Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions

We study the differential inclusion dzₜ∈ F(zₜ)dxₜ, with initial condition z₀=ξ, where F is a nonconvex-valued multifunction, and x a path of bounded q-variation, for some 1 q<2, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of local and global solutions to this inclusion without assuming F bounded. If $z(ξ,x)$ denotes such a solution, we obtain measurability of z with respect to x and $ξ$. To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it. By the way, we prove that a compact-valued $γ$-H{ö}lder map F has, for any $p>1/γ$ and ξ∈ F(0), a selection $f(ξ)$ of bounded p-variation, started at $ξ$, such that f is measurable in $ξ$.