Let us consider $f=f(x,t):\mathbb{R^n}\times \mathbb{R}\to\mathbb{R^n}$ a $C^1$ function which is periodic in $t$. We know that, under the hypothesis $x\cdot f(x,t)<0$ (with $|x|>M$ and $t\in \mathbb{R}$), there exists $x_0\in \mathbb{R^n}$ and a periodic function $x:\mathbb{R}\to\mathbb{R^n}$ which is a periodic solution of $x'(t)=f(x,t),\;x(0)=x_0$.
I was wondering if this result keep being true if we assume $x\cdot f(x,t)>0$ (with $|x|>M$ and $t\in \mathbb{R}$). However, I have not found a counterexample nor proof of this fact. Could somenone give me a hint?
Thanks.
Consider the map $x\mapsto\phi(T;x,0)$ where $T$ is the period of $f$ and $\phi(t;x_0,t_0)$ the flow function of the ODE. Then the condition tells you that $B(0,M)$ is mapped into itself by a continuous function. The general fixed-point theorems now guarantee a fixed point of that map, which then gives a (usually stable) periodic solution.
Changing to $x\cdot f(x,t)>0$ is just a time-reversal away from the original problem, that is, now the map $x\mapsto\phi(-T;x,0)$ is the endomorphism of $B_M(0)$. This again results in a constant or periodic solution which now in general will be unstable.