  We define in this work a notion of Young differential inclusion $$ dz_t \in
F(z_t)dx_t, $$ for an $\alpha$-Holder control $x$, with $\alpha>1/2$, and give
an existence result for such a differential system. As a by-product of our
proof, we show that a bounded, compact-valued, $\gamma$-H\"older continuous
set-valued map on the interval $[0,1]$ has a selection with finite
$p$-variation, for $p>1/\gamma$. We also give a notion of solution to the rough
differential inclusion $$ dz_t \in F(z_t)dt + G(z_t)d{\bf X}_t, $$ for an
$\alpha$-Holder rough path $\bf X$ with $\alpha\in
\left(\frac{1}{3},\frac{1}{2}\right]$, a set-valued map $F$ and a single-valued
one form $G$. Then, we prove the existence of a solution to the inclusion when
$F$ is bounded and lower semi-continuous with compact values, or upper
semi-continuous with compact and convex values.
