There is a function $G:mathbb{R}^dtomathbb{R}^d$ locally lipschitz, suppose there exists $R>0$ such that $xcdot G(x)leq 0 , forall xin mathbb{R}^d$ with $||x||geq R$. Show that every maximal solution to $x’=G(x)$ is defined in an interval of the form $(a,infty)$.

This problem seems complicated to solve, any help is appreciated. The existence of this solution is guaranteed by the Picard Theorem. But I’m confused because the differential equation is $x’=G(x)$ so it means that $x’in mathbb{R}^d$ but at the same time $x$ should be defined on some $(a,infty)$? So the proper statement should be $$x'(t)=G(x(t)) $$ so $x:Isubseteqmathbb{R}tomathbb{R}^d$