期刊菜单

Stability and Boundedness of Time-Changed Delay Stochastic Differential Equations

Abstract: In this paper, we study the stability and boundedness of time-changed delay stochastic differential equations. Three cases of the stability and boundedness of time-changed delay stochastic differen-tial equations are discussed separately, that is, the delay function is constant, bounded function and unbounded function. Using Lyapunov function method and time-changed Ito’s formula, the criteria of stability and boundedness of these three cases are obtained. Finally, three examples are listed to illustrate the effectiveness of our conclusion.

1. 引言

2. 模型介绍和预备知识

$\left(\Omega ,\mathcal{F},\mathcal{P}\right)$ 是关于流 ${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$ 的完备概率空间，也就是说它是右连续的，且 ${\mathcal{F}}_{0}$ 包含所有的零概率集，设 ${\left\{B\left(t\right)\right\}}_{t\ge 0}$ 是在完备带流的概率空间 $\left(\Omega ,\mathcal{F},\mathcal{P},{\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}\right)$ 上定义的实值布朗运动。从属项 ${\left\{{U}_{t}\right\}}_{t\ge 0}$ 是一个递增，具有平稳独立增量，且其样本路径是右连续且左极限存在的随机过程。 $U\left(t\right)$ 的拉普拉斯变换的形式如下：

$\text{E}\left({\text{e}}^{-uU\left(t\right)}\right)={\text{e}}^{-t\phi \left(u\right)},$

$\phi \left(u\right)=\lambda u+{\int }_{0}^{\infty }\left(1-{\text{e}}^{-ux}\right)\nu \left(\text{d}x\right),$

$E\left(t\right)=\mathrm{inf}\left\{s>0:U\left(s\right)>t\right\},t\ge 0.$

$\text{E}{\int }_{0}^{\infty }{|\varphi \left(s\right)|}^{2}\text{d}E\left(s\right)<\infty ,t>0.$

${\int }_{0}^{t}\varphi \left(s\right)\text{d}B\left(E\left(s\right)\right),$

$\begin{array}{c}\text{d}x\left(t\right)=f\left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right)\text{d}t+g\left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right)\text{d}E\left(t\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }+\sigma \left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right)\text{d}B\left(E\left(t\right)\right),\end{array}$ (1)

$\begin{array}{l}f:{R}_{+}×{R}_{+}×R×R\to R,\\ g:{R}_{+}×{R}_{+}×R×R\to R,\\ \sigma :{R}_{+}×{R}_{+}×R×R\to R,\\ \delta :{R}_{+}\to {R}_{+}.\end{array}$

$t<0$ 时，我们令 $E\left(t\right)=0$

(A1) 对任意的 ${t}_{1},{t}_{2}\in {R}_{+}=\left[\text{0,}\infty \right)$${x}_{1},{x}_{2},{y}_{1},{y}_{2}\in R$ ，存在正常数K，使得

$\begin{array}{l}{|f\left({t}_{1},{t}_{2},{x}_{1},{y}_{1}\right)-f\left({t}_{1},{t}_{2},{x}_{2},{y}_{2}\right)|}^{2}\vee {|g\left({t}_{1},{t}_{2},{x}_{1},{y}_{1}\right)-g\left({t}_{1},{t}_{2},{x}_{2},{y}_{2}\right)|}^{2}\\ \vee {|\sigma \left({t}_{1},{t}_{2},{x}_{1},{y}_{1}\right)-\sigma \left({t}_{1},{t}_{2},{x}_{2},{y}_{2}\right)|}^{2}\le K|{\left({x}_{1}-{x}_{2}\right)}^{2}+{\left({y}_{1}-{y}_{2}\right)}^{2}|.\end{array}$

(A2) 如果 $x\left(t\right)$ 是一个右连左极且 ${\mathcal{G}}_{t}$ -适应的过程，那么

$f\left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right),g\left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right),\sigma \left(t,E\left(t\right),x\left(t\right),x\left(t-\delta \left(t\right)\right)\right)\in L\left({\mathcal{G}}_{t}\right),$

$L\left({\mathcal{G}}_{t}\right)$ 指的是右连左极且 ${\mathcal{G}}_{t}$ -适应的过程类。根据引理4.1 [13] ，时变延迟随机微分方程(1)存在唯一解，且该解是 ${\mathcal{G}}_{t}$ -适应的。

3. 稳定性和有界性

(A3) 若存在 $V\in {C}^{2,1}\left({R}_{+}×{R}_{+}×R;{R}_{+}\right)$${U}_{1},{U}_{2}\in C\left(\left[-\tau ,\infty \right)×{R}_{+}×R;{R}_{+}\right)$ ，及常数 ${c}_{1}\ge 0$${c}_{2}>{c}_{3}\ge 0$ ，使得对任意的 $\left(t,E\left(t\right),x\right)\in {R}_{+}×{R}_{+}×R$ 有：

${U}_{1}\left(t,E\left(t\right),x\right)\le V\left(t,E\left(t\right),x\right)\le {U}_{2}\left(t,E\left(t\right),x\right).$ (2)

${L}_{1}V\left(t,E\left(t\right),x,y\right)\le {c}_{1}-{c}_{2}{U}_{2}\left(t,E\left(t\right),x\right)+{c}_{3}{U}_{2}\left(t-\tau ,E\left(t-\tau \right),y\right),$ (3)

${L}_{2}V\left(t,E\left(t\right),x,y\right)\le 0.$ (4)

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)={V}_{{t}_{1}}\left({t}_{1},{t}_{2},x,y\right)+{V}_{x}\left({t}_{1},{t}_{2},x,y\right)f\left({t}_{1},{t}_{2},x,y\right),$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)={V}_{{t}_{2}}\left({t}_{1},{t}_{2},x,y\right)+{V}_{x}\left({t}_{1},{t}_{2},x,y\right)g\left({t}_{1},{t}_{2},x,y\right)+\frac{1}{2}{V}_{xx}\left({t}_{1},{t}_{2},x,y\right){\sigma }^{2}\left({t}_{1},{t}_{2},x,y\right).$

1) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\text{E}{U}_{1}\left(t,E\left(t\right),x\right)\le \frac{{c}_{1}}{\epsilon }.$ (5)

2) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}{\int }_{0}^{t}\text{E}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\le \frac{{c}_{1}}{{c}_{2}-{c}_{3}}.$ (6)

${c}_{1}=0$ 时，有：

3) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left(\text{E}{U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon .$ (7)

4) ${\int }_{0}^{\infty }\text{E}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\le \frac{1}{{c}_{2}-{c}_{3}}\left({U}_{2}\left(0,0,x\left(0\right)\right)+{\int }_{-\tau }^{0}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s.$ (8)

5) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left({U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon \text{a}\text{.s}\text{.}$ (9)

6) ${\int }_{0}^{\infty }{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s<\infty .\text{a}\text{.s}\text{.}$ (10)

${\tau }_{n}=\mathrm{inf}\left\{t\ge 0:|x\left(t\right)|\ge n\right\},$

${\rho }_{k}=k\wedge \mathrm{inf}\left\{t\ge 0:|{\int }_{0}^{{\tau }_{n}\wedge t}{V}_{x}\left(s,E\left(s\right),x\left(s\right),x\left(s-\delta \left(s\right)\right)\right)\sigma \left(s,E\left(s\right),x\left(s\right),x\left(s-\delta \left(s\right)\right)\right)|\ge k\right\},$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}V\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)-V\left(0,0,{x}_{0}\right)\\ =E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}\left(\epsilon V\left(s,E\left(s\right),x\left(s\right)\right)+{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}E\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{V}_{x}\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\sigma \left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}B\left(E\left(s\right)\right).\end{array}$

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)-{U}_{2}\left(0,0,{x}_{0}\right)\\ \le E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}\left({c}_{1}-\left({c}_{2}-\epsilon \right){U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)+{c}_{3}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\right)\text{d}s\\ \le \frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}-\left({c}_{2}-\epsilon \right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{3}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\text{d}s.\end{array}$ (11)

$\begin{array}{l}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\text{d}s\\ ={\text{e}}^{\epsilon \tau }E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon \left(s-\tau \right)}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\text{d}s\\ ={\text{e}}^{\epsilon \tau }E{\int }_{-\tau }^{t\wedge {\tau }_{n}\wedge {\rho }_{k}-\tau }{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \le {\text{e}}^{\epsilon \tau }E{\int }_{-\tau }^{0}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s+{\text{e}}^{\epsilon \tau }E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \le {\text{e}}^{\epsilon \tau }E{\int }_{-\tau }^{0}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s+{\text{e}}^{\epsilon \tau }E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s.\end{array}$ (12)

$E\left({\text{e}}^{\epsilon \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\right)\le {P}_{1}+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}.$

$E\left({\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right)\le {P}_{1}+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t},$ (13)

$\begin{array}{l}EV\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)-V\left(0,0,{x}_{0}\right)\\ =E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}s+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}E\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{V}_{x}\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\sigma \left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}B\left(E\left(s\right)\right).\end{array}$

$\begin{array}{l}E{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\\ \le {U}_{2}\left(0,0,{x}_{0}\right)+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{c}_{1}-{c}_{2}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)+{c}_{3}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\text{d}s\end{array}$ (14)

$\begin{array}{l}\le {U}_{2}\left(0,0,{x}_{0}\right)+{c}_{1}t-{c}_{2}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{3}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s-\tau ,E\left(s-\tau \right),x\left(s-\tau \right)\right)\text{d}s\\ \le {P}_{2}+{c}_{1}t-\left({c}_{2}-{c}_{3}\right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s.\end{array}$ (15)

$\left({c}_{2}-{c}_{3}\right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{2}+{c}_{1}t.$

$k\to \infty$$n\to \infty$ ，再运用Fubini定理，可得

$\left({c}_{2}-{c}_{3}\right){\int }_{0}^{t}E{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{2}+{c}_{1}t.$ (16)

$E\left({U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right)\le {P}_{1}{\text{e}}^{-\epsilon t},$

$\left({c}_{2}-{c}_{3}\right){\int }_{0}^{t}E{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{2}.$

${\int }_{0}^{\infty }E{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le \frac{1}{{c}_{2}-{c}_{3}}\left({U}_{2}\left(0,0,x\left(0\right)\right)+{\int }_{-\tau }^{0}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\right)\text{,}$ (17)

$E{\int }_{0}^{\infty }{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le \frac{1}{{c}_{2}-{c}_{3}}\left({U}_{2}\left(0,0,x\left(0\right)\right)+{\int }_{-\tau }^{0}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\right)\text{,}$

$\begin{array}{l}{\text{e}}^{\epsilon t}V\left(t,E\left(t\right),x\left(t\right)\right)-V\left(0,0,{x}_{0}\right)\\ ={\int }_{0}^{t}{\text{e}}^{\epsilon s}\left(\epsilon V\left(s,E\left(s\right),x\left(s\right)\right)+{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{t}{\text{e}}^{\epsilon s}{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\tau \right)\right)\text{d}E\left(s\right)+M\left(t\right),\end{array}$

${\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\le {P}_{3}+M\left(t\right),$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\left[{\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right]<\infty \text{a}\text{.s}\text{.,}$

$\underset{0\le t<\infty }{\mathrm{sup}}\left[{\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right]\le \lambda \text{a}\text{.s}\text{.,}$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left({U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon \text{a}\text{.s}\text{.,}$

(A4) 若存在 $V\in {C}^{2,1}\left({R}_{+}×{R}_{+}×R;{R}_{+}\right)$${U}_{1},{U}_{2}\in C\left(\left[-\tau ,\infty \right)×{R}_{+}×R;{R}_{+}\right)$ ，及常数 ${c}_{1}\ge 0$${c}_{2}>{c}_{3}\ge 0$ ，使得对任意的 $\left(t,E\left(t\right),x\right)\in {R}_{+}×{R}_{+}×R$ 有：

${U}_{1}\left(t,E\left(t\right),x\right)\le V\left(t,E\left(t\right),x\right)\le {U}_{2}\left(t,E\left(t\right),x\right).$

${L}_{1}V\left(t,E\left(t\right),x,y\right)\le {c}_{1}-{c}_{2}{U}_{2}\left(t,E\left(t\right),x\right)+{c}_{3}\left(1-\mu \right){U}_{2}\left(t-\delta \left(t\right),E\left(t-\delta \left(t\right)\right),y\right),$ (18)

${L}_{2}V\left(t,E\left(t\right),x,y\right)\le 0.$

1) $E{U}_{1}\left(t,E\left(t\right),x\right)<\infty ,\forall t\ge 0.$ (19)

2) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}{\int }_{0}^{t}E{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\le \frac{{c}_{1}}{{c}_{2}-{c}_{3}}.$ (20)

${c}_{1}=0$ 时，有：

3) ${\int }_{0}^{\infty }E{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\le \frac{1}{{c}_{2}-{c}_{3}}\left({U}_{2}\left(0,0,x\left(0\right)\right)+{\int }_{-\delta \left(0\right)}^{0}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\right)\text{.}$ (21)

4) ${\int }_{0}^{\infty }{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s<\infty .\text{a}\text{.s}\text{.}$ (22)

$\begin{array}{l}E{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\\ \le {U}_{2}\left(0,0,{x}_{0}\right)+{c}_{1}t-{c}_{2}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{c}_{3}\left(1-\mu \right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}s.\end{array}$ (23)

$\begin{array}{l}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}s\text{}\\ \le \frac{1}{1-\mu }E{\int }_{-\delta \left(0\right)}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\text{}\\ \le \frac{1}{1-\mu }\left({\int }_{-\delta \left(0\right)}^{0}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\right),\end{array}$ (24)

$\begin{array}{l}E{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\\ \le {P}_{4}+{c}_{1}t-\left({c}_{2}-{c}_{3}\right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s,\end{array}$ (25)

$\left({c}_{2}-{c}_{3}\right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{4}+{c}_{1}t.$

$k\to \infty$$n\to \infty$ ，再运用Fubini定理，可得，

$\left({c}_{2}-{c}_{3}\right){\int }_{0}^{t}E{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{4}+{c}_{1}t.$ (26)

${c}_{1}=0$ 时，根据(26)我们有，

$\left({c}_{2}-{c}_{3}\right){\int }_{0}^{t}E{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\le {P}_{4}.$

${\int }_{0}^{\infty }E{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\le \frac{1}{{c}_{2}-{c}_{3}}\left({U}_{2}\left(0,0,x\left(0\right)\right)+{\int }_{-\delta \left(0\right)}^{0}{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s\right)\text{.}$ (27)

$E{\int }_{0}^{\infty }{U}_{2}\left(s,E\left(s\right),x\right)\text{d}s<\infty \text{.}$

1) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}E{U}_{1}\left(t,E\left(t\right),x\right)\le \frac{{c}_{1}}{\epsilon }.$ (28)

${c}_{1}=0$ 时，有：

2) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left(E{U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon .$ (29)

3) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left({U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon \text{a}\text{.s}\text{.}$ (30)

$\begin{array}{l}E{\text{e}}^{\epsilon \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)-{U}_{2}\left(0,0,{x}_{0}\right)\\ \le E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}\left({c}_{1}-\left({c}_{2}-\epsilon \right){U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{c}_{3}\left(1-\mu \right){U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\right)\text{d}s\\ \le \frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t}-\left({c}_{2}-\epsilon \right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{c}_{3}\left(1-\mu \right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}s,\end{array}$ (31)

$\begin{array}{l}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}s\\ \le {\text{e}}^{\epsilon m}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon \left(s-\delta \left(s\right)\right)}{U}_{2}\left(s-\delta \left(s\right),E\left(s-\delta \left(s\right)\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}s\\ \le \frac{1}{1-\mu }\left({\text{e}}^{\epsilon m}E{\int }_{-\delta \left(0\right)}^{0}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s+{\text{e}}^{\epsilon m}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\text{e}}^{\epsilon s}{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s,\end{array}$ (32)

$E\left({\text{e}}^{\epsilon \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\right)\le {P}_{5}+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t},$ (33)

$n\to \infty ,k\to \infty$ ，可得

$E\left({\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right)\le {P}_{5}+\frac{{c}_{1}}{\epsilon }{\text{e}}^{\epsilon t},$ (34)

${c}_{1}=0$ 时，将(34)两边同除以 ${\text{e}}^{\epsilon t}$ 我们有，

$E\left({U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right)\le {P}_{5}{\text{e}}^{-\epsilon t},$

$\begin{array}{l}{\text{e}}^{\epsilon t}V\left(t,E\left(t\right),x\left(t\right)\right)-V\left(0,0,{x}_{0}\right)\\ ={\int }_{0}^{t}{\text{e}}^{\epsilon s}\left(\epsilon V\left(s,E\left(s\right),x\left(s\right)\right)+{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\delta \left(s\right)\right)\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{\int }_{0}^{t}{\text{e}}^{\epsilon s}{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(s-\delta \left(s\right)\right)\right)\text{d}E\left(s\right)+M\left(t\right).\end{array}$

${\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\le {P}_{5}+M\left(t\right),$ (35)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\left[{\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right]<\infty \text{a}\text{.s}\text{.,}$

$\underset{0\le t<\infty }{\mathrm{sup}}\left[{\text{e}}^{\epsilon t}{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\right]\le \lambda \text{a}\text{.s}\text{.,}$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}\mathrm{log}\left({U}_{1}\left(t,E\left(t\right),x\right)\right)\le -\epsilon \text{a}\text{.s}\text{.,}$

$\eta =1-\omega$ ，则时变延迟随机微分方程(1)变成了

$\begin{array}{c}\text{d}x\left(t\right)=f\left(t,E\left(t\right),x\left(t\right),x\left(\eta t\right)\right)\text{d}t+g\left(t,E\left(t\right),x\left(t\right),x\left(\eta t\right)\right)\text{d}E\left(t\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }+\sigma \left(t,E\left(t\right),x\left(t\right),x\left(\eta t\right)\right)\text{d}B\left(E\left(t\right)\right).\end{array}$ (36)

(A5)若存在 $V\in {C}^{2,1}\left({R}_{+}×{R}_{+}×R;{R}_{+}\right)$${U}_{1},{U}_{2}\in C\left(\left[-\tau ,\infty \right)×{R}_{+}×R;{R}_{+}\right)$ ，及常数 ${c}_{1}\ge 0$${c}_{2}>{c}_{3}\ge 0$ ，使得对任意的 $\left(t,E\left(t\right),x\right)\in {R}_{+}×{R}_{+}×R$ 有：

${U}_{1}\left(t,E\left(t\right),x\right)\le V\left(t,E\left(t\right),x\right)\le {U}_{2}\left(t,E\left(t\right),x\right).$

${L}_{1}V\left(t,E\left(t\right),x,y\right)\le {c}_{1}-{c}_{2}{U}_{2}\left(t,E\left(t\right),x\right)+{c}_{3}\eta {U}_{2}\left(\eta t,E\left(\eta t\right),y\right),$ (37)

${L}_{2}V\left(t,E\left(t\right),x,y\right)\le 0.$

1) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{E{U}_{1}\left(t,E\left(t\right),x\right)}{1+t}\le \frac{{c}_{1}}{1+\epsilon }.$ (38)

${c}_{1}=0$ 时，有：

2) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(E{U}_{1}\left(t,E\left(t\right),x\right)\right)}{\mathrm{log}\left(1+t\right)}\le -\epsilon .$ (39)

3) $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left({U}_{1}\left(t,E\left(t\right),x\right)\right)}{\mathrm{log}\left(1+t\right)}\le -\epsilon \text{a}\text{.s}\text{.}$ (40)

$\begin{array}{l}E\left({\left(1+t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}^{\epsilon }V\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\right)-V\left(0,0,{x}_{0}\right)\\ =E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon -1}\left(\epsilon V\left(s,E\left(s\right),x\left(s\right)\right)\right)+{\left(1+s\right)}^{\epsilon }{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\text{d}E\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }{V}_{x}\left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\sigma \left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\text{d}B\left(E\left(s\right)\right).\end{array}$ (41)

$\begin{array}{l}E\left({\left(1+t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}^{\epsilon }{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\right)-{U}_{2}\left(0,0,{x}_{0}\right)\\ \le E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }\left({c}_{1}-\left({c}_{2}-\epsilon \right){U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)+{c}_{3}\eta {U}_{2}\left(\eta s,E\left(\eta s\right),x\left(\eta s\right)\right)\right)\text{d}s\\ \le \frac{{c}_{1}{\left(1+t\right)}^{1+\epsilon }}{1+\epsilon }-\left({c}_{2}-\epsilon \right)E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }{U}_{2}\left(s,E\left(s\right),x\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{c}_{3}\eta E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }{U}_{2}\left(\eta s,E\left(\eta s\right),x\left(\eta s\right)\right)\text{d}s,\end{array}$ (42)

$\begin{array}{l}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+s\right)}^{\epsilon }{U}_{2}\left(\eta s,E\left(\eta s\right),x\left(\eta s\right)\right)\text{d}s\\ \le E{\int }_{0}^{\eta \left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}{\eta }^{-1}{\left(1+\frac{u}{\eta }\right)}^{\epsilon }{U}_{2}\left(u,E\left(u\right),x\left(u\right)\right)\text{d}u\\ \le {\eta }^{-\left(1+\epsilon \right)}E{\int }_{0}^{t\wedge {\tau }_{n}\wedge {\rho }_{k}}{\left(1+u\right)}^{\epsilon }{U}_{2}\left(u,E\left(u\right),x\left(u\right)\right)\text{d}u,\end{array}$ (43)

$E\left({\left(1+t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)}^{\epsilon }{U}_{1}\left(t\wedge {\tau }_{n}\wedge {\rho }_{k},E\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right),x\left(t\wedge {\tau }_{n}\wedge {\rho }_{k}\right)\right)\right)\le {U}_{2}\left(0,0,{x}_{0}\right)+\frac{{c}_{1}{\left(1+t\right)}^{1+\epsilon }}{1+\epsilon },$ (44)

$n\to \infty$$k\to \infty$ ，可得：

${\left(1+t\right)}^{\epsilon }E{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\le {U}_{2}\left(0,0,{x}_{0}\right)+\frac{{c}_{1}{\left(1+t\right)}^{1+\epsilon }}{1+\epsilon },$ (45)

${c}_{1}=0$ 时，将(45)两边同除以 ${\left(1+t\right)}^{\epsilon }$ ，再将两边同时取对数，再令 $t\to \infty$ ，就能得到(39)。

$\begin{array}{l}{\left(1+t\right)}^{\epsilon }V\left(t,E\left(t\right),x\left(t\right)\right)-V\left(0,0,{x}_{0}\right)\\ ={\int }_{0}^{t}{\left(1+s\right)}^{\epsilon -1}\left(\epsilon V\left(s,E\left(s\right),x\left(s\right)\right)\right)+{\left(1+s\right)}^{\epsilon }{L}_{1}V\left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{\int }_{0}^{t}{\left(1+s\right)}^{\epsilon }{L}_{2}V\left(s,E\left(s\right),x\left(s\right),x\left(\eta s\right)\right)\text{d}E\left(s\right)+M\left(t\right).\end{array}$

${\left(1+t\right)}^{\epsilon }{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)\le {U}_{2}\left(0,0,{x}_{0}\right)+M\left(t\right),$ (46)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}{\left(1+t\right)}^{\epsilon }{U}_{1}\left(t,E\left(t\right),x\left(t\right)\right)<\infty \text{a}\text{.s}\text{.,}$ (47)

4. 例子

$f\left({t}_{1},{t}_{2},x,y\right)=-2{x}^{3}-2x,\text{\hspace{0.17em}}g\left({t}_{1},{t}_{2},x,y\right)=-x{y}^{2},\text{\hspace{0.17em}}\sigma \left({t}_{1},{t}_{2},x,y\right)=xy,$

$V\left({t}_{1},{t}_{2},x\right)={x}^{2},\text{\hspace{0.17em}}{U}_{1}\left({t}_{1},{t}_{2},x\right)={x}^{2},\text{\hspace{0.17em}}{U}_{2}\left({t}_{1},{t}_{2},x\right)={x}^{4}+{x}^{2},$

${U}_{1}\left({t}_{1},{t}_{2},x\right)\le V\left({t}_{1},{t}_{2},x\right)\le {U}_{2}\left({t}_{1},{t}_{2},x\right),$

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)=-4x\left({x}^{3}+x\right)=-4{x}^{2}-4{x}^{4},$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)=-2x\cdot x{y}^{2}+{x}^{2}{y}^{2}=-{x}^{2}{y}^{2},$

${c}_{1}=0$${c}_{2}=0.835$${c}_{3}=0$ 时，因此 $\epsilon =0.835$ 为方程 ${c}_{2}=\epsilon +{c}_{3}{\text{e}}^{\epsilon \tau }$ 的唯一根。

$\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(E{x}^{2}\left(t\right)\right)}{t}\le -\epsilon ,$ (48)

$\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(|x\left(t\right)|\right)}{t}\le -\frac{\epsilon }{2}\text{a}\text{.s}\text{.,}$ (49)

${\int }_{0}^{\infty }E\left({x}^{4}\left(t\right)\right)\text{d}t<\infty ,$ (50)

${\int }_{0}^{\infty }{x}^{4}\left(t\right)\text{d}t<\infty \text{a}\text{.s}\text{.}$ (51)

$m=\underset{t\ge 0}{\mathrm{sup}}\delta \left(t\right)=1.4.$

$f\left({t}_{1},{t}_{2},x,y\right)=-2{x}^{3}-2x+\frac{1}{x},\text{\hspace{0.17em}}g\left({t}_{1},{t}_{2},x,y\right)=-\frac{1}{x}{y}^{2},\text{\hspace{0.17em}}\sigma \left({t}_{1},{t}_{2},x,y\right)=y,$

$V\left({t}_{1},{t}_{2},x\right)={x}^{2},\text{\hspace{0.17em}}{U}_{1}\left({t}_{1},{t}_{2},x\right)={x}^{2},\text{\hspace{0.17em}}{U}_{2}\left({t}_{1},{t}_{2},x\right)={x}^{4}+{x}^{2},$

${U}_{1}\left(x\right)\le V\left(x\right)\le {U}_{2}\left(x\right),$

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)=-4x\left({x}^{3}+x\right)+2=-4{x}^{4}-4{x}^{2}+2,$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)=-{y}^{2}.$

${c}_{1}=2$${c}_{2}=0.365$${c}_{3}=0$ ，且 $\epsilon =0.365$${c}_{2}=\epsilon +{c}_{3}{\text{e}}^{\epsilon m}$ 的唯一根，可得

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)=-4{x}^{4}-4{x}^{2}+2\le {c}_{1}-{c}_{2}{U}_{2}\left(x\right)+{c}_{3}\left(1-0.8\right){U}_{2}\left(y\right),$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)=-{y}^{2}\le 0,\text{\hspace{0.17em}}\forall \left({t}_{1},{t}_{2},x,y\right)\in {R}_{+}×{R}_{+}×R×R.$

$E\left({x}^{4}\left(t\right)\right)<\infty ,\text{\hspace{0.17em}}\forall t\ge 0,$ (52)

$\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(|x\left(t\right)|\right)}{t}\le -0.1825\text{a}\text{.s}\text{.}$ (53)

$f\left({t}_{1},{t}_{2},x,y\right)=-{x}^{3}-x,\text{\hspace{0.17em}}g\left({t}_{1},{t}_{2},x,y\right)=-3\frac{{y}^{2}}{x},\text{\hspace{0.17em}}\sigma \left({t}_{1},{t}_{2},x,y\right)=y,$

$V\left({t}_{1},{t}_{2},x\right)={x}^{4},\text{\hspace{0.17em}}{U}_{1}\left({t}_{1},{t}_{2},x\right)={x}^{4},\text{\hspace{0.17em}}{U}_{2}\left({t}_{1},{t}_{2},x\right)={x}^{6}+{x}^{4},$

${U}_{1}\left(x\right)\le V\left(x\right)\le {U}_{2}\left(x\right),$

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)=-4{x}^{3}\left({x}^{3}+x\right)=-4{x}^{6}-4{x}^{4},$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)=-4{x}^{3}\left(3\frac{{y}^{2}}{x}\right)+6{x}^{2}{y}^{2}=-6{x}^{2}{y}^{2},$

${c}_{1}=0$${c}_{2}=0.2825$${c}_{3}=0$ 时， $\epsilon =0.2825$${c}_{2}=\epsilon +{c}_{3}{\eta }^{-\epsilon }$ 的唯一根。可得

${L}_{1}V\left({t}_{1},{t}_{2},x,y\right)=-4{x}^{3}\left({x}^{3}+x\right)=-4{x}^{6}-4{x}^{4}\le {c}_{1}-{c}_{2}{U}_{2}\left(t,E\left(t\right),x\right)+{c}_{3}\eta {U}_{2}\left(\eta t,E\left(\eta t\right),y\right),$

${L}_{2}V\left({t}_{1},{t}_{2},x,y\right)=-4{x}^{3}\left(3\frac{{y}^{2}}{x}\right)+6{x}^{2}{y}^{2}=-6{x}^{2}{y}^{2}\le 0,\text{\hspace{0.17em}}\forall \left({t}_{1},{t}_{2},x,y\right)\in {R}_{+}×{R}_{+}×R×R.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(E{x}^{4}\left(t\right)\right)}{\mathrm{log}\left(1+t\right)}\le -0.2825,$ (54)

$\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{log}\left(|x\left(t\right)|\right)}{t}\le -0.070625\text{a}\text{.s}\text{.}$ (55)

5. 结论

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