  Under certain initial conditions, we prove the existence of set-valued
selectors of univariate compact-valued multifunctions of bounded (Jordan)
variation when the notion of variation is defined taking into account only the
Pompeiu asymmetric excess between compact sets from the target metric space.
For this, we study subtle properties of the directional variations. We show by
examples that all assumptions in the main existence result are essential. As an
application, we establish the existence of set-valued solutions $X(t)$ of
bounded variation to the functional inclusion of the form $X(t)\subset
F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$.
