Prove by Brouwer's fixed point theorem that there is an integer m such that the equation has a periodic solution of period m. Consider the two dimensional system $\dot{x}=f(t,x)$,
$f(t + 1, x) = f (t, x)$, where $f$ has continuous first derivatives with respect to $x$. Suppose $\Omega$ is a subset of $\mathbb{R}^2$ which is homeomorphic to the closed unit disk. Also, for any solution $x(t, x_0)$, $x(0, x_0) = x_0$, suppose there is a $T(x_0)$ such that $x(t, x_0)$ is in $\Omega$ for all $t\geq T(x_0)$. Prove by Brouwer's fixed point theorem that there is an integer $m$ such that the equation has a periodic solution of period $m$. Does there exist a periodic solution of period $1$ ?
Suppose there is a $\lambda >0$ such
that $$x'f(t, x) \leq -\lambda|x|^2 $$
for all $t, x$.
If $g(t) = g(t + 1)$ is a continuous function, prove the equation $\dot{x} = f(t, x) + g(t)$ has a periodic solution of period $1$.
 A: By compactness of $Ω$, the map $x\mapsto T(x)$ is bounded on $Ω$. Let $m\in\Bbb N$ be such a bound. Then the map $x\mapsto \phi(m,x)$ maps $Ω$ into itself and is continuous. By Brouwer a fixed point $x^*$ exists, and the solution $\phi(t,x^*)$ has period $m$. The solution might have smaller periods, and it is also possible that this solution is stationary, constant.

As example take $f(t,x)=-x$ which has the contraction property for the unit disk and only one equilibrium point without any non-trivial periodic solutions.

As for the second problem, the function $g$ is obviously has a bound $K$, so that any solution satisfies the inequality
$$
\frac12\frac{d}{dt}\|x(t)\|^2=x(t)^T\dot x(t)\le -L\|x(t)\|^2+K\|x\|.
$$
This is strictly contracting towards the origin for $\|x\|>R=\frac{K}{L}$, so with $Ω=\overline{\Bbb D}(0,R)$ the previous claim applies, and the construction is already valid for $m=1$ (assuming that $f$ is still $1$-periodic in $t$). Still, the result of Brouwer might be a constant solution.
