Stability in measure for uncertain delay differential equations based on new Lipschitz conditions

Abstract

Uncertain delay differential equations (UDDEs) charactered by Liu process can be employed to model an uncertain control system with a delay time. The stability of its solution always be a significant matter. At present, the stability in measure for UDDEs has been proposed and investigated based on the strong Lipschitz condition. In reality, the strong Lipschitz condition is so strictly and hardly applied to judge the stability in measure for UDDEs. For the sake of solving the above issue, the stability in measure based on new Lipschitz condition as a larger scale of applications is verified in this paper. In other words, if it satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz conditions. Conversely, it may not be established. An example is provided to show that it is stable in measure based on the new Lipschitz conditions, but it becomes invalid based on the strong Lipschitz condition. Moreover, a special class of UDDEs is verified to be stable in measure without any limited condition. Besides, some examples are investigated in this paper.

Keywords

Stability in measure Liu process uncertain process uncertain delay differential equations

1 Introduction

In our real world, it always exist a delay time in our daily lives, for example, we talk to each other by using the mobile phone, which has a delay time between saying and listening. Delay differential equations were often applied to describe a system with a delay time, such as the cancer-immune system [7], the network system [1], the chaos control system [3], the epidemic system [18], and the population dynamic system [19]. As a system with a delay time was affected by uncertain phenomenon, which was modeled by the stochastic delay differential equation involving Wiener process, such as biological system [6], finance [20], signal transmission system [26].

Apart from the Wiener process, a Liu process was defined and employed to describe the uncertain phenomenon by Liu [15]. Meanwhile, uncertain differential equations (UDEs) concerning the Liu process were first established by Liu [15]. After that, the theorem of its solution existence and the theorem of its solution uniqueness based on different conditions were verified by Chen and Liu [4] and Gao [9], respectively. Moreover, many different kinds of stability for UDEs were explored by many scholars, it included the stability in measure [14], the exponential stability [22], the stability in moment [21], the moment exponential stability [5], the almost sure stability [16], the stability in inverse distribution [34]. Furthermore, the numerical methods for UDEs contained the 99-method [29], the Runge-Kutta method [30], the Adams method [31], the Milne method [10], the Hamming method [37]. Besides, the application of UDEs has been used successfully in many subjects, like finance [13 , 36], game theory [27, 33], biology [23, 38].

Uncertain delay differential equations (UDDEs) as a special style of UDEs were first suggested by Barbacioru [2], who also verified the theorem of existence and uniqueness about its solution. Thereafter, the stability of UDDEs were investigated by many scholars, such as Wang and Ning [24] (stability in measure), Wang and Ning [25] (Almost sure stability), Jia and Sheng [12] (Stability in distribution). The above stability in measure for UDDEs was studied based on strong Lipschitz condition, which was so strictly and hardly used to judge the stability. Therefore, two kinds of new Lipschitz conditions more weaker than the strong Lipschitz condition are proposed, and the stability in measure for UDDEs based on these new Lipschitz conditions are discussed in this paper.

The remainder of this article structure is arranged as below. Section 2 introduces some related contents for UDDEs. Section 3 investigates the first sufficient condition of stability in measure based on the new Lipschitz condition for UDDEs. Section 4 presents the second sufficient condition of stability in measure based on another new Lipschitz condition for UDDEs. Section 5 proves that a class of UDDEs always be stable in measure without any limited conditions. Section 6 discusses the results of this article.

2 Preliminaries

In Liu’s book [17], the Liu process was established to described the uncertain phenomena by Liu [14]. Based on the Liu process, the following principal theorems were demonstrated.

Theorem 2.1. (Yao et al. [14]) Let ℳ stands for the uncertain measure in uncertainty theory [17], C_t represents a Liu process, for each γ ∈ Γ, Γ be a nonempty set, C_t (γ) is termed as a sample path. Then the following equality holds, $lim_{y \to + \infty} ℳ {γ \in Γ | Z (γ) \leq y} = 1,$ where Z (γ) denotes the Lipschitz constant of C_t (γ).

Theorem 2.2. (Chen and Liu [4]) Assume that U_t stands for an integrable uncertain process, t ∈ [a₁, a₂]. Then the inequality $| \int_{a_{1}}^{a_{2}} U_{t} (γ) d C_{t} (γ) | \leq Z (γ) \int_{a_{1}}^{a_{2}} | U_{t} (γ) | d t$ holds, where C_t is a Liu process, Z (γ) denotes its Lipschitz constant.

Moreover, Barbacioru [2] defined an uncertain delay differential equation by means of the Liu process.

Definition 2.1. (Barbacioru [2]) Let the real-valued functions F and G satisfy the following differential equation $d U_{t} = F (t, U_{t}, U_{t - τ}) d t + G (t, U_{t}, U_{t - τ}) d C_{t},$ (1) then the equation (1) is termed as an UDDE, where C_t and the positive number τ stand for the Liu process and a time delay, respectively.

Besides, Wang and Ning [24] gave a definition of stability in measure for UDDEs, and provided a sufficient theorem based on the strong Lipschitz condition.

Definition 2.2. (Wang and Ning [24]) Assume that A_t and B_t are the solution of the UDDE (1) with different initial states, then the UDDE (1) is stable in measure satisfying the following condition, $lim_{sup_{s \in [- τ, 0]} | A_{s} - B_{s} | \to 0} ℳ {| A_{t} - B_{t} | \leq ε} = 1, \forall t > 0,$ (2) where ε is the any given positive number.

Theorem 2.3. (Wang and Ning [24]) Assume the UDDE (1) has a unique solution as we give a initial state. If the coefficients of the UDDE (1) satisfy the strong Lipschitz condition $\begin{matrix} | F (t, a_{1}, b) - F (t, a_{2}, b) | \lor | G (t, a_{1}, b) - \\ G (t, a_{2}, b) | \leq P_{t} | a_{1} - a_{2} |, \forall a_{1}, a_{2}, b \in ℜ, t > 0, \end{matrix}$ where the bounded function P_t satisfies $\int_{0}^{+ \infty} P_{t} dt < + \infty,$ then the UDDE (1) is stable in measure.

Theorem 2.4. (Gronwall’s inequality [28]) Set I denotes [c₁, + ∞), [c₁, c₂] or [c₁, c₂) with c₁ < c₂. Let ρ and σ defined on I are two real-valued non-negative continuous functions, ρ is integrable on every closed and bounded subinterval of I. If the following inequality $σ (s) \leq ρ (s) + \int_{c}^{s} ρ (u) σ (u) d u, \forall s \in I$ holds and ρ is non-decreasing, then we have $σ (s) \leq ρ (s) exp (\int_{c}^{s} ρ (u) d u), \forall s \in I .$

3 The first sufficient condition

In this section, the first sufficient condition of stability in measure for UDDEs based on new Lipschitz condition is investigated.

Theorem 3.1. If the coefficients of UDDEs (1) with a unique solution giving a initial state satisfy the new Lipschitz condition $| F (t, a_{1}, b_{1}) - F (t, a_{2}, b_{2}) | \lor | G (t, a_{1}, b_{1}) - G (t, a_{2}, b_{2}) | \leq N_{1 t} | a_{1} - a_{2} | + N_{2 t} | b_{1} - b_{2} |,$ (3) ∀a₁, a₂, b₁, b₂ ∈ ℜ, t > 0, where the real-valued functions N_it satisfy $\int_{0}^{+ \infty} N_{it} d t < + \infty, i = 1, 2 .$ (4)Then, the UDDEs (1) is stable in measure.

Proof, As we give the initial states a_θ and b_θ, where -τ ≤ θ ≤ 0, then we obtain that the corresponding solutions of the UDDE (1) are A_t and B_t. So, ${\begin{matrix} d A_{t} = F (t, A_{t}, A_{t - τ}) d t + G (t, A_{t}, A_{t - τ}) d C_{t}, t \in (0, + \infty) \\ A_{t} = a_{t}, t \in [- τ, 0] \end{matrix}$ and ${\begin{matrix} d B_{t} = F (t, B_{t}, B_{t - τ}) d t + G (t, B_{t}, B_{t - τ}) d C_{t}, t \in (0, + \infty) \\ B_{t} = b_{t}, t \in [- τ, 0] . \end{matrix}$ Assume that C_t (γ) stands for the Lipschitz continuous sample of C_t, we have $A_{t} (γ) = A_{0} + \int_{0}^{t} F (u, A_{u} (γ), A_{u - τ} (γ)) d u + \int_{0}^{t} G (u, A_{u} (γ), A_{u - τ} (γ)) d C_{u} (γ)$ and $B_{t} (γ) = B_{0} + \int_{0}^{t} F (u, B_{u} (γ), B_{u - τ} (γ)) d u + \int_{0}^{t} G (u, B_{u} (γ), B_{u - τ} (γ)) d C_{u} (γ) .$ Moreover, the Lipschitz constant of C_t (γ) is termed as Z (γ). Applying the new Lipschitz condition (3) and the Theorem 2.2, $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | = & | {A_{0} + \int_{0}^{t} F (u, A_{u} (γ), A_{u - τ} (γ)) d u + \int_{0}^{t} G (u, A_{u} (γ), A_{u - τ} (γ)) d C_{u} (γ)} \\ - {B_{0} + \int_{0}^{t} F (u, B_{u} (γ), B_{u - τ} (γ)) d u + \int_{0}^{t} G (u, B_{u} (γ), B_{u - τ} (γ)) d C_{u} (γ)} | \\ \leq & | A_{0} - B_{0} | + | \int_{0}^{t} {F (u, A_{u} (γ), A_{u - τ} (γ)) - F (u, B_{u} (γ), B_{u - τ} (γ))} d u | \\ + | \int_{0}^{t} {G (u, A_{u} (γ), A_{u - τ} (γ)) - G (u, B_{u} (γ), B_{u - τ} (γ))} d C_{u} (γ) | \\ \leq & | A_{0} - B_{0} | + \int_{0}^{t} {| F (u, A_{u} (γ), A_{u - τ} (γ)) - F (u, B_{u} (γ), B_{u - τ} (γ)) |} d u \\ + Z (γ) \int_{0}^{t} {| G (u, A_{u} (γ), A_{u - τ} (γ)) - G (u, B_{u} (γ), B_{u - τ} (γ)) |} d u \\ \leq & | A_{0} - B_{0} | + \int_{0}^{t} {N_{1 u} | A_{u} (γ) - B_{u} (γ) | + N_{2 u} | A_{u - τ} (γ) - B_{u - τ} (γ) |} d u \\ + \int_{0}^{t} Z (γ) {N_{1 u} | A_{u} (γ) - B_{u} (γ) | + N_{2 u} | A_{u - τ} (γ) - B_{u - τ} (γ) |} d u \\ = & | A_{0} - B_{0} | + (1 + Z (γ)) \int_{0}^{t} {N_{1 u} | A_{u} (γ) - B_{u} (γ) | + N_{2 u} | A_{u - τ} (γ) - B_{u - τ} (γ) |} d u . \end{matrix}$ Based on the condition (4), the terms $\int_{0}^{+ \infty} N_{1 t} d t < + \infty, \int_{0}^{+ \infty} N_{2 t} d t < + \infty,$ so it exist a positive number H, and have $\int_{0}^{+ \infty} N_{1 t} d t < H, \int_{0}^{+ \infty} N_{2 t} d t < H .$ Meanwhile, we set η = u - τ, and obtain $\begin{matrix} \int_{0}^{t} N_{2 u} & | A_{u - τ} (γ) - B_{u - τ} (γ) | d u \\ = \int_{- τ}^{t - τ} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ = \int_{- τ}^{0} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η + \int_{0}^{t - τ} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ \leq sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{- τ}^{0} N_{2 (η + τ)} d η + \int_{0}^{t - τ} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ \leq sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{0}^{τ} N_{2 η} d η + \int_{0}^{t} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ \leq H sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} + \int_{0}^{t} N_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η . \end{matrix}$ So, we obtain $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | & \leq | A_{0} - B_{0} | + (1 + Z (γ)) H sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ + (1 + Z (γ)) \int_{0}^{t} (N_{1 u} + N_{2 (u + τ)}) | A_{u} (γ) - B_{u} (γ) | d u \\ \leq {1 + (1 + Z (γ)) H} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ + (1 + Z (γ)) \int_{0}^{t} (N_{1 u} + N_{2 (u + τ)}) | A_{u} (γ) - B_{u} (γ) | d u . \end{matrix}$ (5) According to the Theorem 2.4, then we have $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | & \leq {1 + (1 + Z (γ)) H} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} exp ((1 + Z (γ)) \int_{0}^{t} (N_{1 u} + N_{2 (u + τ)}) d u) \\ = {1 + (1 + Z (γ)) H} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} exp ((1 + Z (γ)) {\int_{0}^{t} N_{1 u} d u + \int_{τ}^{t + τ} N_{2 u} d u}) \\ \leq {1 + (1 + Z (γ)) H} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} exp ((1 + Z (γ)) {\int_{0}^{+ \infty} N_{1 u} d u + \int_{0}^{+ \infty} N_{2 u} d u}) \\ \leq exp ((1 + Z (γ)) H) sup_{s \in [- τ, 0]} {| X_{s} (γ) - Y_{s} (γ) |} exp ((1 + Z (γ)) {\int_{0}^{+ \infty} N_{1 u} d u + \int_{0}^{+ \infty} N_{2 u} d u}) \\ \leq sup_{s \in [- τ, 0]} {| X_{s} (γ) - Y_{s} (γ) |} exp (3 H (1 + Z (γ))) . \end{matrix}$ By using the Theorem 2.1, ∀ε > 0, it exists a number R > 0 satisfying $ℳ {γ | Z (γ) \leq R} \geq 1 - ε .$ Set $δ = exp (- 3 H (1 + R)) ε .$ Then, we have $\begin{matrix} ℳ {| A_{t} - B_{t} | \leq ε} & \geq ℳ {sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} exp (3 H (1 + Z (γ))) \leq ε} \\ \geq ℳ {γ | Z (γ) \leq R} \geq 1 - ε \end{matrix}$ for $sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \leq δ .$ It also means $ℳ {| A_{t} - B_{t} | \leq ε} \to 1, \forall t > 0$ for $sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0 .$ Therefore, we obtain $lim_{sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0} ℳ {| A_{t} - B_{t} | \leq ε, \forall t \geq 0} = 1 .$

Remark 3.1. If the Theorem 2.3 holds, we set the term N_2t = 0 in the condition (3), then the Theorem 3.1 holds, if the Theorem 3.1 is established, then the Theorem 2.3 is not sure, the example 3.1 is given as follows to illustrate this point.

Example 3.1. Consider the UDDE $d U_{t} = \frac{1}{1 + t^{2}} U_{t - 1} d t + exp (- t) U_{t} d C_{t},$ (6) then we set $F = \frac{1}{1 + t^{2}} b_{1}$ and G = exp(- t) a₂, if we apply the Theorem 2.3, 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 $N_{1 t} = exp (- t), N_{2 t} = \frac{1}{1 + t^{2}} .$ Meanwhile, we obtain $\int_{0}^{+ \infty} N_{1 t} d t = \int_{0}^{+ \infty} exp (- t) d t = 1 < + \infty,$ and $\int_{0}^{+ \infty} N_{2 t} d t = \int_{0}^{+ \infty} \frac{1}{1 + t^{2}} d t = \frac{π}{2} < + \infty .$ Therefore, the UDDE (6) based on the Theorem 3.1 is stable in measure.

Corollary 3.1 Consider a linear UDDE $d U_{t} = (n_{1 t} U_{t} + n_{2 t} U_{t - τ} + n_{3 t}) d t + (n_{1 t} U_{t} + n_{2 t} U_{t - τ} + n_{4 t}) d C_{t}$ (7) satisfying $\int_{0}^{+ \infty} n_{1 t} d t < + \infty, \int_{0}^{+ \infty} n_{2 t} d t < + \infty .$ (8)Then, the UDDE (7) is stable in measure.

Proof, Set $F (t, a, b) = n_{1 t} a + n_{2 t} b + n_{3 t}, G (t, a, b) = n_{1 t} a + n_{2 t} b + n_{4 t},$ then we obtain $| F (t, a_{1}, b_{1}) - F (t, a_{2}, b_{2}) | \lor | G (t, a_{1}, b_{1}) - G (t, a_{2}, b_{2}) | \leq n_{1 t} | a_{1} - a_{2} | + n_{2 t} | b_{1} - b_{2} | .$ Because of that it is satisfy the Theorem 3.1 based on the condition (8), so the corollary is proved.

Example 3.2 Consider the UDDE $\begin{matrix} d & U_{t} = (exp (- t^{2}) U_{t} + \frac{1}{t (1 + (ln t)^{2})} U_{t - 4} + sin (\frac{t}{1 + t^{2}})) d t + (exp (- t^{2}) U_{t} + \frac{1}{t (1 + (ln t)^{2})} U_{t - 4} + tan (t)) d C_{t}, \end{matrix}$ (9) and we set $\begin{matrix} F (t, a, b) = exp (- t^{2}) a + \frac{1}{t (1 + (ln t)^{2})} b + sin (\frac{t}{1 + t^{2}}), G (t, a, b) = exp (- t^{2}) a + \frac{1}{t (1 + (ln t)^{2})} b + tan (t) . \end{matrix}$ Thus, we obtain $n_{1 t} = exp (- t^{2}), n_{2 t} = \frac{1}{t (1 + (ln t)^{2})} .$ However, $\int_{0}^{+ \infty} n_{1 t} d t = \int_{0}^{+ \infty} exp (- t^{2}) d t = \frac{\sqrt{π}}{2} < + \infty, \int_{0}^{+ \infty} n_{2 t} d t = \int_{0}^{+ \infty} \frac{1}{t (1 + (ln t)^{2})} d t = π < + \infty .$ By means of the Corollary 3.1, the UDDE (9) is stable in measure.

4 The second sufficient condition

In this section, the second sufficient condition of stability in measure for UDDEs based on another new Lipschitz condition is explored.

Theorem 4.1. If the coefficients of UDDEs (1) with a unique solution giving a initial state satisfy the new Lipschitz condition $\begin{matrix} | F (t, a_{1}, b_{1}) - F (t, a_{2}, b_{2}) | \leq M_{1 t} | a_{1} - a_{2} | + M_{2 t} | b_{1} - b_{2} | \\ | G (t, a_{1}, b_{1}) - G (t, a_{2}, b_{2}) | \leq M_{3 t} | a_{1} - a_{2} | + M_{4 t} | b_{1} - b_{2} | \end{matrix}$ (10) ∀a_i, b_i ∈ ℜ, i = 1, 2, t > 0, where the positive real-valued functions M_jt satisfy the following conditions, $\int_{0}^{+ \infty} M_{jt} d t < + \infty, j = 1, 2, 3, 4 .$ (11)Then the UDDEs (1) is stable in measure.

Proof, As we give the initial states a_θ and b_θ, where -τ ≤ θ ≤ 0, then we assume that the corresponding solutions of the UDDE (1) are A_t and B_t. Thus, we have ${\begin{matrix} d A_{t} = F (t, A_{t}, A_{t - τ}) d t + G (t, A_{t}, A_{t - τ}) d C_{t}, t \in (0, + \infty) \\ A_{t} = a_{t}, t \in [- τ, 0] \end{matrix}$ and ${\begin{matrix} d B_{t} = F (t, B_{t}, B_{t - τ}) d t + G (t, B_{t}, B_{t - τ}) d C_{t}, t \in (0, + \infty) \\ B_{t} = b_{t}, t \in [- τ, 0] . \end{matrix}$ Assume the Lipschitz continuous sample of C_t denotes C_t (γ), and Z (γ) stands for the Lipschitz constant of C_t (γ), we obtain $\begin{matrix} A_{t} (γ) = A_{0} + \int_{0}^{t} F (u, A_{u} (γ), A_{u - τ} (γ)) d u + \int_{0}^{t} G (u, A_{u} (γ), A_{u - τ} (γ)) d C_{u} (γ) \end{matrix}$ and $B_{t} (γ) = B_{0} + \int_{0}^{t} F (u, B_{u} (γ), B_{u - τ} (γ)) d u + \int_{0}^{t} G (u, B_{u} (γ), B_{u - τ} (γ)) d C_{u} (γ) .$ Applying the new Lipschitz condition (10) and the Theorem 2.2, we have $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | = & | (A_{0} + \int_{0}^{t} F (u, A_{u} (γ), A_{u - τ} (γ)) d u + \int_{0}^{t} G (u, A_{u} (γ), A_{u - τ} (γ)) d C_{u} (γ)) \\ - (B_{0} + \int_{0}^{t} F (u, B_{u} (γ), B_{u - τ} (γ)) d u + \int_{0}^{t} G (u, B_{u} (γ), B_{u - τ} (γ)) d C_{u} (γ)) | \\ \leq & | A_{0} - B_{0} | + | \int_{0}^{t} F (u, A_{u} (γ), A_{u - τ} (γ)) - F (u, B_{u} (γ), B_{u - τ} (γ)) d u | \\ + | \int_{0}^{t} G (u, A_{u} (γ), A_{u - τ} (γ)) - G (u, B_{u} (γ), B_{u - τ} (γ)) d C_{u} (γ) | \\ \leq & | A_{0} - B_{0} | + | \int_{0}^{t} | F (u, A_{u} (γ), A_{u - τ} (γ)) - F (u, B_{u} (γ), B_{u - τ} (γ)) | d u \\ + Z (γ) \int_{0}^{t} | G (u, A_{u} (γ), A_{u - τ} (γ)) - G (u, B_{u} (γ), B_{u - τ} (γ)) | d u \\ \leq & | A_{0} - B_{0} | + \int_{0}^{t} M_{1 u} | A_{u} (γ) - B_{u} (γ) | + M_{2 u} | A_{u - τ} (γ) - B_{u - τ} (γ) | d u \\ + Z (γ) \int_{0}^{t} M_{3 u} | A_{u} (γ) - B_{u} (γ) | + M_{4 u} | A_{u - τ} (γ) - B_{u - τ} (γ) | d u . \end{matrix}$ Meanwhile, we set η = u - τ, and obtain $\begin{matrix} \int_{0}^{t} M_{2 u} | A_{u - τ} (γ) - B_{u - τ} (γ) | d u \\ = \int_{- τ}^{t - τ} M_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ = \int_{- τ}^{0} M_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η + \int_{0}^{t - τ} M_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ \leq sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{- τ}^{0} M_{2 (η + τ)} d η + \int_{0}^{t - τ} M_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η \\ \leq sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{0}^{τ} M_{2 η} d η + \int_{0}^{t} M_{2 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η . \end{matrix}$ Similarly, we have $\int_{0}^{t} M_{4 η} | A_{η - τ} (γ) - B_{η - τ} (γ) | d η \leq sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{0}^{τ} M_{4 η} d η + \int_{0}^{t} M_{4 (η + τ)} | A_{η} (γ) - B_{η} (γ) | d η .$ Because of that the terms $\int_{0}^{+ \infty} M_{jt} d t < + \infty, j = 1, 2, 3, 4,$ then it exist a positive number D, $\int_{0}^{+ \infty} M_{jt} d t < D, j = 1, 2, 3, 4 .$ Therefore $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | & \leq | A_{0} - B_{0} | + sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{0}^{τ} M_{2 u} d u \\ + \int_{0}^{t} (M_{1 u} + M_{2 (u + τ)}) | A_{u} (γ) - B_{u} (γ) | d u \\ + Z (γ) sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \int_{0}^{τ} M_{4 u} d u \\ + \int_{0}^{t} (M_{3 u} + M_{4 (u + τ)}) | A_{u} (γ) - A_{u} (γ) | d u \\ \leq | A_{0} - A_{0} | + D (1 + Z (γ)) sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ + \int_{0}^{t} (M_{1 u} + M_{2 (u + τ)} + Z (γ) {M_{3 u} + M_{4 (u + τ)}}) \cdot | A_{u} (γ) - B_{u} (γ) | d u \\ \leq {1 + D (1 + Z (γ))} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ + \int_{0}^{t} (M_{1 u} + M_{2 (u + τ)} + Z (γ) {M_{3 u} + M_{4 (u + τ)}}) \cdot | A_{u} (γ) - B_{u} (γ) | d u . \end{matrix}$ According the Theorem 2.4, $\begin{matrix} | A_{t} (γ) - B_{t} (γ) | & \leq {1 + D (1 + Z (γ))} sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ \cdot exp (\int_{0}^{t} M_{1 u} + M_{2 (u + τ)} + Z (γ) {M_{3 u} + M_{4 (u + τ)}} d u) \\ \leq exp (D (1 + Z (γ))) sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \\ \cdot exp (\int_{0}^{+ \infty} M_{1 u} + M_{2 (u + τ)} + Z (γ) {M_{3 u} + M_{4 (u + τ)}} d u) \\ \leq exp (D (1 + Z (γ))) sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \cdot exp (2 D (1 + Z (γ))) \\ = sup_{s \in [- τ, 0]} {| A_{s} (γ) - B_{s} (γ) |} \cdot exp (3 D (1 + Z (γ))) . \end{matrix}$ By using the Theorem 2.1, ∀ε > 0, it exists a number W > 0 satisfying $ℳ {γ | Z (γ) \leq W} \geq 1 - ε .$ (12) Take $δ = exp (- 3 D (1 + W)) ε .$ Then, we have $\begin{matrix} ℳ {| A_{t} - B_{t} | \leq ε} & \geq ℳ {sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} exp (3 D (1 + Z (γ))) \leq ε} \\ \geq ℳ {γ | Z (γ) \leq W} \geq 1 - ε \end{matrix}$ for $sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \leq δ .$ It also means $ℳ {| A_{t} - B_{t} | \leq ε} \to 1, \forall t > 0$ for $sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0 .$ Therefore, we obtain $lim_{sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0} ℳ {| A_{t} - B_{t} | \leq ε, \forall t \geq 0} = 1 .$

Remark 4.1. In fact, Theorem 3.1 and Theorem 4.1 have a equivalence relation. If the condition (3) holds, we set M_1t = M_3t = N_1t and M_2t = M_4t = N_2t, then the condition (10) holds. In contrast, if the condition (10) holds, we set N_1t = M_1t + M_3t and N_2t = M_2t + M_4t, then the condition (3) holds.

Remark 4.2. Because of that Theorem 3.1 and Theorem 4.1 have a equivalence relation, thus, if the UDDEs satisfy the new Lipschitz condition (10), it must satisfy the strong Lipschitz condition in Theorem 2.3. Conversely, it may not be established.

Corollary 4.1. Assume a linear UDDE $d U_{t} = (m_{1 t} U_{t} + m_{2 t} U_{t - τ} + m_{3 t}) d t + (m_{4 t} U_{t} + m_{5 t} U_{t - τ} + m_{6 t}) d C_{t}_$ (13) satisfying $\int_{0}^{+ \infty} m_{it} dt < + \infty, i = 1, 2, 4, 5 .$ (14)Then the UDDE (13) is stable in measure.

Proof, we set $F (t, a, b) = m_{1 t} a + m_{2 t} b + m_{3 t}, G (t, a, b) = m_{4 t} a + m_{5 t} b + c_{6 t},$ then we have $M_{1 t} = m_{1 t}, M_{2 t} = m_{2 t}, M_{3 t} = m_{4 t}, M_{4 t} = m_{5 t} .$ By use of the condition (14) and Theorem 4.1, we obtain that the UDDE (13) is stable in measure.

Example 4.1. Consider the UDDE $\begin{matrix} d U_{t} = & (exp (- t) U_{t} + \frac{1}{1 + t^{2}} U_{t - 1} + arcsin (\frac{t}{1 + exp (t^{2})})) d t \\ + (t exp (- t) U_{t} + t exp (- t^{2}) U_{t - 1} + \frac{t}{1 + arccos t}) d C_{t} . \end{matrix}$ (15) Set $F (t, a, b) = exp (- t) a + \frac{1}{1 + t^{2}} b + arcsin (\frac{t}{1 + exp (t^{2})})$ and $G (t, a, b) = t exp (- t) a + t exp (- t^{2}) b + \frac{t}{1 + arccos t},$ then we have $m_{1 t} = exp (- t), m_{2 t} = \frac{1}{1 + t^{2}}, m_{4 t} = t exp (- t), m_{5 t} = t exp (- t^{2}),$ and $\int_{0}^{+ \infty} m_{1 t} d t = \int_{0}^{+ \infty} exp (- t) d t = 1 < + \infty, \int_{0}^{+ \infty} m_{2 t} d t = \int_{0}^{+ \infty} \frac{1}{1 + t^{2}} d t = \frac{π}{2} < + \infty,$ $\int_{0}^{+ \infty} m_{4 t} d t = \int_{0}^{+ \infty} t exp (- t) d t = 1 < + \infty, \int_{0}^{+ \infty} m_{5 t} d t = \int_{0}^{+ \infty} t exp (- t^{2}) d t = \frac{1}{2} < + \infty .$ By using the Corollary 4.1, the UDDE (15) is stable in measure.

5 Stability in measure for a special UDDEs

In this section, a special class of UDDEs are investigated to be stable in measure without any limited condition.

Theorem 5.1. Consider the UDDE $d U_{t} = K_{t} U_{t - τ} d t + a d C_{t},$ (16) where K_t is a real-valued function, then the UDDE (16) is stable in measure.

Proof, For any given two different initial states a_t (t ∈ [- τ, 0]) and b_t (t ∈ [- τ, 0]), we assume that A_t and B_t are the corresponding solutions of the UDDE (16). That is ${\begin{matrix} d A_{t} = K_{t} A_{t - τ} d t + a d C_{t}, t > 0 \\ A_{t} = a_{t}, - τ \leq t \leq 0 \end{matrix}$ and ${\begin{matrix} d B_{t} = K_{t} B_{t - τ} d t + a d C_{t}, t > 0 \\ B_{t} = b_{t}, - τ \leq t \leq 0, \end{matrix}$ we have $A_{t} = {\begin{matrix} a_{t}, - τ \leq t \leq 0 \\ a_{0} + \int_{0}^{t} K_{s} a_{s - τ} d s + a C_{t}, 0 < t \leq τ \\ A_{τ} + \int_{τ}^{t} K_{s} A_{s - τ} d s + a (C_{t} - C_{τ}), τ < t \leq 2 τ \\ \dots \end{matrix}$ and $B_{t} = {\begin{matrix} b_{t}, t \in - τ \leq t \leq 0 \\ b_{0} + \int_{0}^{t} K_{s} b_{s - τ} d s + a C_{t}, 0 < t \leq τ \\ B_{τ} + \int_{τ}^{t} K_{s} B_{s - τ} d s + a (C_{t} - C_{τ}), τ < t \leq 2 τ \\ \dots \end{matrix}$ Then, $| A_{t} - B_{t} | \leq {\begin{matrix} | a_{t} - b_{t} |, - τ \leq t \leq 0 \\ | a_{0} - b_{0} | + \int_{0}^{t} | K_{s} | | a_{s - τ} - b_{s - τ} | d s, 0 < t \leq τ \\ | A_{τ} - B_{τ} | + \int_{τ}^{t} | K_{s} | | A_{s - τ} - B_{s - τ} | d s, τ < t \leq 2 τ \\ \dots \end{matrix}$ When $sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0,$ we have $| a_{s - τ} - b_{s - τ} | \to 0, \forall t \in (0, τ] .$ In other words, $| A_{t} - B_{t} | \to 0, \forall t \in (0, τ] .$ Meanwhile, we obtain $| A_{t} - B_{t} | \to 0, \forall t \in (τ, 2 τ] .$ As we always repeat the above process, and obtain $| A_{t} - B_{t} | \to 0, \forall t \in (0, + \infty) .$ Therefore, $lim_{sup_{s \in [- τ, 0]} {| A_{s} - B_{s} |} \to 0} ℳ {| A_{t} - B_{t} | \leq ε, \forall t \geq 0} = 1 .$

Remark 5.1. Theorem 5.1 is different from the Theorem 3.1 and the Theorem 4.1. The following example shows that it is stable in measure, but it becomes invalid by using the Theorem 3.1 and Theorem 4.1.

Example 5.1. Consider the UDDE $d U_{t} = t^{2} U_{t - 0.1} d t + d C_{t} .$ (17) According to the Theorem 5.1, the UDDE (17) is stable in measure. But due to the fact that $\int_{0}^{+ \infty} t^{2} d t ≮ + \infty,$ so the Theorem 3.1 and Theorem 4.1 are invalid for this example.

6 Conclusion

The stability in measure based on the strong Lipschitz condition for UDDEs had been investigated, but it was so strictly and hardly employed to judge the stability in measure for UDDEs. In order to solve this issue, the stability in measure based on new Lipschitz condition more weaker than the strong Lipschitz condition was verified in this paper. An example was given to show that the UDDEs based on the new Lipschitz condition was stability in measure, but it loses effectiveness based on the strong Lipschitz condition. Under the new Lipschitz conditions, two sufficient theorems of the stability in measure for UDDEs were provided. Besides, the theorem for a special class of UDDEs always be stable in measure without any limited condition that was verified, meanwhile, an example was shown that it was stable in measure, but it was not available for the theorem established by the new Lipschitz conditions. In future, it can be extended to investigate the stability in p-moment and the almost sure stability for UDDEs.

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