## Gronwall 不等式

$v(t)\leq C+\int_a^t v(s)u(s)\rd s,\quad t\in [a,b],$

$v(t)\leq C\exp\left( \int_a^t u(s)\rd s \right).$

# Summer School 2005

Clay Mathematics Institute 2005 Summer School. June 20 – July 15 at the Mathematical Sciences Research Institute (MSRI) Berkeley, California

## Overview

The Clay Mathematics Institute will hold its 2005 summer school at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California.
Let $\set{x_n}_{n=1}^\infty$ be a real sequence defined by $$x_{n+1}=\frac{C}{2}+\frac{x_n^2}{2},$$  with $x_1=C/2$, where $C$ is a constant, try to show that
1. If $C>1$, then $\set{x_n}_{n=1}^\infty$ is divergent;
2. If $0< C\leq 1$, then $\set{x_n}_{n=1}^\infty$ is convergent;
3. If $-3\leq C < 0$, then $\set{x_n}_{n=1}^\infty$ is convergent;
Try to discuss the case of $C< -3$, is $\set{x_n}_{n=1}^\infty$ divergent?