If f is continuous and locally Lipschitz there is a soultion for ODE for each t

Question

I need help with the following proof-

Let f be a continuous and locally Lipschitz such that:

$sup|f(t,u)-f(t,v)|≤L(t)|u-v|$ (L is continuous and defind by t only).

Prove that for each $(t_0,u_0)$ there is a solution of $u'=f(t,u)$,$u(t_0)=u_0$, for each t.

I absolutely have no idea how to solve this...

Answer 1

The classical trick is to show that $F(u)(t)=\int_{t_0}^t f(t,u(t))dt+u_0$ is a contracting function, thus there exists $v$ such that $F(v)=v$. Thus, $v$ is a solution.