Suppose for $x\in \mathbb{R}^m$ we have a well behaved differential inclusion $\dot x \in F(x)$, where $F(x)$ is a Marchaud map, i.e. u.s.c. with compact convex values and linear growth condition: there exists $c>0$ s.t. $$\sup\{\|y\|:\, y\in F(x) \}\le c(1+\|x\|)$$ for all $x$. So absolutely continuous solutions exist, but may not be unique. We also assume $F$ is $C^1$ whenever it is single-valued.
Now suppose there is a rest point $x^*$ in a neighborhood $U$, s.t. $F(x)$ is single valued on that neighborhood, $F(x^*)=0$, and $x^*$ is hyperbolic. Can we still apply Hartman-Grobman in a neighborhood around that rest point? I would think yes, since the construction is local, but I'm not sure if somehow the non-uniqueness of solutions may give us an issue. Could someone give me a hint please?
