdot (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \end{matrix}$

where K (γ) is the Lipschitz constant of C_t (γ).

Thus we have $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + \\ sup_{t \in [0, T]} \int_{0}^{t} (F_{s} + K (γ) G_{s}) \\ \cdot (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \end{matrix}$

$\begin{matrix} \leq | X_{0} - Y_{0} | \\ + 2 sup_{t \in [0, T]} \int_{0}^{t} (F_{s} + K (γ) G_{s}) sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + 2 \int_{0}^{T} (F_{s} + K (γ) G_{s}) sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s . \end{matrix}$

By using Gronwall’s inequality [11], we have $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp {\int_{0}^{T} 2 (F_{s} + K (γ) G_{s}) d s} \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{T} 2 F_{s} d s) \\ \cdot exp (\int_{0}^{T} 2 K (γ) G_{s} d s) . \end{matrix}$ (3)

By using Theorem 2.3, we know that K is a nonnegative uncertain variable such that $\begin{matrix} ℳ {γ \in Γ | K (γ) \geq x} \\ = 1 - ℳ {γ \in Γ | K (γ) \leq x} \\ \leq 1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1) \\ = 1 - \frac{2}{1 + exp (\frac{- π x}{\sqrt{3}})} + 1 \\ = 2 (1 - \frac{1}{1 + exp (\frac{- π x}{\sqrt{3}})}) \\ = 2 (\frac{exp (\frac{- π x}{\sqrt{3}})}{1 + exp (\frac{- π x}{\sqrt{3}})}) \\ = 2 {(1 + exp (\frac{π x}{\sqrt{3}}))}^{- 1} . \end{matrix}$ Taking expected value on both sides of expression (3), we have

$\begin{matrix} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{T} 2 F_{s} d s) E [exp (K \int_{0}^{T} 2 G_{s} d s)] . \end{matrix}$

Since F_s satisfies the condition $\int_{0}^{T} F_{s} d s < + \infty$ , we have $exp (\int_{0}^{T} F_{s} d s) < + \infty .$

In addition, according to the definition of expected value and the condition $\int_{0}^{T} G_{s} d s < \frac{π}{2 \sqrt{3}},$ we have $\begin{matrix} E [exp (K \int_{0}^{T} 2 G_{s} d s)] \\ = \int_{0}^{+ \infty} ℳ {γ | exp (K (γ) \int_{0}^{T} 2 G_{s} d s) \geq x} d x \\ = \int_{0}^{+ \infty} ℳ {γ | K (γ) \geq \frac{ln x}{\int_{0}^{T} 2 G_{s} d s}} d x \\ \leq 2 \int_{0}^{+ \infty} \frac{1}{1 + x^{π / (\sqrt{3} \int_{0}^{T} 2 G_{s} d s)}} d x < + \infty . \end{matrix}$

Overall, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] \\ \leq lim_{| X_{0} - Y_{0} | \to 0} | X_{0} - Y_{0} | exp (\int_{0}^{T} 2 F_{s} d s) \\ \cdot E [exp (K \int_{0}^{T} 2 G_{s} d s)] \\ = 0 . \end{matrix}$ In addition, $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] \geq 0 .$

That is, $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] = 0$ holds and uncertain pantograph differential Equation (1) is stable in mean under Condition (2). □

Example 3.1. Consider an uncertain pantograph differential equation $\begin{matrix} d X_{t} = (exp (t) X_{t} + \frac{1}{\sqrt{3} T} X_{0.5 t}) d t \\ + \frac{1}{2 \sqrt{3} T} X_{t} d C_{t} \end{matrix}$ (4) with 0 ≤ t ≤ T, where the parameter T > 0.

Take $f (t, x, y) = exp (t) x + \frac{1}{\sqrt{3} T} y,$ $g (t, x, y) = \frac{1}{2 \sqrt{3} T} x$ and $L = max {\frac{1}{\sqrt{3} T}, | exp (T) |} .$ The inequalities $\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ \leq exp (t) | x_{1} - x_{2} | + \frac{1}{\sqrt{3} T} | y_{1} - y_{2} | \\ \leq L (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ and $\begin{matrix} | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq \frac{1}{2 \sqrt{3} T} | x_{1} - x_{2} | \\ \leq \frac{1}{2 \sqrt{3} T} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ hold for $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T], and Condition (2) in Theorem 3.1 is satisfied.

In addition, two inequalities $\int_{0}^{T} L d t < + \infty$ and $\int_{0}^{T} \frac{1}{2 \sqrt{3} T} d t = \frac{1}{2 \sqrt{3}} < \frac{π}{2 \sqrt{3}}$ hold.

Hence, uncertain pantograph differential Equation (4) is stable in mean by using Theorem 3.1. □

In what follows, the relationship between stability in mean and stability in measure for uncertain pantograph differential equations is discussed and the result is shown by the following theorem.

Theorem 3.2. If uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 is stable in mean, then it is stable in measure.

Proof. Let X_t and Y_t be two solutions of uncertain pantograph differential Equation (1) with different initial values x₀ (X₀ = x₀) and y₀ (Y₀ = y₀), respectively. According to Definition 3.1, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] = 0 .$

Then for any ɛ > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} ℳ {sup_{t \in [0, T]} | X_{t} - Y_{t} | \geq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] / ɛ = 0 \end{matrix}$ by Theorem 2.1. Thus, for uncertain pantograph differential equation, stability in mean could lead to stability in measure. □

4 Almost sure stability

In this section, we study another type of stability called almost sure stability for uncertain pantograph differential equation. Meanwhile, two sufficient conditions for uncertain pantograph differential equation and linear uncertain pantograph differential equation being stable almost surely are provided, respectively.

Definition 4.1. Uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 is said to be stable almost surely if for any two solutions X_t and Y_t with different initial states, we have $ℳ {lim_{| X_{0} - Y_{0} | \to 0} sup_{t \in [0, T]} | X_{t} - Y_{t} | = 0} = 1 .$

Theorem 4.1. Assume that uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 has a unique solution for each given initial state. Then it is stable almost surely if the coefficients f (t, x, y) and g (t, x, y) satisfy $\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ + | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq L_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ (5) for $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T], where L_t is a bounded positive function satisfying $\int_{0}^{T} L_{t} d t < + \infty .$

Proof. Let X_t and Y_t be two solutions of uncertain pantograph differential Equation (1) with different initial values x₀ (X₀ = x₀) and y₀ (Y₀ = y₀), respectively. Then for any Lipschitz continuous sample C_t (γ), we have $\begin{matrix} X_{t} (γ) = X_{0} + \int_{0}^{t} f (s, X_{s} (γ), X_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{qs} (γ)) d C_{s} (γ) \end{matrix}$ and $\begin{matrix} Y_{t} (γ) = Y_{0} + \int_{0}^{t} f (s, Y_{s} (γ), Y_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, Y_{s} (γ), Y_{qs} (γ)) d C_{s} (γ) . \end{matrix}$

By Condition (5) and Theorem 2.2, we have

$\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s \\ + \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s | \end{matrix}$ $\begin{matrix} + | \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ + K (γ) \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ = | X_{0} - Y_{0} | + (1 + K (γ)) \\ \cdot \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \end{matrix}$

where K (γ) is the Lipschitz constant of C_t (γ).

Thus we have

$\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + (1 + K (γ)) \\ sup_{t \in [0, T]} \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ \leq | X_{0} - Y_{0} | \\ + 2 (1 + K (γ)) sup_{t \in [0, T]} \int_{0}^{t} L_{s} sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + 2 (1 + K (γ)) \int_{0}^{T} L_{s} sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s . \end{matrix}$

It follows from Gronwall’s inequality [11] that $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp {2 (1 + K (γ)) \int_{0}^{T} L_{s} d s} . \end{matrix}$

Since $\int_{0}^{T} L_{s} d s < + \infty$ and K (γ) is finite, we can obtain that $sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \to 0$ as long as |X₀ - Y₀|→0 which implies that $ℳ {lim_{| X_{0} - Y_{0} | \to 0} sup_{t \in [0, T]} | X_{t} - Y_{t} | = 0} = 1 .$

According to Definition 4.1, uncertain pantograph differential equation is almost surely stable under Condition (5). □

Example 4.1. Consider an uncertain pantograph differential equation $d X_{t} = (exp (t) X_{t} + μ X_{0.1 t}) d t + σ X_{t} d C_{t}$ (6) with 0 ≤ t ≤ T, where the parameters μ, σ > 0.

Take $f (t, x, y) = exp (t) x + μ y$ and $g (t, x, y) = σ x .$ Let L denote a common upper bound of |μ|, |σ| and | exp(t) | with 0 ≤ t ≤ T. The inequalities $\begin{matrix} | f (t, x_{1}, y) - f (t, x_{2}, y) | \\ + | g (t, x_{1}, y) - g (t, x_{2}, y) | \\ \leq exp (t) | x_{1} - x_{2} | + μ | y_{1} - y_{2} | + σ | x_{1} - x_{2} | \\ = (exp (t) + σ) | x_{1} - x_{2} | + μ | y_{1} - y_{2} | \\ \leq 2 L (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ hold for $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T], and Condition (5) in Theorem 4.1 is satisfied.

In addition, the condition $\int_{0}^{T} 2 L d t < + \infty$ holds. Hence, uncertain pantograph differential Equation (6) is stable almost surely by using Theorem 4.1. □

In accordance with Theorem 4.1, we immediately get a corollary about a sufficient condition for a linear uncertain pantograph differential equation.

Corollary 4.1. Suppose that u_it, v_it and w_it for i = 1, 2 are real-valued functions. Then a linear uncertain pantograph differential equation $\begin{matrix} d X_{t} = (u_{1 t} X_{t} + v_{1 t} X_{qt} + w_{1 t}) d t \\ + (u_{2 t} X_{t} + v_{2 t} X_{qt} + w_{2 t}) d C_{t} \end{matrix}$ (7) with 0 ≤ t ≤ T and 0 < q < 1 is almost surely stable if u_it, v_it and w_it for i = 1, 2 are bounded.

Proof. Take f (t, x, y) = u_1tx + v_1ty + w_1t and g (t, x, y) = u_2tx + v_2ty + w_2t, and let M denote a common upper bound of |u_it|, |v_it| and |w_it| (i = 1, 2). The inequality $\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ + | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ = | u_{1 t} | | x_{1} - x_{2} | + | v_{1 t} | | y_{1} - y_{2} | \\ + | u_{2 t} | | x_{1} - x_{2} | + | v_{2 t} | | y_{1} - y_{2} | \\ \leq 2 M (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ for any $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T] holds, and Condition (5) in Theorem 4.1 is satisfied.

According to Theorem 4.1, we obtain that linear uncertain pantograph differential Equation (7) is almost surely stable. □

Similarly to Section 3, we will explore the relationship between almost sure stability and stability in measure for uncertain pantograph differential equations and show the result by the following theorem.

Theorem 4.2. If uncertain pantograph differential equation $d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t}$ with 0 ≤ t ≤ T and 0 < q < 1 is almost surely stable, then it is stable in measure.

$ℳ {γ \in Γ | lim_{| X_{0} - Y_{0} | \to 0} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | = 0} = 1 .$

That is, there exists a set Γ₀ in Γ with ℳ {Γ₀} =1 such that for any γ ∈ Γ₀, $lim_{| X_{0} - Y_{0} | \to 0} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | = 0 .$ (8) According to Equation (8), for any ε > 0 and γ ∈ Γ₀, we have $sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \leq ε$ as long as |X₀ - Y₀|→0. It directly leads to

$lim_{| X_{0} - Y_{0} | \to 0} ℳ {γ \in Γ | sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \leq ε} = 1 .$

Hence, for any ε > 0, we have $lim_{| X_{0} - Y_{0} | \to 0} ℳ {sup_{t \in [0, T]} | X_{t} - Y_{t} | > ε} = 0$

Thus almost sure stability leads to stability in measure for uncertain pantograph differential equation.

□

5 Conclusion

This paper discussed two types of stability for uncertain pantograph differential equations. In detail, the concepts of stability in mean and almost sure stability for uncertain pantograph differential equations were proposed. Moreover, this paper deduced the sufficient conditions for uncertain pantograph differential equations being stable in mean and almost surely, respectively. Meanwhile, we also proposed a theorem on the almost sure stability of the linear uncertain pantograph differential equations. Finally, we analyzed the relationships among stability in mean, almost sure stability and stability in measure, and found that both stability in mean and almost sure stability could lead to stability in measure for uncertain pantograph differential equations.

In our future work, we will focus on two directions. First, we conduct research on the numerical methods for solving uncertain pantograph differential equations. Second, we attempt to discuss pantograph differential equations in the fuzzy case.

Footnotes

Acknowledgment

This work was funded by National Natural Science Foundation of China (11701338, 71601022), China Postdoctoral Science Foundation (2019M650551) and Youth Top Talent Cultivation Plan Project of Beijing (CIT&TCD201804036).

In addition, the authors would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions to improve presentation of this paper.

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