Some theoretical results on the stability of uncertain pantograph differential equations

Abstract

Uncertain pantograph differential equation is a new type of differential equations in uncertainty theory. In this paper, we first give the concept of uncertain pantograph differential equation. In addition, this paper aims at the stability of this type of differential equations. The concepts of stability in measure and stability in pth moment for uncertain pantograph differential equations are presented and the sufficient conditions for uncertain pantograph differential equations being stable in measure and in pth moment are derived. Finally, this paper also discusses the relationship between these two types of stability.

Keywords

Uncertain pantograph differential equation stability in measure uncertainty theory

1 Introduction

Differential equation is an important tool to deal with dynamic systems. Since nondeterminacy can be seen everywhere in the real world, many authors began to explore new type of differential equations with non-deterministic input. So far, non-deterministic input is mainly divided into three categories: random input, fuzzy input and uncertain input. In these cases, to construct a mathematical model, we can use stochastic differential equation [8], fuzzy differential equation [18, 21] or uncertain differential equation [22, 26].

In addition, many physical systems do not only depend on present states but also involves past states [3 , 23]. At this time, delay differential equation is widely used to model such systems. As a special stochastic delay differential equation, stochastic pantograph differential equation has attracted a great deal of attention. In particular, special attention has been focused on numerical methods [13 , 30] and stability analysis [4 , 25] for stochastic pantograph differential equation.

As we all know, stochastic delay differential equation was established based on probability theory, while uncertain delay differential equation was established within the framework of uncertainty theory. On the basis of uncertain measure [9], uncertainty theory was established, where uncertain measure satisfies normality, duality, subadditivity and product measure axioms. In order to represent the quantity with uncertainty in uncertainty theory, uncertain variable was defined as a measurable function from an uncertainty space to a set of real numbers. In addition, to describe the evolution of an uncertain phenomenon, Liu [10] proposed uncertain process and designed a Liu process [11]. Meanwhile, Liu [11] founded uncertain calculus to handle the integral and differential of an uncertain process. Driven by a Liu process, uncertain differential equation was first proposed by Liu [10]. After that, many researchers have done a lot of work in theory [12 , 19] and practice [14 , 31] about uncertain differential equations.

Back to uncertain delay differential equation, such type of differential equation can be seen an extended uncertain differential equation and [5] was always used to describe such dynamic system, in which the rate of change of the system depends on the present state and its past states. In the last few years, uncertain delay differential equations have attracted a great deal of attention due to its importance for many applications in economics, biology, physics and other sciences, and some theoretical research was presented to the public, including existence of solutions, uniqueness of solutions as well as stability analysis of solutions. Barbacioru [1] first discussed a local existence and uniqueness of solutions for a special type of uncertain delay differential equations. Subsequently, under the global Lipschitz condition, Ge and Zhu [5] provided an existence and uniqueness theorem of solutions for uncertain delay differential equations. Considering the rigour of the global Lipschitz condition, Wang and Ning [24] recently proved a general existence and uniqueness theorem for uncertain delay differential equations under the one-sided local Lipschitz condition.

In aspect of stability analysis, the concepts of stability in measure, stability in mean and stability in pth moment of uncertain delay differential equations were first given and the corresponding stability theorems were proved in [24]. Recently, the stability in distribution for uncertain delay differential equations was discussed in [15].

In this paper, we aim at a special unbounded uncertain delay differential equation, which is called uncertain pantograph differential equation. In the field of engineering, natural science and social science, uncertain pantograph differential equation maybe widely used. To the best of author’s knowledge, there is no literature concerned with uncertain pantograph differential equation yet. Hence, this paper shall first give an accurate definition of uncertain pantograph differential equation. Then, we focus on the stability of this type of differential equation. The main contribution of this paper includes four aspects as follow. (1) Define the concept of uncertain pantograph differential equation. (2) Define the concepts of stability in measure and stability in pth moment for uncertain pantograph differential equation. (3) Deduce some stability theorems. (4) Analyze the relationship between stability in measure and stability in pth moment.

The structure of this paper is organized as follows. Section 2 introduces some basic concepts and theorems in uncertainty theory. Section 3 provides the concept of uncertain pantograph differential equation. In Section 4, stability in measure and stability in pth moment for this type of uncertain differential equations are explored. Next, we analyze the relationship between these two types of stability in Section 5. Finally, we make a brief conclusion in Section 6.

2 Preliminary

In this section, we make a brief introduction of some concepts and theorems in uncertainty theory. First of all, uncertain measure in uncertainty theory is defined to provide a quantitative measurement that an uncertain phenomenon will occur.

Definition 2.1. (Liu [9, 11]) Let eulerL be a σ-algebra on a nonempty set Γ. A set function M is called an uncertain measure if it satisfies four axioms:

M {Γ} =1.

M {Λ} + M {Λ^c} =1, where Λ ∈ eulerL and Λ^c is the complementary set of Λ.

For every countable sequence of Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}}$ where $⋃_{i = 1}^{2} a_{i}$ is the union of a₁ and a₂, and $\sum_{i = 1}^{2} a_{i} = a_{1} + a_{2}$ .

The triplet (Γ, eulerL, M) is usually called an uncertainty space, and Λ ∈ eulerL is called an event.

Let (Γ_i, eulerL_i, _Mi) be uncertainty spaces for i = 1, 2, ⋯ The product uncertain measure M is an uncertain measure satisfying $M {\prod_{i = 1}^{\infty} Λ_{i}} = ⋀_{i = 1}^{\infty} M_{i} {Λ_{i}}$ where $\prod_{i = 1}^{2} a_{i} = a_{1} \times a_{2}$ , $⋀_{i = 1}^{2} a_{i} = min (a_{1}, a_{2})$ , and Λ_i ∈ eulerL_i for i = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [9]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, eulerL, M) to the set of real numbers.

Definition 2.3. (Liu [11]) 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.

Definition 2.4. (Liu [9]) Let ξ, p be an uncertain variable and a positive integer, respectively. Then E [ξ^p] is called the pth moment of ξ if the expected value E [ξ^p] exists.

Theorem 2.1. (Liu [9]) Let ξ be an uncertain variable. Then for any x > 0 and p > 0, we have $M {| ξ | \geq x} \leq \frac{E [| ξ |^{p}]}{x^{p}}$ where M is an uncertain measure.

Uncertain process was defined as a sequence of uncertain variables driven by time[10]. In addition, as an important uncertain process, a Liu process was defined as below.

Definition 2.5. (Liu [11]) An uncertain process C_t with t ≥ 0 is called a Liu process if

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

C_t is a stationary independent increment process.

C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t² for any s > 0.

Theorem 2.2. (Chen and Liu [2]) Suppose that C_t is a Liu process on uncertainty space (Γ, eulerL, M), and X_t is an integrable uncertain process on time interval [a, b]. 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).

Theorem 2.3. (Yao et al. [28]) Suppose that C_t is a Liu process on uncertainty space (Γ, eulerL, 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 $M {γ \in Γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$

3 Uncertain pantograph differential equation

In this section, we will discuss a new type of differential equation called uncertain pantograph differential equation within the framework of uncertainty theory. First of all, uncertain pantograph differential equation is defined as below.

Definition 3.1. Suppose that C_t is a Liu process, X_t is an uncertain 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.

Given an initial value X₀ = x₀, ${\begin{matrix} d X_{t} = f (t, X_{t}, X_{qt}) d t + g (t, X_{t}, X_{qt}) d C_{t} \\ X_{0} = x_{0} \end{matrix}$ is called an uncertain pantograph differential equation with the initial value x₀.

Definition 3.2. Suppose that u_1t, u_2t, v_1t and v_2t are integrable uncertain processes on time interval [0, T], X_t is an uncertain process and C_t is a Liu process. Uncertain pantograph differential equation $d X_{t} = (u_{1 t} X_{t} + v_{1 t} X_{qt}) d t + (u_{2 t} X_{t} + v_{2 t} X_{qt}) d C_{t}$ with 0 ≤ t ≤ T is called a linear uncertain pantograph differential equation, where 0 < q < 1.

For example, ${\begin{matrix} d X_{t} = exp (- t) X_{t} d t + [X_{t} + X_{0.1 t}] d C_{t} \\ X_{0} = 1 \end{matrix}$ with 0 ≤ t ≤ 1, where q = 0.1, is a linear uncertain pantograph differential equation, while ${\begin{matrix} d X_{t} = 3 X_{t} d t + sin (X_{0.5 t}) d C_{t} \\ X_{0} = 1 \end{matrix}$ with 0 ≤ t ≤ 2, where q = 0.5, is not an uncertain pantograph differential equation.

For being convenient to study, we focus on the integral form of (1) instead of its differential form, and the integral form of (1) can be expressed as $\begin{matrix} X_{t} = X_{0} + \int_{0}^{t} f (s, X_{s}, X_{qs}) d s \\ + \int_{0}^{t} g (s, X_{s}, X_{qs}) d C_{s} . \end{matrix}$

On the one hand, almost all sample paths of C_t are Lipschitz continuous functions on uncertainty space (Γ, eulerL, M). It shows that there exists a set Γ₀ in Γ with M {Γ₀} =1 such that for any γ ∈ Γ₀, C_t (γ) is Lipschitz continuous. For convenient research, Γ₀ is seen as Γ. Thus, for each γ ∈ Γ, there exists a positive number K (γ) such that $| C_{x} (γ) - C_{y} (γ) | \leq K (γ) | x - y |, \forall x, y \geq 0 .$

On the other hand, uncertain integral of C_t is equivalent to Riemann-Stieltjes integral from the point of each sample path. Hence, we focus our attention on the following uncertain pantograph integral equation $\begin{matrix} X_{t} (γ) = X_{0} (γ) + \int_{0}^{t} f (s, X_{s} (γ), X_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{qs} (γ)) d C_{s} (γ) \end{matrix}$ with 0 ≤ t ≤ T for any γ ∈ Γ, where 0 < q < 1.

4 Main results

This section mainly researches on the stability of uncertain pantograph differential equation. To be specific, two types of stability, namely stability in measure and stability in pth moment are discussed. Moreover, some stability results in the from of theorems are provided.

4.1 Stability in measure

To begin with, the concept of stability in measure for uncertain pantograph differential equation is put forward. And then, a sufficient condition for an uncertain pantograph differential equation being stable in measure is provided.

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 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, where M is an uncertain measure and ‘sup’ is short for supremum.

Theorem 4.1. 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 measure if the coefficients f (t, x, y) and g (t, x, y) satisfy ${\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ \leq L_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \\ | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq L_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ (2) for any $x_{1}, x_{2}, y_{1}, y_{2} \in R, t \in [0, T]$ and L_t satisfies $\int_{0}^{T} L_{t} d t < + \infty .$

Proof: Let X_t and Y_t be two solutions of uncertain pantograph differential equation (1) with two different initial values x₀ (X₀ = x₀) and y₀ (Y₀ = y₀), respectively. For a Lipschitz continuous sample C_t (γ), we immediately obtain $\begin{matrix} X_{t} (γ) = X_{0} + \int_{0}^{t} f (s, X_{s} (γ), X_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{qs} (γ)) d C_{s} (γ) \end{matrix}$ 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 $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s \\ + \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} [f (s, X_{s} (γ), X_{qs} (γ)) - f (s, Y_{s} (γ), Y_{qs} (γ))] d s | \\ + | \int_{0}^{t} [g (s, X_{s} (γ), X_{qs} (γ)) - g (s, Y_{s} (γ), Y_{qs} (γ))] d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ + K (γ) \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ = | X_{0} - Y_{0} | + (1 + K (γ)) \\ \cdot \int_{0}^{t} L_{s} (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ).

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

In addition, by Gronwall’s inequality [6], we have $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp {\int_{0}^{T} 2 L_{s} (1 + K (γ)) d s} \\ = | X_{0} - Y_{0} | exp {2 (1 + K (γ)) \int_{0}^{T} L_{s} d s} . \end{matrix}$

At the same time, by using Theorem 2.3, we know that K is a nonnegative uncertain variable and $lim_{x \to + \infty} M {γ \in Γ | K (γ) \leq x} = 1 .$

Then, for any ɛ > 0, there exists a positive number H such that $M {γ | K (γ) \leq H} \geq 1 - ɛ .$ Taking $δ = exp {- 2 (1 + H) \int_{0}^{T} L_{s} d s} ɛ,$ we immediately get $sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \leq ɛ$ when |X₀ - Y₀| ≤ δ and K (γ) ≤ H.

Based on the above conclusions, we then have $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \in [0, T]} | X_{t} - Y_{t} | > ɛ} = 0 .$ The proof is completed. □

Example 4.1. Consider an uncertain pantograph differential equation $d X_{t} = (exp (t) X_{t} + μ X_{0.1 t}) d t + σ X_{t} d C_{t}$ (3) with 0 ≤ t ≤ T, where the parameters μ, σ > 0.

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

In addition, $\int_{0}^{T} L d t < + \infty$ holds. Hence, uncertain pantograph differential equation (3) is stable in measure by using Theorem 4.1. □

4.2 Stability in pth moment

This section is concerned with the pth moment stability for uncertain pantograph differential equation, where p > 0. First of all, stability in pth moment for uncertain pantograph differential equation is defined as follows.

Definition 4.2. 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 pth moment (p > 0) if for any two different solutions X_t and Y_t, we have $lim_{| X_{0} - Y_{0} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] = 0$ where E [ξ] denotes the expected value of ξ.

In the case of p = 1, stability in 1th moment of uncertain pantograph differential equation (1) is usually called stability in mean. That is to say, uncertain pantograph differential equation (1) is said to be stable in mean if for any two different solutions X_t and Y_t, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \in [0, T]} | X_{t} - Y_{t} |] = 0 .$

Since stability in mean is a special case of pth moment stability, we only pay attention to stability in pth moment for uncertain pantograph differential equation in this paper.

Theorem 4.2. 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 pth moment if the coefficients f (t, x, y) and g (t, x, y) satisfy ${\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ \leq F_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \\ | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq G_{t} (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ (4) for any $x_{1}, x_{2}, y_{1}, y_{2} \in R, t \in [0, T]$ , where F_t and G_t are two bounded functions satisfying $\int_{0}^{T} F_{t} d t < + \infty, \int_{0}^{T} G_{t} d t < \frac{π}{2 \sqrt{3} p} .$

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 $\begin{matrix} Y_{t} (γ) = Y_{0} + \int_{0}^{t} f (s, Y_{s} (γ), Y_{qs} (γ)) d s \\ + \int_{0}^{t} g (s, Y_{s} (γ), Y_{qs} (γ)) d C_{s} (γ) . \end{matrix}$

With the aid of Condition (4) 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 $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + sup_{t \in [0, T]} \int_{0}^{t} (F_{s} + K (γ) G_{s}) \\ \cdot (| X_{s} (γ) - Y_{s} (γ) | + | X_{qs} (γ) - Y_{qs} (γ) |) d s \\ \leq | X_{0} - Y_{0} | \\ + 2 sup_{t \in [0, T]} \int_{0}^{t} (F_{s} + K (γ) G_{s}) sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + 2 \int_{0}^{T} (F_{s} + K (γ) G_{s}) sup_{u \in [0, s]} | X_{u} (γ) - Y_{u} (γ) | d s . \end{matrix}$

By using Gronwall’s inequality [6], we have $\begin{matrix} sup_{t \in [0, T]} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp {\int_{0}^{T} 2 (F_{s} + K (γ) G_{s}) d s} \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{T} 2 F_{s} d s) \\ \cdot exp (\int_{0}^{T} 2 K (γ) G_{s} d s) . \end{matrix}$ (5)

By Theorem 2.3, K is a nonnegative uncertain variable such that $M {γ \in Γ | K (γ) \geq x} \leq 2 {(1 + exp (\frac{π x}{\sqrt{3}}))}^{- 1} .$ Taking pth moment on both sides of (5), we get $\begin{matrix} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] \\ \leq | X_{0} - Y_{0} |^{p} exp (2 p \int_{0}^{T} F_{s} d s) \\ \cdot E [exp (pK \int_{0}^{T} 2 G_{s} d s)] . \end{matrix}$

Since F_s satisfies $\int_{0}^{T} F_{s} d s < + \infty$ , we immediately have $exp (2 p \int_{0}^{T} F_{s} d s) < + \infty$ . According to the definition of expected value and inequality $\int_{0}^{T} G_{s} d s < \frac{π}{2 \sqrt{3} p},$ we have $\begin{matrix} E [exp (pK \int_{0}^{T} 2 G_{s} d s)] \\ = \int_{0}^{+ \infty} M {γ | exp (pK (γ) \int_{0}^{T} 2 G_{s} d s) \geq x} d x \\ = \int_{0}^{+ \infty} M {γ | K (γ) \geq \frac{ln x}{p \int_{0}^{T} 2 G_{s} d s}} d x \\ \leq 2 \int_{0}^{+ \infty} \frac{1}{1 + x^{π / (2 p \sqrt{3} \int_{0}^{T} G_{s} d s)}} d x < + \infty . \end{matrix}$

Hence, we have $lim_{| X_{0} - Y_{0} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] = 0$ and uncertain pantograph differential equation (1) is stable in pth moment. The proof is completed. □

Example 4.2. Consider an uncertain pantograph differential equation $d X_{t} = (t + exp (t) X_{t}) d t + {tX}_{0.5 t} d C_{t}$ (6) with 0 ≤ t ≤ T.

Take f (t, x, y) = t + exp(t) x and g (t, x, y) = ty. The inequalities $\begin{matrix} | f (t, x_{1}, y_{1}) - f (t, x_{2}, y_{2}) | \\ \leq exp (t) | x_{1} - x_{2} | \\ \leq exp (t) (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ and $\begin{matrix} | g (t, x_{1}, y_{1}) - g (t, x_{2}, y_{2}) | \\ \leq t | y_{1} - y_{2} | \\ \leq t (| x_{1} - x_{2} | + | y_{1} - y_{2} |) \end{matrix}$ hold for any $x_{1}, x_{2}, y_{1}, y_{2} \in R$ and t ∈ [0, T]. Take F_t = exp(t) and G_t = t in Theorem 4.2, and Condition (4) in Theorem 4.2 is satisfied.

Obviously, $\int_{0}^{T} F_{t} d t = \int_{0}^{T} exp (t) d t = exp (T) - 1 < + \infty$ holds. In addition, to make the inequality $\int_{0}^{T} G_{t} d t = \int_{0}^{T} t d t = \frac{1}{2} T^{2} < \frac{π}{2 \sqrt{3} p}$ hold, p must satisfy $p < \frac{π}{\sqrt{3} T^{2}}$ .

Hence, uncertain pantograph differential equation (6) is stable in pth moment when $p < \frac{π}{\sqrt{3} T^{2}}$ . □

5 Comparison of stability

In this section, we mainly analyze the relationship between stability in measure and stability in pth moment for uncertain pantograph differential equation. Meanwhile, we also discuss the relationship between stability in pth moment and stability in rth moment for uncertain pantograph differential equation, where p ≠ r.

Theorem 5.1. 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 pth moment (p > 0), then it is stable in measure.

Proof: Let X_t and Y_t be two solutions of uncertain pantograph differential equation (1) with the initial values x₀ (X₀ = x₀) and y₀ (Y₀ = y₀), respectively. According to the definition of stability in pth moment, we have $lim_{| X_{0} - Y_{0} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] = 0 .$

By using Theorem 2.1, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \in [0, T]} | X_{t} - Y_{t} | \geq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] / ɛ^{p} = 0 \end{matrix}$ for any ɛ ≥ 0.

Thus, for uncertain pantograph differential equation, stability in pth moment leads to stability in measure. □

Theorem 5.2. For any two real numbers p and r (0 < p < r), 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 rth moment, then it is stable in pth moment.

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 the definition of stability in rth moment, we have $lim_{| X_{0} - Y_{0} | \to 0} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{r}] = 0 .$

In addition, by using Hölder’s inequality, we have $\begin{matrix} E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p}] \\ = E [| {(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{p} \cdot 1 |] \\ \leq \sqrt[r / p]{E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{pr / p}]} \sqrt[r / (r - p)]{E [1^{r / (r - p)}]} \\ = \sqrt[r / p]{E [{(sup_{t \in [0, T]} | X_{t} - Y_{t} |)}^{r}]} . \end{matrix}$

Hence, for uncertain pantograph differential equation, stability in rth moment leads to stability in pth moment when p < r. □

6 Conclusions

In this paper, a new type of differential equations within the framework of uncertainty theory was discussed for the first time. First of all, we gave the definition of uncertain pantograph differential equation. And then, this paper was concerned with the stability analysis for this type of differential equations, where the concepts of stability in measure and stability in pth moment were proposed. Meanwhile, this paper also deduced the sufficient conditions for uncertain pantograph differential equations being stable in measure and in pth moment. At last, we analyzed the relationships between these two types of stability and found that stability in pth moment can lead to stability in measure.

In the future work, we will try to explore some application fields of this type of differential equations.

Footnotes

Acknowledgment

This work was funded by National Natural Science Foundation of China (Nos. 11701338, 61663035) and China Postdoctoral Science Foundation (No.2019M650551). The author would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions to improve presentation of the paper.

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