ABSTRACT
Our main result is the following: let be a multifunction, and assume that there exists a neglegible subset
, satisfying a certain geometrical condition, such that the restriction of F to
is bounded, lower semicontinuous with non-empty closed values, and its range belongs to a certain family
defined below. Then, there exists a bounded multifunction
such that G is upper semicontinuous with non-empty compact convex values, and every generalized solution of
is a solution of
. Such a result improves a celebrated result by A. Bressan, valid for lower semicontinuous multifunctions. We point out that a multifunction F satisfying our assumptions can fail to be lower semicontinuous even at all points
. We derive some existence and qualitative results for the Cauchy problem associated to such a class of multifunctions. As an application, we prove existence and qualitative results for the implicit Cauchy problem
,
, with f discontinuous in u.
2010 MATHEMATICS SUBJECT CLASSIFICATION:
Disclosure statement
No potential conflict of interest was reported by the authors.