New sufficient conditions for p -th moment stability of uncertain delay differential equation

Abstract

Uncertain delay differential system is an important mathematical model. Stability is a basic problem of uncertain delay differential system. Delay and uncertain interference often lead to changes in the stability of the system. Establishing the judgment of the stability of uncertain delay differential system conditions is very important. Based on the strong Lipschitz condition, the judgment of p-th moment stability for uncertain delay differential equations (UDDEs) has been investigated. Actually, the strong Lipschitz condition is assumed that it only relates to the current state, it is difficult to be employed to determine the stability in p-th moment for the UDDEs. In this paper, we consider two kinds of new Lipschitz conditions containing the current state and the past state, which are more weaker than the strong Lipschitz condition. Meanwhile, new sufficient theorems and corollaries under the new Lipschitz conditions as the tools to judge the p-th moment stability for the UDDEs are proved. Some examples explain the rationality of the corresponding theorems and corollaries.

Keywords

Liu process new Lipschitz conditions uncertain delay differential equations

1 Introduction

Stochastic differential equations (SDEs) are an active branch of mathematics that emerged in the 1940s. Many mathematicians in the world have devoted themselves to this new scientific field and achieved remarkable results. As early as 1951, Itô [1] defined stochastic integral for the first time, also known as Itô integral, and proposed SDEs on this basis. Later, researchers discovered that there is a hysteresis phenomenon in many actual dynamic systems. The evolution of the system depends on the current state of the system as well as the past state of the system. From this, the study of delay differential equations is derived. Among them, stability is a basic problem of differential systems. Since then, the p-th moment stability began to emerge as a special kind of stability. For example, Kolmanouskii and Nosoul [2] adopted some new methods for determining the asymptotic stability of p-th moments of zero solutions of constant stochastic delay differential equations under the Lipshitz hypothesis and linear growth hypothesis as early as 1982. After that, Zhu et al. [3] studied a special variable delay stochastic differential equation, and gave several necessary and sufficient conditions for the p-th moment exponential stability of the equation. In 2004, Wu et al. [4] studied uniform p-th moment stability and asymptotic p-th moment stability for a special kind of stochastic differential equations with jumps. Later, scholars obtained a stochastic differential equation with jumps by combining jumps and delays. Peng et al. [5] obtained the sufficient conditions for p-th moment stability of stochastic delay differential equations by constructing appropriate Lyapunov functions. Peng et al. [6] changed the coefficients of the system to time-varying coefficients on the basis of Ref [5], and using the principle of comparison and Razumikhin technique to obtain the p-th moment stability and p-th moment asymptotic stability of this kind of stochastic delay differential equation sufficient conditions, expand the scope of application. Zhu et al. [7] further considered p-th moment exponential stabilisation of hybrid stochastic delay differential equations. In addition, there are many theoretical and applied researches on the asymptotic stability of p-th moments related to stochastic differential equations, which are not stated one by one here.

It is well known that all results about SDEs are based on an axiomatic probability theory, and large amounts of sample data is needed to obtain the frequency of their random disturbances. Furthermore, their distribution functions can be obtained. 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, and we need to invite some domain experts to evaluate the belief degree that each event may happen in these situations. Human uncertainty with respect to belief degrees can play a crucial role in addressing the issue of indeterminate phenomenon. Uncertainty theory is one of the most important achievements in dealing with belief degrees. Many scholars have conducted a lot of research around its basic theories [8] and application fields [9]. Hitherto, uncertainty theory has been widely used in control [10] and game [11], prediction [12], biology [13], finance [14] and other fields. In the field of uncertain system, there have been many studies on the methods of using uncertain differential equations [15 –17], and most scholars investigated uncertain differential equations and their improved forms driven by Liu process. For uncertain system, stability is the prerequisite to ensure the normal operation of the control system. Stability theory is one of the research hotspots in the application of uncertain differential equations, which has been extended into many widely used versions. The standard stability theorem of UDEs developed by Yao et al. [18] is the first one, which is suitable to nonlinear and linear uncertain system but is limited to a necessary condition for an UDE being stable. Stability in p-th moment proposed by Sheng and Wang [19] for UDE is the second, which is applicable to suitable to nonlinear and linear uncertain system but is limited to a necessary condition for an UDE being stable in p-th moment. Recent years, researches on stability in p-th moment for uncertain system are emerging one after another. For example, stability in p-th moment for multifactor UDE [20], UDE with jumps [21], multi-dimensional UDE [22], uncertain heat equations [23] and uncertain spring vibration equation [24] were successively investigated.

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 [25]; Ge et al. [26]; Wang et al. [27]). Following that, Wang et al. [28, 29] put forward several kinds of stabilities including stability in p-th moment of UDDEs. Jia et al. [30] discussed the stability in distribution. Recently, Wang et al. [31 –33] and Jia et al. [34, 35] considered some stabilities about other class of UDDEs. Gao and Jia [36, 37] proved stability in measure and stability in mean for UDDEs based on new Lipschitz conditions. The delay equation in this paper is introduced as follows:

${\begin{matrix} {dZ}_{k} = & h (k, Z_{k}, Z_{k - τ}) dk \\ + q (k, Z_{k}, Z_{k - τ}) d C_{k}, k \in [0, + \infty) \\ Z_{0} = & φ = {φ (k), k \in [- τ, 0]}, \end{matrix}$ (1) where C_k is a Liu process with respect to time k, and h and q are two continuous functions, Z₀ is the initial value, τ is a delay. Its equivalent integral form is as follows:

${\begin{matrix} Z_{k} = & Z_{0} + \int_{0}^{k} h (r, Z_{r}, Z_{r - τ}) dr \\ + \int_{0}^{k} q (r, Z_{r}, Z_{r - τ}) d C_{r}, k \in [0, + \infty) \\ Z_{0} = & φ = {φ (k), k \in [- τ, 0]} . \end{matrix}$ (2)

The objective of this paper is to study the new p-th moment stability for UUDE driven by Liu processs. The contributions of this paper are: (1) to derive the first sufficient condition; (2) to obtain the second sufficient condition; (3) counterexample illustrates the sufficient and unnecessary conditions. The stability in p-th moment of Ref [28] for UDDEs was studied based on the strong Lipschitz condition, which was so strictly and hardly used to judge the stability in p-th moment. Inspired by Refs [36, 37], we consider two kinds of new Lipscitz conditions more weaker than the strong Lipschitz condition, then two theorems of stability in p-th moment for UDDEs based on these new Lipschitz conditions are verified in this paper.

The structure of this paper is organized as follows. Section 2 recall some preliminary results about uncertain variable, uncertain expected value, Liu process, UDDEs and so on, which are essential for our analysis. Section 3 present the first sufficient condition and the second sufficient condition of stability in p-th moment based on the new Lipschitz condition for UDDEs and give some examples. In Section 4, we give a conclusion.

2 Preliminaries

In this section, we will recall some preliminary results about uncertain variable, uncertain expected value, Liu process, UDDEs and so on.

Definition 2.1. [9] 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 2.2. [9] Let ξ be an uncertain variable. Then the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} dx - \int_{- \infty}^{0} M {ξ \leq x} dx$ provided that at least one of the two integrals is finite.

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

Definition 2.3. [9] Let T be an index set and let $(Γ, L, M)$ be an uncertainty space. An uncertain process 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.

Definition 2.4. [9] An uncertain process C_k with respect to time k is said to be a canonical process if

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

(ii) C_k has stationary and independent increments,

(iii) every increment C_r+k - C_r is a normal uncertain variable with expected value 0 and variance k², whose uncertainty distribution is $\begin{matrix} Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1}, x \in ℝ . \end{matrix}$

Definition 2.5. [9] Let Z_k be an uncertain process and C_k be a Liu process. For any partition of closed interval [a, b] with a = k₁ < k₂ < ⋯ < k_n+1 = b, the mesh is written as $\begin{matrix} Δ = max_{1 \leq i \leq n} | k_{i + 1} - k_{i} | . \end{matrix}$ Then uncertain integral of Z_k with respect to C_k is $\begin{matrix} \int_{a}^{b} Z_{k} d C_{k} = lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{k_{i}} \cdot (C_{k_{i + 1}} - C_{k_{i}}) \end{matrix}$ provided that the limit exists almost surely and is finite. In this case, the uncertain process Z_k is said to be integrable.

Definition 2.6. [28] The UDDE (1) is said to be stable in p-th moment if

$lim_{sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \to 0} E [| Z_{k} - {\hat{Z}}_{k} |^{p}] = 0, \forall p > 0,$ (3) where E is uncertain expected value.

Theorem 2.1. [28] Assume that UDDE (1) has an unique solution for each given initial state. Then it is stable in p-th moment if the coefficients $h (k, z, \hat{z})$ and $q (k, z, \hat{z})$ satisfy

${\begin{matrix} | h (k, z_{1}, \hat{z}) - h (k, z_{2}, \hat{z}) | \leq N_{k} | z_{1} - z_{2} | \\ , \forall z_{1}, z_{2}, \hat{z} \in ℝ, k \geq 0 \\ | q (k, z_{1}, \hat{z}) - q (k, z_{2}, \hat{z}) | \leq G_{k} | z_{1} - z_{2} | \\ , \forall z_{1}, z_{2}, \hat{z} \in ℝ, k \geq 0, \end{matrix}$ (4) where N_k and G_k are two bounded functions satisfying

$\begin{matrix} \int_{0}^{+ \infty} N_{k} dk < \infty, \int_{0}^{+ \infty} G_{k} dk < \frac{π}{\sqrt{3} p} . \end{matrix}$

3 Main results

3.1 The first sufficient condition

Theorem 3.1. Assume that UDDE (1) has an unique solution for each given initial state. Then it is stable in p-th moment if the coefficients $h (k, z, \hat{z})$ and $q (k, z, \hat{z})$ satisfy

$\begin{matrix} | h (k, z_{1}, {\hat{z}}_{1}) - h (k, z_{2}, {\hat{z}}_{2}) | \lor | q (k, z_{1}, {\hat{z}}_{1}) - q ((k, z_{2}, {\hat{z}}_{2})) | \\ \leq A_{1 k} | z_{1} - z_{2} | + A_{2 k} | {\hat{z}}_{1} - {\hat{z}}_{2} |, \forall z_{1}, z_{2}, \hat{z} \in ℝ, k \geq 0, \end{matrix}$ (5)

where $\forall z_{1}, z_{2}, {\hat{z}}_{1}, {\hat{z}}_{2} \in ℝ, k > 0,$ where A_1k and A_2k are the real-valued functions satisfying

$\int_{0}^{+ \infty} A_{1 k} + 2 A_{2 k} dk < \frac{π}{\sqrt{3} p}, \forall p > 0 .$ (6)

Proof. Suppose that Z_k and $\hat{Z}$ _k are two solutions of the equation (1) with different initial states φ (k) and ψ (k) (k ∈ [- τ, 0]), respectively. Then,

$\begin{matrix} {\begin{matrix} {dZ}_{k} = h (k, Z_{k}, Z_{k - τ}) dk + q (k, Z_{k}, Z_{k - τ}) d C_{k}, k \in [0, + \infty) \\ Z_{0} = φ = {φ (k), k \in [- τ, 0]}, \end{matrix} \end{matrix}$

and

$\begin{matrix} {\begin{matrix} d {\hat{Z}}_{k} = h (k, {\hat{Z}}_{k}, {\hat{Z}}_{k - τ}) dk + q (k, {\hat{Z}}_{k}, {\hat{Z}}_{k - τ}) d C_{k}, k \in [0, + \infty) \\ {\hat{Z}}_{0} = ψ = {ψ (k), k \in [- τ, 0]} . \end{matrix} \end{matrix}$

Then for a Lipschitz continuous sample C_k (γ), we have

$\begin{matrix} {\begin{matrix} Z_{k} (γ) = & Z_{0} + \int_{0}^{k} h (r, Z_{r} (γ), Z_{r - τ} (γ)) dr \\ + \int_{0}^{k} q (r, Z_{r} (γ), Z_{r - τ} (γ)) d C_{r} (γ), k \in [0, + \infty) \\ Z_{0} = & φ = {φ (k), k \in [- τ, 0]}, \end{matrix} \end{matrix}$

and

$\begin{matrix} {\begin{matrix} {\hat{Z}}_{k} (γ) = & {\hat{Z}}_{0} + \int_{0}^{k} h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) dr \\ + \int_{0}^{k} q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) d C_{r} (γ), k \in [0, + \infty) \\ {\hat{Z}}_{0} = & ψ = {ψ (k), k \in [- τ, 0]} . \end{matrix} \end{matrix}$

By condition (5) and Lemma 4.1 in [15], we have

$\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} + \int_{0}^{k} h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) dr \\ + \int_{0}^{k} q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) {dC}_{r} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + | \int_{0}^{k} h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) dr | \\ + | \int_{0}^{k} q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) {dC}_{r} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + \int_{0}^{k} | h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) | dr \\ + H (γ) \int_{0}^{k} | q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) | dr \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + \int_{0}^{k} A_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | + A_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ + H (γ) \int_{0}^{k} A_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | + A_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ = & | Z_{0} - {\hat{Z}}_{0} | + (1 + H (γ)) \int_{0}^{k} {A_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | \\ + A_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) |} dr, \end{matrix}$

where H (γ) is the Lipschitz constant of C_k (γ). Meanwhile, we set μ = r - τ, and obtain $\begin{matrix} \int_{0}^{k} A_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ = \int_{- τ}^{k - τ} A_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ = \int_{- τ}^{0} A_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ + \int_{0}^{k - τ} A_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{- τ}^{0} A_{2 (μ + τ)} d μ \\ + \int_{0}^{k - τ} A_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} A_{2 μ} d μ \\ + \int_{0}^{k} A_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ . \end{matrix}$

Then, we denote by $W = \int_{0}^{τ} A_{2 μ} d μ$ , and obtain

$\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + (1 + H (γ)) \int_{0}^{k} {A_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | \\ + A_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) |} dr \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + (1 + H (γ)) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} A_{2 μ} d μ \\ + (1 + H (γ) \int_{0}^{k} (A_{1 μ} + A_{2 (μ + τ)}) | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq & {1 + (1 + H (γ)) W} sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | + (1 + H (γ)) \\ \int_{0}^{k} (A_{1 μ} + A_{2 (μ + τ)}) | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ . \end{matrix}$

Gronwall inequality implies that $\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & {1 + (1 + H (γ)) W} sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) \int_{0}^{k} (A_{1 μ} + A_{2 (μ + τ)}) d μ) \\ = & {1 + (1 + H (γ)) W} sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) {\int_{0}^{k} A_{1 μ} d μ + \int_{τ}^{k + τ} A_{2 μ} d μ}) \\ \leq & {1 + (1 + H (γ)) W} sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) {\int_{0}^{+ \infty} A_{1 μ} d μ + \int_{0}^{+ \infty} A_{2 μ} d μ}) \\ \leq & exp (1 + (1 + H (γ)) W) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) {\int_{0}^{+ \infty} A_{1 μ} d μ + \int_{0}^{+ \infty} A_{2 μ} d μ}) \\ \leq & sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | exp ((1 + H (γ) \\ {W + \int_{0}^{+ \infty} A_{1 μ} + A_{2 μ} d μ}) \\ \leq & sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | exp ((1 + H (γ) \\ {\int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}) . \end{matrix}$ Thus we have $\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) |^{p} \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} |^{p} \\ exp (p (1 + H (γ) {\int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}) . \end{matrix}$ By the properties of expected value for uncertain variables, we have $\begin{matrix} E [| Z_{k} (γ) - {\hat{Z}}_{k} (γ) |^{p}] \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} |^{p} exp (p {\int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}) \\ E [exp (pH {\int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ})] . \end{matrix}$ Thus, It follows from the definition of uncertain expected value and Theorem 2 in [18] that

$\begin{matrix} E [exp (pH \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ)] \\ = & \int_{0}^{+ \infty} M {exp (pH \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ) \geq x} dx \\ \leq & 1 + \int_{1}^{+ \infty} M {exp (pH \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ) \geq x} dx \\ = & 1 + \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ \int_{0}^{+ \infty} \\ exp (p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ x) M {H \geq x} dx \\ = & 1 + \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ \int_{0}^{+ \infty} \\ exp (p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ x) (1 - M {H \leq x}) dx \\ \leq & 1 + \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ \int_{0}^{+ \infty} exp (p \int_{0}^{+ \infty} A_{1 μ} \\ + 2 A_{2 μ} d μ x) (1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1} - 1)) dx \\ = & 1 + 2 \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ \int_{0}^{+ \infty} exp (p \int_{0}^{+ \infty} A_{1 μ} \\ + 2 A_{2 μ} d μ x) {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1} dx \\ = & 1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}}}^{- 1} dx . \end{matrix}$

Because $\int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ < \frac{π}{\sqrt{3} p}$ . Let $\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ} = a$ , so a > 1. Then, $\begin{matrix} 1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}}}^{- 1} dx \\ = 1 + 2 \int_{1}^{+ \infty} \frac{1}{1 + x^{a}} dx, a > 1 . \end{matrix}$ Since $\frac{1}{1 + x^{a}} < \frac{1}{x^{a}}, a > 1,$ while $\int_{1}^{+ \infty} \frac{1}{x^{a}} dx, a > 1$ is convergent, so $\int_{1}^{+ \infty} \frac{1}{1 + x^{a}} dx, a > 1$ is convergent. So $1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ}}}^{- 1} dx < + \infty .$ That is, $\begin{matrix} E [exp (pH \int_{0}^{+ \infty} A_{1 μ} + 2 A_{2 μ} d μ)] < + \infty . \end{matrix}$ In addition, $\begin{matrix} \int_{0}^{+ \infty} p (A_{1 μ} + 2 A_{2 μ}) d μ < \frac{π}{\sqrt{3}} < + \infty . \end{matrix}$ Thus, we can get $\begin{matrix} lim_{sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \to 0} E [| Z_{k} - {\hat{Z}}_{k} |^{p}] = 0, \forall p > 0 . \end{matrix}$

Example 3.1. Consider an UDDE

${dZ}_{k} = \frac{2}{5 (1 + k^{2})} Z_{k - 1} dk + \frac{π}{10} exp (- k) Z_{k} {dC}_{k} .$ (7) Firstly, we set $h = \frac{1}{5 (1 + k^{2})} \hat{z}$ and $q = \frac{π}{10} exp (- k) z$ , if we apply the Theorem 2, we can easily find that it not follows the strong Lipschitz condition, thus it becomes invalid, but if we use the Theorem 3.1, we have $\begin{matrix} A_{1 k} = \frac{2}{5 (1 + k^{2})}, A_{2 k} = \frac{π}{10} exp (- k) \end{matrix}$ Moreover, we obtain $\begin{matrix} \int_{0}^{+ \infty} A_{1 k} dk = \int_{0}^{+ \infty} \frac{2}{5 (1 + k^{2})} dk = \frac{π}{5}, \\ \int_{0}^{+ \infty} A_{2 k} dk = \int_{0}^{+ \infty} \frac{π}{10} exp (- k) dk = \frac{π}{10} . \end{matrix}$ and $\begin{matrix} \int_{0}^{+ \infty} A_{1 k} + 2 A_{2 k} dk < \frac{π}{\sqrt{3} p}, \forall 0 < p < \frac{5}{2 \sqrt{3}} . \end{matrix}$ Therefore, the UDDE (7) is stable in p-th moment based on the Theorem 3.1.

Remark 3.1. The Theorem 2 is different to the Theorem 3.1. The strong Lipschitz condition of Theorem 2 only involves the present state, but the coefficients of UDDEs must contain the past state, so it is difficult to judge the stability in p-th moment for UDDEs, the new Lipschitz condition of the Theorem 3.1 concerns not only the present state but also the past state.

Corollary 3.1. Supposing that u_ik, v_ik, and η_ik (i = 1,2) are real-valued functions, then the linear UDDE

$\begin{matrix} d Z_{k} = (u_{1 k} Z_{k} + u_{2 k} Z_{k - τ} + u_{3 k}) dk \\ + (u_{1 k} Z_{k} + u_{2 k} Z_{k - τ} + u_{4 k}) d C_{k} \end{matrix}$ (8) is stable in p-th moment if u_ik (i = 1, 2) satisfying $\begin{matrix} \int_{0}^{+ \infty} u_{1 k} + 2 u_{2 k} dk < \frac{π}{\sqrt{3} p}, \forall p > 0 . \end{matrix}$

Proof. Take $h (k, z, \hat{z}) = u_{1 k} z + u_{2 k} \hat{z} + u_{3 k}$ and $q (k, z, \hat{z}) = u_{1 k} z + u_{2 k} \hat{z} + u_{4 k}$ , then we obtain $\begin{matrix} | h (k, z_{1}, {\hat{z}}_{1}) - h (k, z_{2}, {\hat{z}}_{2}) | \lor | q (k, z_{1}, {\hat{z}}_{1}) \\ - q ((k, z_{2}, {\hat{z}}_{2})) | \leq u_{1 k} | z_{1} - z_{2} | + u_{2 k} | {\hat{z}}_{1} - {\hat{z}}_{2} | . \end{matrix}$ Because of that they satisfy the Theorem 3.1 based on the condition (5), so this corollary is proved.

Example 3.2. Consider an UDDE

$\begin{matrix} {dZ}_{k} = (\frac{1}{2 π (k + 1)^{2}} Z_{k} + \frac{1}{8} exp (- k) Z_{k - 1}) dk \\ + (\frac{1}{2 π (k + 1)^{2}} Z_{k} + \frac{1}{8} exp (- k) Z_{k - 1} + tan (k)) {dC}_{k} \end{matrix}$ (9) and set $\begin{matrix} h = \frac{1}{2 π (k + 1)^{2}} z + \frac{1}{8} exp (- k) \hat{z}, \\ q = \frac{1}{2 π (k + 1)^{2}} z + \frac{1}{8} exp (- k) \hat{z} + tan (k) . \end{matrix}$ Thus, we obtain $\begin{matrix} u_{1 k} = \frac{1}{2 π (k + 1)^{2}}, u_{2 k} = \frac{1}{8} exp (- k) \end{matrix}$ and $\begin{matrix} \int_{0}^{+ \infty} u_{1 k} dk = \int_{0}^{+ \infty} \frac{1}{2 π (k + 1)^{2}} dk = \frac{1}{4}, \\ \int_{0}^{+ \infty} u_{2 k} dk = \int_{0}^{+ \infty} \frac{1}{8} exp (- k) dk = \frac{1}{8} \end{matrix}$ $\begin{matrix} \int_{0}^{+ \infty} u_{1 k} + 2 u_{2 k} dk = \frac{1}{2} < \frac{π}{\sqrt{3} p}, \forall 0 < p < \frac{2 π}{\sqrt{3}} . \end{matrix}$ By means of the Theorem 3.1 or the Corollary 3.1, the UDDE (9) is stable in p-th moment.

3.2 The second sufficient condition

Theorem 3.2. Assume that UDDE (1) has an unique solution for each given initial state. Then it is stable in p-th moment if the coefficients $h (k, z, \hat{z})$ and $q (k, z, \hat{z})$ satisfy

$\begin{matrix} | h (k, z_{1}, {\hat{z}}_{1}) - h (k, z_{2}, {\hat{z}}_{2}) | \leq B_{1 k} | z_{1} - z_{2} | \\ + B_{2 k} | {\hat{z}}_{1} - {\hat{z}}_{2} |, \forall z_{1}, z_{2}, {\hat{z}}_{1}, {\hat{z}}_{2} \in ℝ, k \geq 0, \end{matrix}$ (10) and

$\begin{matrix} | q (k, z_{1}, {\hat{z}}_{1}) - q ((k, z_{2}, {\hat{z}}_{2})) | \leq B_{3 k} | z_{1} - z_{2} | \\ + B_{4 k} | {\hat{z}}_{1} - {\hat{z}}_{2} |, \forall z_{1}, z_{2}, {\hat{z}}_{1}, {\hat{z}}_{2} \in ℝ, k \geq 0, \end{matrix}$ (11) where B_1k and B_2k, B_3k and B_4k are the real-valued functions satisfying

$\begin{matrix} \int_{0}^{+ \infty} B_{1 k} dk < + \infty, \int_{0}^{+ \infty} B_{2 k} dk < + \infty, \\ \int_{0}^{+ \infty} B_{3 k} + 2 B_{4 k} dk < \frac{π}{\sqrt{3} p}, \forall p > 0 . \end{matrix}$ (12)

Proof. Suppose that Z_k and $\hat{Z}$ _k are two solutions of the equation (1) with different initial states φ (k) and ψ (k) (k ∈ [- τ, 0]), respectively. Then,

$\begin{matrix} {\begin{matrix} {dZ}_{k} = h (k, Z_{k}, Z_{k - τ}) dk + q (k, Z_{k}, Z_{k - τ}) d C_{k}, k \in [0, + \infty) \\ Z_{0} = φ = {φ (k), k \in [- τ, 0]}, \end{matrix} \end{matrix}$

and

Then for a Lipschitz continuous sample C_k (γ), we have

and

By condition (10), (11) and Lemma 4.1 in [15], we have

$\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} + \int_{0}^{k} h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) dr \\ + \int_{0}^{k} q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) {dC}_{r} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + | \int_{0}^{k} h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) dr | \\ + | \int_{0}^{k} q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) {dC}_{r} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + \int_{0}^{k} | h (r, Z_{r} (γ), Z_{r - τ} (γ)) - h (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) | dr \\ + H (γ) \int_{0}^{k} | q (r, Z_{r} (γ), Z_{r - τ} (γ)) - q (r, {\hat{Z}}_{r} (γ), {\hat{Z}}_{r - τ} (γ)) | dr \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + \int_{0}^{k} B_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | + B_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ + H (γ) \int_{0}^{k} B_{3 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | + B_{4 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr, \end{matrix}$

where H (γ) is the Lipschitz constant of C_k (γ).Meanwhile, we set μ = r - τ, and obtain $\begin{matrix} \int_{0}^{k} B_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ = \int_{- τ}^{k - τ} B_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ = \int_{- τ}^{0} B_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ + \int_{0}^{k - τ} B_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{- τ}^{0} B_{2 (μ + τ)} d μ \\ + \int_{0}^{k - τ} B_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} B_{2 μ} d μ \\ + \int_{0}^{k} B_{2 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ . \end{matrix}$ By the same means, $\begin{matrix} \int_{0}^{k} B_{4 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} B_{4 μ} d μ \\ + \int_{0}^{k} B_{4 (μ + τ)} | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ . \end{matrix}$ It holds that $\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + \int_{0}^{k} B_{1 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | \\ + B_{2 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ + H (γ) \int_{0}^{k} B_{3 r} | Z_{r} (γ) - {\hat{Z}}_{r} (γ) | \\ + B_{4 r} | Z_{r - τ} (γ) - {\hat{Z}}_{r - τ} (γ) | dr \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} B_{2 μ} d μ \\ + \int_{0}^{k} (B_{1 μ} + B_{2 (μ + τ)}) | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ + H (γ) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \int_{0}^{τ} B_{4 μ} d μ \\ + \int_{0}^{k} (B_{3 μ} + B_{4 (μ + τ)}) | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq & | Z_{0} - {\hat{Z}}_{0} | + {\int_{0}^{τ} (B_{2 μ} + H (γ) B_{4 μ}) d μ} \\ sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | + \int_{0}^{k} (B_{1 μ} + B_{2 (μ + τ)} \\ + H (γ) {B_{3 μ} + B_{4 (μ + τ)}}) | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ \\ \leq & (1 + {\int_{0}^{τ} (B_{2 μ} + H (γ) B_{4 μ}) d μ}) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ + \int_{0}^{k} (B_{1 μ} + B_{2 (μ + τ)} + H (γ) {B_{3 μ} + B_{4 (μ + τ)}}) \\ | Z_{μ} (γ) - {\hat{Z}}_{μ} (γ) | d μ . \end{matrix}$ Gronwall inequality implies that

$\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) | \\ \leq & (1 + {\int_{0}^{τ} (B_{2 μ} + H (γ) B_{4 μ}) d μ}) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ \cdot exp (\int_{0}^{k} (B_{1 μ} + B_{2 (μ + τ)} + H (γ) {B_{3 μ} + B_{4 (μ + τ)}}) d μ) \\ \leq & exp (\int_{0}^{τ} (B_{2 μ} + H (γ) B_{4 μ}) d μ) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) {\int_{0}^{+ \infty} A_{1 μ} d μ + \int_{0}^{+ \infty} A_{2 μ} d μ}) \\ \cdot exp (\int_{0}^{+ \infty} (B_{1 μ} + B_{2 (μ + τ)} + H (γ) {B_{3 μ} + B_{4 (μ + τ)}}) d μ) \\ \leq & exp (\int_{0}^{+ \infty} (B_{2 μ} + H (γ) B_{4 μ}) d μ) sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \\ exp ((1 + H (γ) {\int_{0}^{+ \infty} A_{1 μ} d μ + \int_{0}^{+ \infty} A_{2 μ} d μ}) \\ \cdot exp (\int_{0}^{+ \infty} (B_{1 μ} + B_{2 (μ + τ)} + H (γ) {B_{3 μ} + B_{4 (μ + τ)}}) d μ) \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | exp (\int_{0}^{+ \infty} (B_{1 μ} + 2 B_{2 μ} \\ + H (γ) {B_{3 μ} + 2 B_{4 μ}}) d μ) . \end{matrix}$

Thus we have

$\begin{matrix} | Z_{k} (γ) - {\hat{Z}}_{k} (γ) |^{p} \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} |^{p} exp (p {\int_{0}^{+ \infty} B_{1 μ} + 2 B_{2 μ} d μ}) \\ exp (pH (γ) {\int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ}) . \end{matrix}$ By the definition of expected value for uncertain variables, we have $\begin{matrix} E [| Z_{k} (γ) - {\hat{Z}}_{k} (γ) |^{p}] \\ \leq sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} |^{p} exp (p {\int_{0}^{+ \infty} B_{1 μ} + 2 B_{2 μ} d μ}) \\ E [exp (pH (γ) {\int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ})] . \end{matrix}$ Thus, it follows from the definition of uncertain expected value and Theorem 2 in [18] that

$\begin{matrix} E [exp (pH \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ)] \\ = & \int_{0}^{+ \infty} M {exp (pH \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ) \geq x} dx \\ \leq & 1 + \int_{1}^{+ \infty} M {exp (pH \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ) \geq x} dx \\ = & 1 + \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ \int_{0}^{+ \infty} \\ exp (p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ x) M {H \geq x} dx \\ = & 1 + \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ \int_{0}^{+ \infty} \\ exp (p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ x) (1 - M {H \leq x}) dx \\ \leq & 1 + \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ \int_{0}^{+ \infty} exp (p \int_{0}^{+ \infty} B_{3 μ} \\ + 2 B_{4 μ} d μ x) (1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1} - 1)) dx \\ = & 1 + 2 \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ \int_{0}^{+ \infty} \\ exp (p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ x) {(1 + exp (\frac{- π x}{\sqrt{3} k}))}^{- 1} dx \\ = & 1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ}}}^{- 1} dx . \end{matrix}$

Because $\int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ < \frac{π}{\sqrt{3} p}$ . Let $\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ} = a$ , so a > 1. Then, $\begin{matrix} 1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ}}}^{- 1} dx \\ = 1 + 2 \int_{1}^{+ \infty} \frac{1}{1 + x^{a}} dx, a > 1 . \end{matrix}$ Since $\frac{1}{1 + x^{a}} < \frac{1}{x^{a}}, a > 1,$ while $\int_{1}^{+ \infty} \frac{1}{x^{a}} dx, a > 1$ is convergent, so $\int_{1}^{+ \infty} \frac{1}{1 + x^{a}} dx, a > 1$ is convergent. So $1 + 2 \int_{1}^{+ \infty} {1 + x^{\frac{π}{\sqrt{3} p \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ}}}^{- 1} dx < + \infty .$ That is, $\begin{matrix} E [exp (pH \int_{0}^{+ \infty} B_{3 μ} + 2 B_{4 μ} d μ)] < + \infty . \end{matrix}$ In addition, by the condition $\begin{matrix} \int_{0}^{+ \infty} B_{1 k} dk < + \infty, \int_{0}^{+ \infty} B_{2 k} dk < + \infty . \end{matrix}$ Thus, we can get $\begin{matrix} lim_{sup_{r \in [- τ, 0]} | Z_{r} - {\hat{Z}}_{r} | \to 0} E [| Z_{k} - {\hat{Z}}_{k} |^{p}] = 0, \forall p > 0 . \end{matrix}$

Remark 3.2. In fact, if the new Lipschitz condition of Theorem 3.1 holds, we set B_1k = B_3k = A_1k and B_2k = B_4k = A_2k, then the Lipschitz condition of Theorem 3.2 holds. Conversely, it may not be established, we give the following counter-example.

Example 3.3. Consider an UDDE

$\begin{matrix} {dZ}_{k} = (\frac{2 k}{(1 + k^{4})} Z_{k} + \frac{1}{2 (1 + k^{2})} Z_{k - 1}) dk \\ + (\frac{k}{(1 + k^{4})} Z_{k} + \frac{1}{4 (1 + k^{2})} Z_{k - 1}) {dC}_{k} . \end{matrix}$ (13) Firstly, we set $h = \frac{2 k}{(1 + k^{4})} z + \frac{1}{2 (1 + k^{2})} \hat{z}$ and $q = \frac{k}{(1 + k^{4})} z + \frac{1}{4 (1 + k^{2})} \hat{z}$ , then, we have $\begin{matrix} B_{1 k} = \frac{2 k}{(1 + k^{4})}, B_{2 k} = \frac{1}{2 (1 + k^{2})}, \\ B_{3 k} = \frac{k}{(1 + k^{4})}, B_{4 k} = \frac{1}{4 (1 + k^{2})} . \end{matrix}$ Moreover, we obtain $\begin{matrix} \int_{0}^{+ \infty} B_{1 k} dk = \int_{0}^{+ \infty} \frac{2 k}{(1 + k^{4})} dk = \frac{π}{2}, \\ \int_{0}^{+ \infty} B_{2 k} dk = \int_{0}^{+ \infty} \frac{1}{2 (1 + k^{2})} dk = \frac{π}{4} < + \infty, \end{matrix}$ and $\begin{matrix} \int_{0}^{+ \infty} B_{3 k} + 2 B_{4 k} dk \\ = \int_{0}^{+ \infty} \frac{k}{(1 + k^{4})} + 2 \times \frac{1}{4 (1 + k^{2})} dk \\ = \frac{π}{4} + \frac{π}{4} = \frac{π}{2} < \frac{π}{\sqrt{3} p}, \forall \frac{1}{\sqrt{3}} \leq p < \frac{2}{\sqrt{3}} . \end{matrix}$ According to the Theorem 3.2, the UDDE (13) is stable in p-th moment. If we want to use the Theorem 3.1, we have $\begin{matrix} \int_{0}^{+ \infty} A_{1 k} + 2 A_{2 k} dk \\ = \frac{π}{2} + \frac{π}{2} > \frac{π}{\sqrt{3} p}, \forall \frac{1}{\sqrt{3}} \leq p < \frac{2}{\sqrt{3}} . \end{matrix}$ Therefore, the Theorem 3.1 becomes ineffectiveness.

Remark 3.3. If the UDDEs satisfy the strong Lipschitz condition of Theorem 2, we can set B_2t = 0 and B_4t = 0, so it must follow the Lipschitz condition of Theorem 3.2. Conversely, it is obvious that it not be established.

Corollary 3.2. Supposing that u_ik, v_ik, and η_ik (i = 1,2) are real-valued functions, then the linear UDDE

$\begin{matrix} d Z_{k} = & (u_{1 k} Z_{k} + u_{2 k} Z_{k - τ} + u_{3 k}) dk \\ + (u_{4 k} Z_{k} + u_{5 k} Z_{k - τ} + u_{6 k}) d C_{k} \end{matrix}$ (14) is stable in p-th moment if u_ik (i = 1, 2, 4, 5) satisfying $\begin{matrix} \int_{0}^{+ \infty} u_{1 k} < + \infty, \int_{0}^{+ \infty} u_{2 k} < + \infty, \\ \int_{0}^{+ \infty} u_{4 k} + 2 u_{5 k} dk < \frac{π}{\sqrt{3} p}, \forall p > 0 . \end{matrix}$

Proof. Take $h (k, z, \hat{z}) = u_{1 k} z + u_{2 k} \hat{z} + u_{3 k}$ and $q (k, z, \hat{z}) = u_{4 k} z + u_{5 k} \hat{z} + u_{6 k}$ , then we obtain $\begin{matrix} L_{1 k} = u_{1 k}, L_{2 k} = u_{2 k}, L_{3 k} = u_{4 k}, L_{4 k} = u_{5 k} . \end{matrix}$ Because of that they satisfy the Theorem 3.2 based on the condition (10), (11), so this corollary is proved.

Example 3.4. Consider an UDDE

$\begin{matrix} {dZ}_{k} = & (exp (- k) Z_{k} + \frac{1}{(1 + k^{2})} Z_{k - 1} \\ + \frac{k}{1 + arccos k}) dk + (k^{2} exp (- k^{3}) Z_{k} \\ + \frac{1}{(1 + k)^{4}} Z_{k - 1} + exp (tan (k))) {dC}_{k}, \end{matrix}$ (15) and set $\begin{matrix} h = exp (- k) z + \frac{1}{(1 + k^{2})} \hat{z} + \frac{k}{1 + arccos k}, \\ q = k^{2} exp (- k^{3}) z + \frac{1}{(1 + k)^{4}} \hat{z} + exp (tan (k)) . \end{matrix}$ Thus, we obtain $\begin{matrix} u_{1 k} = exp (- k), u_{2 k} = \frac{1}{(1 + k^{2})}, \\ u_{4 k} = k^{2} exp (- k^{3}), u_{5 k} = \frac{1}{(1 + k)^{4}} \end{matrix}$ and $\begin{matrix} \int_{0}^{+ \infty} u_{1 k} dk = \int_{0}^{+ \infty} exp (- k) dk = 1 < + \infty, \\ \int_{0}^{+ \infty} u_{2 k} dk = \int_{0}^{+ \infty} \frac{1}{(1 + k^{2})} dk = \frac{π}{2} < + \infty, \end{matrix}$ $\begin{matrix} \int_{0}^{+ \infty} u_{4 k} + 2 u_{5 k} dk = \int_{0}^{+ \infty} k^{2} exp (- k^{3}) \\ + \frac{2}{(1 + k)^{4}} dk = 1 < \frac{π}{\sqrt{3} p}, \forall 0 < p < \frac{π}{\sqrt{3}} . \end{matrix}$ By means of the Theorem 3.2 or the Corollary 3.2, the UDDE (15) is stable in p-th moment.

4 Conclusion

This study set out to propose new sufficient conditions for p-th moment stability of UDDE. By using the new Lipschitz condition, we presented the stability in p-th moment for the UDDE. After each theorem and corollary, some examples are given to demonstrate the reasonableness of our results. In the future, researchers can further consider the parameter estimation of uncertain delay differential equations, the influence of the size of delay on the stability of uncertain differential equations, etc.

Footnotes

Acknowledgments

The research was supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_-0170).

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