On the Cauchy problem for a class of differential inclusions with applications

On the unified approach between upper and lower semicontinuous differential inclusions

“…As far as we know, there are not many results in this direction, even for the special case k = 1. In this latter case, some existence and qualitative results can be found in [2] and [11].…”

“…Recently, in [2], Theorem 1.1 was extended to a larger class of multifunctions F : R × R n → 2 R n , which is wider than the one of lower semicontinuous multifunctions. In order to state the main result of [2], we need to introduce some notations.…”

“…Recently, in [2], Theorem 1.1 was extended to a larger class of multifunctions F : R × R n → 2 R n , which is wider than the one of lower semicontinuous multifunctions. In order to state the main result of [2], we need to introduce some notations. First, we denote by D the class of all subsets U ⊆ R × R n that can be expressed as a finite union of subsets of R × R n , each with at least one projection of null Lebesgue measure.…”

“…As a matter of fact, a multifunction F : R × R n → 2 R n satisfying the assumptions of Theorem 1.2 can fail to be lower semicontinuous even at all points (t, x) ∈ R × R n (see [2]). Such a weaker regurarity is paid by assuming that the set F((R × R n ) \ U) belongs to the family G n .…”

“…As showed in [2, Example 3.3], Theorem 1.2 does not hold without the latter assumption. As an application of Theorem 1.2, some new existence and qualitative results were obtained for the first-order differential inclusion 1.1 associated with a multifunction F, which may not have any semicontinuity property, as well as for first-order implicit differential equations of the type g(u ) = f (t, u) associated with a discontinuous function f (see, e.g., [2,3]). Now, fix n, k ∈ N, let I ⊆ R be a compact interval, and let p ∈ [1, +∞].…”