Let $f:\mathbb{R}^2\to\mathbb{R}$ a locally lipschitz continuous function in its second variable and let $y(x)$ a solution of $y'(x)=f(x,y(x))$ which verifies $y(x_0+T)=y(x_0)$ for some $x_0\in\mathbb{R}$. Check if $y(x)$ is a periodic function with period $T$.
Let $\psi(x)=y(x+T)~\forall x\in\mathbb{R}$. Furthermore, it verifies $y(x_0+T)=\psi(x_0)$. I want to see if $\psi$ is a solution for the following IVP: \begin{cases} z'(x)=f(x,z(x))\\ z(x_0)=y_0 \end{cases}
We have that $$\psi'(x)=y'(x+T)\stackrel{y\text{ is sol for IVP}}{=}f(x+T,y(x+T))=f(x+T,\psi(x))$$ but we can't assure that the previous equation is equal to $f(x,\psi(x)$. It seems analytically that this is false, but i'm struggling to find a counterexample. Any suggestions?