A Filippov-Type Lemma for Stieltjes Differential Inclusions with Supremum

The aim of this paper is to generalize the classical Filippov lemma to the framework of Cauchy differential set-valued problems involving the supremum of the unknown function on a past interval and its Stieltjes derivative with respect to a left-continuous non-decreasing function. To highlight the wide spectrum of the result (coming from the fact that this setting covers the theories of ordinary differential or difference problems, impulsive problems or problems on time scales), as a consequence, a Filippov-type lemma for dynamic inclusions on time scales with supremum is obtained.