New stability theorem for uncertain pantograph differential equations

Abstract

Uncertain pantograph differential equation (UPDE for short) is a special unbounded uncertain delay differential equation. Stability in measure, stability almost surely and stability in p-th moment for uncertain pantograph differential equation have been investigated, which are not applicable for all situations, for the sake of completeness, this paper mainly gives the concept of stability in distribution, and proves the sufficient condition for uncertain pantograph differential equation being stable in distribution. In addition, the relationships among stability almost surely, stability in measure, stability in p-th moment, and stability in distribution for the uncertain pantograph differential equation are also discussed.

Keywords

uncertainty theory uncertain pantograph differential equation stability in distribution the relationships among stabilities

1 Introduction

More than half a century ago, when the Itô’s [1] landmark work comes out, the stochastic differential equations (SDEs), as a new branch of mathematics, have aroused great interest in academies. In recent decades, Various SDEs have been widely applied in many fields such as biological, engineering, social sciences, finance and other models hidden in the observed data. One of the best important works in this research field is to discuss the stability of such systems. For example, as an important class of stochastic delay systems, stochastic pantograph differential equations (SPDEs) have received considerable attention about their stabilities and numerical solutions. Over the past few years, Zhou [2] investigated almost surely exponential stability of numerical solutions in 2014. You et al. [3] analysed exponential stability of hybrid SPDEs with highly nonlinear coefficients in 2015. In the same year, Iftikhar et al. [4] considered stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model. In 2019, Guo et al. [5] proved Razumikhin-type theorems on the moment stability of the exact and numerical solutions for the SPDEs.

In most existing works under the system of probability theory including SDEs, it was implicitly assumed that the estimated probability distribution is closed enough to the real frequencies. If we want to obtain the estimated probability distribution by statistic method, we should obtain large amounts of historical data. However, in reality, people seem to lack data or the size of sample data applied in practice are less in some cases, such as the emerging infectious disease model, the new stock model and so on. Although sometimes we have a lot of available sample data, the frequency obtained by sample data is, unfortunately, not close enough to the distribution function obtained in some practical problems. In order to overcome these limitations, we need to invite some domain experts to evaluate the belief degree that each event may happen in these situations. Different from random phenomenon, human uncertainty associated with belief degrees is another different type of indeterminate phenomenon. For describing human uncertainty, Liu [6] introduced uncertainty theory as a new branch of mathematics in 2007, and updated it in 2015 [7]. He discovered belief degree should be modeled by uncertainty theory rather than subjective probability or fuzzy set theory. Until now, uncertainty theory have received extensive attention and scholars from many fields are attempting to link uncertainty theory with their own fields (for example, Gao et al. [8]; Wang et al. [9]; Li et al. [10]; Zhang [11]; Huang [12]).

In 2008, Liu [13] first proposed uncertain differential equations (UDEs) which are driven by canonical Liu process. Later in 2009, Liu [14] claimed Liu process is an uncertain process with stationary and independent normal uncertain increments. Chen and Liu [15] analysed the solution of a linear uncertain differential equation and proved existence and uniqueness theorem with Lipschitz condition in 2010. After that, The achievements about the special nonlinear UDEs were produced one after another in 2013. For example, Liu et al. [16] proposed analytic method about how to solve the special UDEs. Yao et al. [17] analysed a type of UDEs with analytic solution. Liu et al. [18] considered semi-linear UDE with its analytic solution. Wang et al. [19] considered analytic solution for a general type of UDE. More importantly, Yao and Chen [20, 21] provided a numerical method to obtain the uncertain distributions of solution of an uncertain differential equation. Based on above theoretical development, uncertain differential equation has been successfully applied in many fields such as uncertain differential game [22], uncertain heat conduction [23], uncertain wave equation [24 –26], uncertain differential equation with jump [27, 28] and so on.

Since 2009, after the concept of stability for an UDE was first addressed by Liu [14], there have been more interest in the stabilities of UDEs. For example, Yao et al. [29] proved some stability theorems of uncertain differential equations. Following that, Sheng et al. [30], Liu et al. [31], Yao et al. [32], Sheng et al. [33] and Yang et al. [34] discussed stability in mean, stability in p-th moment, exponential stability, almost sure stability and stability in distribution of uncertain differential equations, respectively. In some cases, researchers often use uncertain delay differential equations to describe such physical conditions that are related to both the current state and the past state (for example, Barbacioru [35]; Ge et al. [36]; Wang et al. [37]). Following that, Wang et al. [38, 39] put forward the stability in measure, stability in mean and stability in p-th moment for uncertain delay differential equations, and proved the corresponding stability theorems. Jia and Sheng [40] discussed the stability in distribution. It is also worth noting that uncertain pantograph differential equations (UPDEs) are particular cases of uncertain unbounded delay differential equations, when one considers electrodynamics, astrophysics, nonlinear dynamical systems and cell growth, one speaks of UPDEs. In 2020, Wang et al. [41, 42] proposed UPDEs and defined the concepts of stability in measure, stability in p-th moment, stability almost surely, and stability in mean for UPDE, which are not applicable for all situations. As we all known, stability in distribution plays an important role in various types of differential equations [34, 40]. However, as we can know, the issue of stability in distribution for uncertain pantograph differential equations have not been investigated yet, which motivates us for current work. The UPDE is described as follows: ${\begin{matrix} {dZ}_{k} = & h (k, Z_{k}, Z_{mk}) dk \\ + q (k, Z_{k}, Z_{mk}) d C_{k}, k \in [0, T] \\ Z_{0} = & z_{0}, \end{matrix}$ (1) where 0 < m < 1, T is a fixed time, C_k is a canonical Liu process with respect to time k, and h and q are two continuous functions, Z₀ = z₀ is the initial value. Its equivalent integral form is as follows: ${\begin{matrix} Z_{k} = & Z_{0} + \int_{0}^{k} h (r, Z_{r}, Z_{mr}) dr \\ + \int_{0}^{k} q (r, Z_{r}, Z_{mr}) d C_{r}, k \in [0, T] \\ Z_{0} = & z_{0} . \end{matrix}$ (2)

The main contribution of this paper includes three aspects as follow. (i) Define the stability in distribution for uncertain pantograph differential equation. (ii) Prove the theorem on stability in distribution. (iii) Prove some relationships about stabilities. The structure of this paper is organized as follows. Section 2 recalls some important concepts including stability almost surely, stability in measure, stability in p-th moment, and stability in mean. In Section 3, firstly, a sufficient condition of stability in distribution of uncertain pantograph differential equation are presented. Secondly, we discuss the relationship among stability almost surely, stability in measure, stability in p-th moment, and stability in distribution for the uncertain pantograph differential equation. Finally, Section 4 makes a brief conclusion.

2 Preliminaries

In this section, we will recall some related concepts including canonical Liu process, stability almost surely, stability in measure, stability in p-th moment, stability in mean and Gronwall inequality for UPDE (1).

Let Γ be a nonempty set, and $L$ a σ-algebra over Γ. Each element Λ in $L$ is called an event and assigned a number $M {Λ}$ to indicate the belief degree with which we believe Λ will happen. In order to deal with belief degrees rationally, Liu [7] suggested the following four axioms:

Axiom 1. (Normality Axiom) $M {Λ} = 1$ for the universal set Γ; Axiom 2. (Duality Axiom) $M {Λ} + M {Λ^{c}} = 1$ for any event Λ; Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M_{k} {Λ_{i}};$

Axiom 4. (Product Axiom) Let $(Γ_{k}, L_{k}, M_{k})$ 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 are arbitrary events chosen from $L_{k}$ for k = 1, 2, ⋯, respectively.

Definition 1. [7] An uncertain variable is a measurable function ξ from an uncertainty space $(Γ, L, M)$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Definition 2. [7] The uncertainty distribution Φ of an uncertain variable ξ is defined by $\begin{matrix} Φ (x) = M {ξ \leq x} \end{matrix}$ for any real number x.

Definition 3. [7] Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

An uncertain process is essentially a sequence of uncertain variables indexed by time k.

Definition 4. [7] Let T be an index set and let $(Γ, L, M)$ be an uncertainty space. An uncertain process Z_k is a measurable function from $T \times (Γ, L, M)$ to the set of real numbers such that {Z_k ∈ B} is an event for any Borel set B for each time k.

Different from the Wiener process, the definition of canonical Liu process C_k is as follows.

Definition 5. [7]

An uncertain process C_k is said to be a canonical Liu process if

(a) C₀ = 0 and almost all sample paths are Lipschitz continuous,

(b) C_k has stationary and independent increments,

(c) every increment C_r+k - C_r is a normal uncertain variable with expected value 0 and variance k², whose uncertainty distribution is $Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1}, x \in R,$ and inverse distribution with belief degree α (α ∈ (0, 1)) is $Φ_{k}^{- 1} (α) = \frac{\sqrt{3} Δ k}{- π} ln (\frac{1}{α} - 1) .$

Definition 6. [7] Let Z_k be an uncertain process and C_k be a canonical Liu process. If there exist two processes μ_k and σ_k such that $Z_{k} = Z_{0} + \int_{0}^{k} μ_{r} dr + \int_{0}^{k} σ_{r} d C_{r}, \forall k \geq 0 .$ Then Z_k is called a Liu process with drift μ_k and diffusion σ_k. Furthermore, Z_k has an uncertain differential $d Z_{k} = μ_{k} dk + σ_{k} d C_{k} .$

Definition 7. [7] Suppose that C_k is a canonical Liu process, h and q are continuous functions. Given an initial value Z₀, the uncertain differential equation $d Z_{k} = h (k, Z_{k}) dk + q (k, Z_{k}) d C_{k}$ is called an uncertain differential equation with an initial value Z₀.

Definition 8. [41] The UPDE (1) is said to be stable in measure if for any two solutions Z_k and ${\hat{Z}}_{k}$ with different initial states, respectively, we have

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} M {sup_{k \in [0, T]} | Z_{k} - {\hat{Z}}_{k} | > ε} = 0,$ (3) for any ε > 0, where $M$ is uncertain measure defined in [7], T is a fixed time, ’sup’ is short for supremum.

Definition 9. [41] The UPDE (1) is said to be stable in p-th moment if for any two solutions Z_k and ${\hat{Z}}_{k}$ with different initial states, respectively, we have

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} E [{(sup_{k \in [0, T]} | Z_{k} - {\hat{Z}}_{k} |)}^{p}] = 0,$ (4) where E [ξ] denotes the expected value of ξ defined in [7], 0< p < + ∞.

Definition 10. [42] The UPDE (1) is said to be stable almost surely if for any two solutions Z_k and ${\hat{Z}}_{k}$ with different initial states, respectively, we have

$M {lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Z_{k} - {\hat{Z}}_{k} | = 0} = 1 .$ (5)

Definition 11. [42] The UPDE (1) is said to be stable in mean if for any two solutions Z_k and ${\hat{Z}}_{k}$ with different initial states, respectively, we have

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} E [sup_{k \in [0, T]} | Z_{k} - {\hat{Z}}_{k} |] = 0 .$ (6)

Remark 1. It is obvious that when p = 1, stability in p-th moment become stability in mean for the UPDE (1).

Theorem 1. (Gronwall inequality [7]). Let I denote an interval of the real line of the form [a, + ∞), [a, b] or [a, b) with a < b. Let α, β and μ be real-valued functions defined on I. Assume that β and μ are continuous and that the negative part of α is integrable on every closed and bounded subinterval of I. If β is non-negative, μ satisfies the integral inequality $μ (k) \leq α (k) + \int_{a}^{k} β (r) μ (r) dr, \forall k \in I$ and α is non-decreasing, then we have $μ (k) \leq α (k) exp (\int_{a}^{k} β (r) dr), \forall k \in I .$

3 Stability analysis

3.1 Stability in distribution

Definition 12. Let Z_k and Ẑ_k be two solutions of UPDE (1) with different initial states. Assume the uncertain distributions of Z_k and Ẑ_k are Φ_k (z) and Ψ_k (z), respectively, then the UPDE (1) is said to be stable in distribution if

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k} (z) - Ψ_{k} (z) | = 0$ (7) ∀z at which Φ and Ψ are continuous.

Theorem 2. Assume the UPDE (1) has an unique solution for each given initial state. Then it is stable in distribution if the coefficients $h (k, z, \hat{z})$ and $q (k, z, \hat{z})$ satisfy ${\begin{matrix} | h (k, z_{1}, {\hat{z}}_{1}) - h (k, z_{2}, {\hat{z}}_{2}) | \\ \leq N_{k} (| z_{1} - z_{2} | + | {\hat{z}}_{1} - {\hat{z}}_{2} |) \\ | q (k, z_{1}, {\hat{z}}_{1}) - q ((k, z_{2}, {\hat{z}}_{2})) | \\ \leq N_{k} (| z_{1} - z_{2} | + | {\hat{z}}_{1} - {\hat{z}}_{2} |), \end{matrix}$ (8) where $\forall z_{1}, z_{2}, {\hat{z}}_{1}, {\hat{z}}_{2} \in R, k \in [0, T],$ N_k is a bounded function satisfying $\int_{0}^{T} N_{k} dk < + \infty .$

Proof. Suppose that Z_k and Ẑ_k are two solutions of the equation (1) with different initial states z₀ (Z₀ = z₀) and ${\hat{z}}_{0}$ ( ${\hat{Z}}_{0} = {\hat{z}}_{0}$ ), respectively, then,

$\begin{matrix} {dZ}_{k} = & h (k, Z_{k}, Z_{mk}) dk \\ + q (k, Z_{k}, Z_{mk}) d C_{k}, k \in [0, T] \end{matrix}$ (9) and

$\begin{matrix} d Ẑ_{k} = & h (k, {\hat{Z}}_{k}, {\hat{Z}}_{mk}) dk \\ + q (k, {\hat{Z}}_{k}, {\hat{Z}}_{mk}) d C_{k}, k \in [0, T] . \end{matrix}$ (10) Then for a Lipschitz continuous sample C_k (γ), we have ${\begin{matrix} Z_{k} (γ) & = Z_{0} + \int_{0}^{k} h (r, Z_{r} (γ), Z_{mr} (γ)) dr \\ + \int_{0}^{k} q (r, Z_{r} (γ), Z_{mr} (γ)) d C_{r} (γ), k \in [0, T] \\ Z_{0} & = z_{0}, \end{matrix}$ (11) and ${\begin{matrix} Ẑ_{k} (γ) & = {\hat{Z}}_{0} + \int_{0}^{k} h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{mr} (γ)) dr \\ + \int_{0}^{k} q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{mr} (γ)) d C_{r} (γ), k \in [0, T] \\ {\hat{Z}}_{0} & = {\hat{z}}_{0} . \end{matrix}$ (12) By Theorem 4.1 in [34], the inverse uncertain distributions $Φ_{k}^{- 1} (α)$ and $Ψ_{k}^{- 1} (α)$ of Z_k and Ẑ_k satisfy the ordinary pantograph differential equation

$\begin{matrix} d Φ_{k}^{- 1} (α) & = h (k, Φ_{k}^{- 1} (α), Φ_{mk}^{- 1} (α)) dk + \\ | q (k, Φ_{k}^{- 1} (α), Φ_{mk}^{- 1} (α)) | ϒ^{- 1} (α) dk, \end{matrix}$ (13)

$\begin{matrix} d Ψ_{k}^{- 1} (α) & = h (k, Ψ_{k}^{- 1} (α), Ψ_{mk}^{- 1} (α)) dk + \\ | q (k, Ψ_{k}^{- 1} (α), Ψ_{mk}^{- 1} (α)) | ϒ^{- 1} (α) dk, \end{matrix}$ (14) respectively, where $ϒ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1 .$ It follows that $\begin{matrix} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | \\ \leq & | (Z_{0} + \int_{0}^{k} (h (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) \\ + | q (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) | ϒ^{- 1} (α)) dr) \\ - & (Ẑ_{0} + \int_{0}^{k} (h (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α)) \\ + | q (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α)) | ϒ^{- 1} (α) dr)) | \\ \leq & | Z_{0} - Ẑ_{0} | + | \int_{0}^{k} (h (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) \\ - h (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α))) dr | \\ + & | \int_{0}^{k} (| q (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) | ϒ^{- 1} (α) \\ - | q (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α)) | ϒ^{- 1} (α)) dr | \\ \leq & | Z_{0} - Ẑ_{0} | + \int_{0}^{k} | (h (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) \\ - h (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α))) | dr \\ + & | ϒ^{- 1} (α) | \int_{0}^{k} (| q (r, Φ_{r}^{- 1} (α), Φ_{mr}^{- 1} (α)) | \\ - | q (r, Ψ_{r}^{- 1} (α), Ψ_{mr}^{- 1} (α)) |) dr \\ \leq & | Z_{0} - Ẑ_{0} | + \int_{0}^{k} N_{r} (| Φ_{r}^{- 1} (α) - Ψ_{r}^{- 1} (α) | \\ + | Φ_{mr}^{- 1} (α) - Ψ_{mr}^{- 1} (α) |) dr \\ + | ϒ^{- 1} (α) | \int_{0}^{k} N_{r} (| Φ_{r}^{- 1} (α) - Ψ_{r}^{- 1} (α) | \\ + | Φ_{mr}^{- 1} (α) - Ψ_{mr}^{- 1} (α) |) dr \\ = & | Z_{0} - Ẑ_{0} | \\ + (1 + | ϒ^{- 1} (α) |) \int_{0}^{k} N_{r} (| Φ_{r}^{- 1} (α) - Ψ_{r}^{- 1} (α) | \\ + | Φ_{mr}^{- 1} (α) - Ψ_{mr}^{- 1} (α) |) dr . \end{matrix}$ Thus we have $\begin{matrix} sup_{k \in [0, T]} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | \\ \leq & | Z_{0} - Ẑ_{0} | + (1 + | ϒ^{- 1} (α) |) sup_{k \in [0, T]} \int_{0}^{k} N_{r} \\ \cdot (| Φ_{r}^{- 1} (α) - Ψ_{r}^{- 1} (α) | \\ + | Φ_{mr}^{- 1} (α) - Ψ_{mr}^{- 1} (α) |) dr \\ \leq & | Z_{0} - Ẑ_{0} | + 2 (1 + | ϒ^{- 1} (α) |) sup_{k \in [0, T]} \int_{0}^{k} N_{r} \\ \cdot sup_{v \in [0, r]} | Φ_{v}^{- 1} (α) - Ψ_{v}^{- 1} (α) | dr \\ = & | Z_{0} - Ẑ_{0} | + 2 (1 + | ϒ^{- 1} (α) |) \int_{0}^{T} N_{r} \\ \cdot sup_{v \in [0, r]} | Φ_{v}^{- 1} (α) - Ψ_{v}^{- 1} (α) | dr . \end{matrix}$ By the Gronwall’s inequality, we have $\begin{matrix} sup_{k \in [0, T]} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | \\ \leq & | Z_{0} - Ẑ_{0} | exp (\int_{0}^{T} 2 (1 + | ϒ^{- 1} (α) |) N_{r} dr) \\ = & | Z_{0} - Ẑ_{0} | exp (2 (1 + | ϒ^{- 1} (α) |) \int_{0}^{T} N_{r} dr) . \end{matrix}$ Since $\int_{0}^{T} N_{k} dk < + \infty,$ there exists a real number Q > 0 such that $exp (2 (1 + | ϒ^{- 1} (α) |) \int_{0}^{T} N_{r} dr) < Q$ for any k ≥ 0. Thus, for any given ε > 0, we set δ = ε/Q such that $\begin{matrix} sup_{k \in [0, T]} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | \\ \leq & | Z_{0} - Ẑ_{0} | exp ((1 + | ϒ^{- 1} (α) |) \int_{0}^{k} N_{r} dr) \\ < & δ Q = ε \end{matrix}$ for any k ≥ 0 provided |Z₀ - Ẑ₀| < δ. Then we have

$\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | = 0 \\ , \forall α \in (0, 1) . \end{matrix}$ (15) By Theorem 3.1 in [34], we obtain

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k} (z) - Ψ_{k} (z) | = 0, \forall z \in R .$ (16) So, the UPDE (1) is stable in distribution.

Example 1. Consider an UPDE

${dZ}_{k} = Z_{mk} dk + σ {dC}_{k}, k \in [0, T] .$ (17) The solutions of Eq. (17) with two initial states z₀ and ${\hat{z}}_{0}$ are ${\begin{matrix} Z_{k} & = Z_{0} + \int_{0}^{k} Z_{mr} dr + σ C_{r}, k \in [0, T] \\ Z_{0} & = z_{0}, \end{matrix}$ (18) and ${\begin{matrix} {\hat{Z}}_{k} & = {\hat{Z}}_{0} + \int_{0}^{k} {\hat{Z}}_{mr} dr + σ C_{r}, k \in [0, T] \\ {\hat{Z}}_{0} & = {\hat{z}}_{0}, \end{matrix}$ (19) whose inverse uncertain distributions are

$\begin{matrix} Φ_{k}^{- 1} (α) \\ = & Z_{0} + \int_{0}^{k} Φ_{mr}^{- 1} (α) + σ \frac{r \sqrt{3}}{π} ln \frac{α}{1 - α} dr, \end{matrix}$ (20) and

$\begin{matrix} Ψ_{k}^{- 1} (α) \\ = & {\hat{Z}}_{0} + \int_{0}^{k} Ψ_{mr}^{- 1} (α) + σ \frac{r \sqrt{3}}{π} ln \frac{α}{1 - α} dr, \end{matrix}$ (21) respectively. Then we have

$\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k}^{- 1} (α) - Ψ_{k}^{- 1} (α) | = 0 \\ , \forall α \in (0, 1) . \end{matrix}$ (22) Its equivalent form is

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k} (z) - Ψ_{k} (z) | = 0, \forall z \in R .$ (23) So, the UPDE (17) is stable in distribution.

3.2 Comparison of stability

Theorem 3. If the UPDE (1) is stable in measure, then it is stable in distribution.

Proof. Suppose that Z_k and Ẑ_k are two solutions of the equation (1) with different initial states z₀ (Z₀ = z₀) and ${\hat{z}}_{0}$ ( ${\hat{Z}}_{0} = {\hat{z}}_{0}$ ), respectively. According to Definition 4.1 in [41], we have $\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} M {sup_{k \in [0, T]} | Z_{k} (γ) - Ẑ_{k} (γ) | > ε} \\ = 0 . \end{matrix}$ for any given number ε > 0. Suppose z is a given real number. On the one hand, for any $\hat{z} > z$ , we can get $\begin{matrix} {Z_{k} \leq z} \\ = & {Z_{k} \leq z, Ẑ_{k} \leq \hat{z}} \cup {Z_{k} \leq z, Ẑ_{k} > \hat{z}} \\ \subset {Ẑ_{k} \leq \hat{z}} \cup {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq \hat{z} - z} . \end{matrix}$ By the monotonicity theorem and subadditivity axiom in uncertainty theory [7], it holds that $\begin{matrix} Φ_{k} (z) - Ψ_{k} (\hat{z}) \\ \leq & M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq \hat{z} - z} . \end{matrix}$ Thus we have $\begin{matrix} Φ_{k} (z) - Ψ_{k} (\hat{z}) \\ \leq & lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq \hat{z} - z} . \end{matrix}$ We can get $M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq \hat{z} - z} \to 0$ as $| Z_{0} - {\hat{Z}}_{0} | \to 0$ . Then we have $\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} Φ_{k} (z) - Ψ_{k} (\hat{z}) \leq 0 \end{matrix}$ for any $\hat{z} > z$ . Letting $\hat{z} \to z^{+}$ , we have

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} Φ_{k} (z) - Ψ_{k} (z) \leq 0 .$ (24) On the other hand, for any $\tilde{z} < z$ , it is obvious that $\begin{matrix} {Ẑ_{k} \leq \tilde{z}} \\ = & {Z_{k} \leq z, Ẑ_{k} \leq \tilde{z}} \cup {Z_{k} > z, Ẑ_{k} \leq \tilde{z}} \\ \subset {Z_{k} \leq z} \cup {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq z - \tilde{z}} . \end{matrix}$ By the monotonicity theorem and subadditivity axiom in uncertainty theory [7], it holds that $\begin{matrix} Ψ_{k} (\tilde{z}) - Φ_{k} (z) \\ \leq & M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq z - \tilde{z}} . \end{matrix}$ Thus we have $\begin{matrix} Ψ_{k} (\tilde{z}) - Φ_{k} (z) \\ \leq & lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq z - \tilde{z}} . \end{matrix}$ We can get $M {sup_{k \in [0, T]} | Z_{k} - Ẑ_{k} | \geq z - \tilde{z}} \to 0$ as $| Z_{0} - {\hat{Z}}_{0} | \to 0$ . Then we have $\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} Ψ_{k} (\tilde{z}) - Φ_{k} (z) \leq 0 \end{matrix}$ for any $\tilde{z} < z$ . Letting $\tilde{z} \to z^{-}$ , we have

$lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} Ψ_{k} (z) - Φ_{k} (z) \leq 0 .$ (25) By (24) and (25), we obtain $\begin{matrix} lim_{| Z_{0} - {\hat{Z}}_{0} | \to 0} sup_{k \in [0, T]} | Φ_{k} (z) - Ψ_{k} (z) | = 0 . \end{matrix}$ According to Definition 3.1, the UPDE (1) is stable in distribution.

Theorem 4. If the UPDE (1) is almost surely stable, then it is stable in distribution.

Proof. If the UPDE (1) is almost surely stable, by Theorem 4.2 in Ref [42], the UPDE (1) is stable in measure. And by Theorem 3, we obtain the UPDE (1) is stable in distribution.

Theorem 5. If the UPDE (1) is stable in p-th moment (0 < p < + ∞), then it is stable in distribution.

Proof. If the UPDE (1) is stable in p-th moment (0 < p < + ∞), by Theorem 5.1 in Ref [41], the UPDE (1) is stable in measure. And by Theorem 3, we obtain the UPDE (1) is stable in distribution.

Theorem 6. If the UPDE (1) is stable in mean, then it is stable in distribution.

Proof. If the UPDE (1) is stable in mean, by Theorem 3.2 in Ref [42], the UPDE (1) is stable in measure. And by Theorem 3, we obtain the UPDE (1) is stable in distribution.

Remark 2. Theorem 2, 3, 4, 5, and 6 give a sufficient condition but not a necessary condition for an uncertain pantograph differential equations being stable in distribution.

4 Conclusions

This paper mainly discussed a special type of unbounded uncertain delay differential equation. First of all, we proposed a new stability called stability in distribution. Furthermore, some theorems on stability in distribution for uncertain pantograph differential equation were provided. In the future work, we will discuss the numerical methods for solving uncertain pantograph differential equations and make the application research.

Footnotes

Acknowledgments

The research was supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0170) and National Natural Science Foundation of China (61374183).

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