# Existence of a solution; Initial value problem

• December 4th 2010, 12:35 AM
hiddy
Existence of a solution; Initial value problem
Hello

Can somebody help me to solve this problem:
Given an ODE $x'=f(t,x)$ continous and lokally lipschitz for x. $x(t), t \geq t_0$ is a solution to the initial value problem $x(t_0)=x_0$. $x_1(t)$ and $x_2(t)$ are differentiable and:

$x_1(t_0) \leq x_0, x_1' \leq f(t,x_1), t \geq t_0$
$x_2(t_0) \geq x_0, x_2' \geq f(t,x_2), t \geq t_0$

Show that for $t \geq t_0: x_1(t) \leq x(t) \leq x_2(t)$.
Thank you very much!