"Consider the n-dimensional system $x^{'}=f(x)+g(t)$ where $x^{T}f(x)\leq -k|x|^{2}, k>0$, for all $x$ and $|g(t)|\leq M$ for all t. Show that this equation has a w-periodic solution if $g$ is w-periodic.
Hint: Using Brouwer's fixed point theorem."
I do not know how to prove the existence of periodic orbit in this case. This is my idea. I want to prove $x(t+nw,\bar{x})=x(t,\bar{x})$ where $\bar{x}$ is limit point.
We have:
$\dfrac{1}{2}.\dfrac{d|x|^2}{dt}=x^{T}f(x)+x^{T}g(t)\leq -k|x|^{2}+m|x|$.
Hence, $|x|\leq \dfrac{M}{k}$.
Let $x_1=x(w,x_0)$, $x_2=x(w,x_1)$, ..., $x_n=x(w,x_{n-1})$.
The solution $x(t,x_0)$ is unique so $x_n=x(nw,x_0)$.
Since $x$ is bounded, there exists subsequence $x_{n_{k}}$ converges to $\bar{x}$.
Now I need to prove that the system
$\begin{cases} x^{'}(t)=f(x(t))+g(t)\\ x(0)=\bar{x} \end{cases}.$
has w-periodic orbit. However, I do not know where I can apply fixed point theorem. Could you please give me suggestions? Thank you so much!
Consider the map $F(x)=\phi(w;0,x)$ where $\phi(t;t_0,x_0)$ is the flow with values the solution at $t$ of the IVP with initial point $(t_0,x_0)$.
Setting $R=\frac{M}{k}$ you found or at least hinted that you understand that any solution starting in $B(0,R)$ stays in this ball for all times. This means that $F$ is a continuous map from $B(0,R)$ into itself.
Now apply Brouwers fixed-point theorem to get the existence of a fixed point $x^*=F(x^*)$. The corresponding solution $x(t)=\phi(t;0,x^*)$ is then periodic with period $w$.