The following problem has been taken from Differential Equations, Dynamical Systems, and an Introduction to Chaos (Hirsch, Devaney, Smale).
Let $A$ be an annular region in $\mathbb{R}^2$. Let $F$ be a planar vector field that points inward along the two boundary curves of $A$. Suppose also that every radial segment of $A$ is a local section. Prove there is a periodic solution in $A$.
We have been asked to solve this without invoking the Poincaré-Bendixson Theorem. One obvious way is to mimic the proof of the theorem, but I suppose there must be a more direct approach.
I think that I can show that any solution curve will intersect a given local section $S$ infinitely often, by arguing that $|\dot{\theta}|$ has a strictly positive minimum. Thus, any solution curve $\gamma(x)$ starting at some $x \in S$ must cut $S$ infinitely many times, monotonically. This sequence of points must have a limit $y \in S$, so $y$ is an $\omega$-limit point of $\gamma(x)$. I want to show that $y$ is not on the boundary, and that the solution curve $\gamma(y)$ starting at $y$ is the desired periodic orbit.
Here is one way to argue for this using first return maps (in Hirsch-Smale-Devaney there is the analogous Poincaré maps; although they define these for local sections around periodic points):
Let us denote by $\varphi_\bullet:(t,x)\mapsto \varphi_t(x)$ the $C^1$ flow generated by the vector field. Note that since the vector field points inwards on the boundary for any $t\in\mathbb{R}_{>0}$, $\varphi_t$ sends the annulus to its interior.
For any radial segment $S$, there is an $\epsilon_S\in\mathbb{R}_{>0}$ such that $S\times ]-\epsilon_S,\epsilon_S[$ is a flowbox for $\varphi$. Such open sets cover the annulus $A$; since it is compact there are finitely many such open sets that cover $A$. Consequently for any radial segment $S$ the first return map $\rho=\rho_{\varphi,S}:S\to S$ is defined everywhere on $S$. By the Implicit Function Theorem $\rho_{\varphi,S}$ is $C^1$ and the restriction of $\rho_{\varphi,S}$ to the interior of $S$ is a $C^1$ diffeomorphism (see the discussion at Question about the Poincaré first return function for the $C^2$ version of the statement).
Fix a radial segment $S$ and consider its first return map $\rho: S\to S$. Let $x_0\in S$. Then either its forward orbit $\{\rho^n(x_0)\in S\,|\, n\in\mathbb{Z}_{\geq0}\}$ is finite xor its infinite. If it's finite then some iterate of $x_0$ is fixed under $\rho$, i.e. it is a periodic point of $\varphi$. If the forward orbit is not finite since $S$ is compact, for some subsequence $n_k$ there is a point $x^\ast$ in the interior of $S$ such that $\lim_{k\to\infty} \rho^{n_k}(x_0)=x^\ast$. Then $x^\ast$ is a fixed point of $\rho$, that is, a periodic point of $\varphi$.