Monotone with respect to preorder trajectories of differential equations

This article contains an extension and strengthening of a theorem of G. Haddad in which a criterion for existence of monotone trajectories to differential inclusions was given. In the present paper it is proved that contingent cones used in the criterion can be replaced by its convex hulls. As it demonstrated, the criterion remains valid under assumptions which are weaker than those in Haddad's theorem. In particular, it is shown that the continuity of preorder can be replaced by its semicontinuity.