For a differential inclusion $x'(t)\in h(x(t))$, is there any condition (of course, I don't want the map to be single-valued) under which we can say that for any trajectory $x(.)$ satisfying the differential inclusion there exist a continuous function $F_{x(.)}(.)$ (i.e. depending on the trajectory x(.)) s.t. $F_{x(.)}(x) \in h(x)$ and $x'(t) = F_{x(.)}(x(t))$.
I am aware of continuous selection theorems for set-valued maps, but have not seen anything like above.
There are such results. For example, if $-h$ is a maximal monotone operator on a Hilbert space $H$
$$ A: D(A)\subset H\to 2^H, $$
where $2^H$ denotes the collection of subsets of $H$, then the differential inclusion
$$ x'(t)+A x(t)\ni 0,\;\;x(0)=x_0, \;\;t>0, $$
has a unique (appropriately defined) solution. Moreover,
$$x'(t)+A^0 x(t) =0 $$
for almost all $t$, where $A^0 x(t)$ denotes the point in the closed convex set $A x(t)$ closest to the origin, i.e., the shortest vector in $A x(t)$. For details I refer to Theorem 3.1 in H. Brezis' book
Operateurs maximaux monotone et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co. 1973.
