Consider the differential equation $x'=f(t,x)$, where $f(t,x)$ is continuously differentiable in $t$ and $x$.
Suppose that $f(t+T,x)=f(t,x)$ for all $t$ . Suppose there are constants $p$, $q$ such that $f(t,p)\ge0,f(t,q)\le0$ for all $t$. Prove that there is a periodic solution $x(t)$ for this equation with $p<x(0)<q$
I have seen this question here - Prove that there is a periodic solution $x(t)$ with $p<x(0)<q$
My professor claims that the statement is also true if the inequalities $f(t,p)\gt0,f(t,q)\lt0$ are not strict. However, the proof in the link depicts the idea that a solution for the ODE satisfying $x(0)=p$ satisfies $x(t)>p$ for all $t>0$ and uses the fact that the inequality is strict (otherwise, the point $\tau$ satisfying $\varphi(\tau) = p$ could be an inflection point).
I'd appreciate any proof of this claim given the non-strict inequalities or a counterexample if the proposition is false.