# 广义Gronwall不等式及相关注记The Generalized Gronwall Inequalities and Related Notes

• 全文下载: PDF(276KB)    PP.132-135   DOI: 10.12677/PM.2018.82017
• 下载量: 675  浏览量: 2,005

In this paper, the accurate generalized Gronwall inequalities are presented and proved. In addition, we pointed out the mistakes in the literature [1] and [2].

1. 引言

Gronwall不等式是一个重要的积分不等式，在研究和论证诸如Cauchy问题解的唯一性、解对初值和参数的连续性和可微性等方面起着重要的作用。详见 [1] [2] [3] [4] [5] 等诸多文献。本文给出了(准确的)广义Gronwall不等式及其证明，并指出了文献 [1] 和 [2] 中的错误。

$\psi \left(t\right)\le \alpha \left(t\right)+{\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right]$ (1)

$\psi \left(t\right)\le \alpha \left(t\right)+{\int }_{0}^{t}\alpha \left(s\right)\beta \left(s\right)\mathrm{exp}\left({\int }_{s}^{t}\beta \left(r\right)\text{d}r\right)\text{d}s,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right]$ (2)

$\psi \left(t\right)\le |\alpha \left(t\right)|\mathrm{exp}\left({\int }_{0}^{t}\beta \left(s\right)\text{d}s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[0,T\right]$ (3)

2. 不等式(2)的另一种证法

$R\left(t\right)=\alpha \left(t\right)+{\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s$ (4)

${R}^{\prime }\left(t\right)={\alpha }^{\prime }\left(t\right)+\beta \left(t\right)\psi \left(t\right)\le {\alpha }^{\prime }\left(t\right)+\beta \left(t\right)R\left(t\right)$

${R}^{\prime }\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)-\beta \left(t\right)R\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)\le {\alpha }^{\prime }\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)$

$\frac{\text{d}}{\text{d}t}\left[R\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)\right]\le {\alpha }^{\prime }\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)$

$R\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)-R\left(0\right)\le {\int }_{0}^{t}\mathrm{exp}\left(-{\int }_{0}^{s}\beta \left(r\right)\text{d}r\right)\text{d}\alpha \left(s\right)$

$\begin{array}{l}R\left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)-\alpha \left(0\right)\\ \le \alpha \left(t\right)\mathrm{exp}\left(-{\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)-\alpha \left(0\right)+{\int }_{0}^{t}\alpha \left(s\right)\beta \left(s\right)\mathrm{exp}\left(-{\int }_{0}^{s}\beta \left(r\right)\text{d}r\right)\text{d}s\end{array}$

$R\left(t\right)\le \alpha \left(t\right)+{\int }_{0}^{t}\alpha \left(s\right)\beta \left(s\right)\mathrm{exp}\left({\int }_{s}^{t}\beta \left(r\right)\text{d}r\right)\text{d}s$

3. 不等式(3)的证明

$\psi \left(t\right)\ge 0$ 的情形，因 $\beta \left(t\right)\ge 0$ ，所以 ${\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s\ge 0$ 。 再由条件 $\psi \left(t\right)\le R\left(t\right)$ ，我们有

$\frac{{\alpha }^{\prime }\left(t\right)+\beta \left(t\right)\psi \left(t\right)}{\alpha \left(t\right)+{\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s}\le \frac{{\alpha }^{\prime }\left(t\right)}{\alpha \left(t\right)}+\beta \left(t\right)$ (5)

$\mathrm{ln}|\alpha \left(t\right)+{\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s|-\mathrm{ln}|\alpha \left(0\right)|\le \mathrm{ln}|\alpha \left(t\right)|-\mathrm{ln}|\alpha \left(0\right)|+{\int }_{0}^{t}\beta \left(s\right)\text{d}s$

$\mathrm{ln}\frac{|\alpha \left(t\right)+{\int }_{0}^{t}\beta \left(s\right)\psi \left(s\right)\text{d}s|}{|\alpha \left(t\right)|}\le {\int }_{0}^{t}\beta \left(s\right)\text{d}s$

$\mathrm{ln}\frac{\psi \left(t\right)}{|\alpha \left(t\right)|}\le {\int }_{0}^{t}\beta \left(s\right)\text{d}s$

$\frac{\psi \left(t\right)}{|\alpha \left(t\right)|}\le \mathrm{exp}\left({\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)$

$\alpha \left(t\right)=±1$ 时，由上面的过程可推得 $\psi \left(t\right)\le \mathrm{exp}\left({\int }_{0}^{t}\beta \left(s\right)\text{d}s\right)$

$\alpha \left(0\right)=0$ 时，由所设条件(1)知， $\psi \left(0\right)=0=\alpha \left(0\right)$

4. 相关注记

 [1] 盖拉德•泰休. 常微分方程与动力系统[M]. 金成桴, 译. 北京: 机械工业出版社, 2011: 31-33. [2] 周尚仁, 权宏顺. 常微分方程习题集[M]. 北京: 高等教育出版社, 1980: 75-76. [3] 张锦炎, 冯贝叶. 常微分方程几何理论与分支问题[M]. 北京: 北京大学出版社, 2000: 12-15. [4] 陆启韶, 彭临平, 杨卓琴. 常微分方程与动力系统[M]. 北京: 北京航空航天大学出版社, 2010: 16-20. [5] 范进军. 常微分方程续论[M]. 济南: 山东大学出版社, 2009: 30-34.