It‘s a problem from my ode textbook:
Let $f(t,y)$ be a real-valued function of two independent variables $t$ and $y$ which is continuous on a region $$ \mathcal{D}=\left\{ \left( t,y \right) :g_1\left( t \right) \le y\le g_2\left( t \right) ,\tau _1\le t\le \tau _2 \right\} . $$
Suppose that
(i).$g_1,g_1',g_2$,and $g_2'$ are continuous on $\tau_1 \leq t\leq \tau_2$,
(ii).$g_1(t)\text{<}g_2(t)$ on $\tau_1\leq t\leq \tau_2$,
(iii).$$
\begin{cases}
g_1'\left( t \right) \text{<}f\left( t,g_1\left( t \right) \right)\\
g_2'\left( t \right)\text{>}f\left( t,g_2\left( t \right) \right)\\
\end{cases},\qquad \tau _1\le t\le \tau _2
$$
(iv).$\tau _1\text{<}t_0\text{<}\tau _2$ and $g_1(t_0)\text{<}c_0\text{<}g_2(t_0)$.
Show that there exists a function $\varphi(t)$ such that
(1).$\phi$ and $\phi'$ are continuous on $t_0\leq t\leq \tau_2$ and $(t,\phi(t))\in \mathcal{D}$ on $t_0\leq t\leq \tau_2$,
(2).$\phi'(t)=f(t,\phi(t))$ on $t_0\leq t\leq \tau_2$ and $\phi(t_0)=c_0$.
I think we should use Peano's existence theorem to prove that the solution exists locally.And then we show that this solution can be extended.
But I don't know how to prove it. In fact, I have no sense of the usefulness of conditions (III) and (iv).