I've a differential system living in the $(d-1)$-simplex of the following form
$$ \dot{x}_i = \sum_{j=1}^d f_{i,j}(x) \,\times\, \mathbb{I}\left\{ x_k = 0,\,\forall k \in I_{i,j} \right\} $$ where the $f_{i,j}(x)$ are all Lipschitz and bounded (from below and above), and $I_{i,j}\subseteq \{1,\ldots,d\}$. Therefore, the indicator functions make the differential system non-Lipschitz in some convex regions. Note also that $x_i(t)\ge 0$.
I'm interested in proving existence of solutions, once fixed an initial condition, where solution is in the sense of Filippov. Do such solutions exist in this particular case? Which theorems are available out there for this particular case?