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Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuous map. Assume the existence of two solutions $\varphi_1,\varphi_2: [0,1] \to \mathbb{R}$ of $x'=f(t,x)$ such that:

  • Graph$(\varphi_1)$ $\cap$ Graph$(\varphi_2)$ $=\{(0,p),(1,q)\}$, for some $p,q \in \mathbb{R}$;
  • Graph$(\varphi_1)$ $\cup$ Graph$(\varphi_2)$ is the boundary of (a region homeomorphic to) a disc $D$.

Show that for every $(t_0,x_0)\in D$ there is a solution $\varphi$ of $x'=f(t,x)$ such that $\varphi(0)=p$, $\varphi(1)=q$ and $\varphi(t_0)=x_0$.

Since the graphs of the given solutions only intercepts at $0$ and $1$, we have (without loss of generality) $\varphi_1(t) \leq \varphi_2$. So $(t_0,x_0) \in D \Rightarrow \varphi_1(t_0)<x_0<\varphi_2(t_0)$. And one could use Peano theorem to get a (local) solution with initial condition $\varphi(t_0)=x_0$, but I dont know how to extend this solution or how to ensure that this extended solution would meet the other requirements.

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