For $$ \dot{x}=x+f(x,y),\quad \dot{y}=-y+g(x,y) $$ where $f,g\in$ Lip $(\mathbb{R}^2,\mathbb{R})$ and $$ |f(x,y)|\leq c(x^2+y^2),\quad |g(x,y)|\leq c(x^2+y^2) $$ in a neighborhood of the origin, consider the stability of $(0,0)$.
After I tried several special $f,g$, I believe the origin is Lyapunov stable, but I do not know how to prove the general case. I tried to constuct some Lyapunov function but failed. I have no idea how to use the Lipschitz condition.
Could you give me some hint to find a Lyapunov function? Or you can suggest your own way. Appreciate any help!