On the Cauchy problem for a class of differential inclusions with applications - Cubiotti, Paolo; Yao, Jen-Chih (Applicable analysis, 2020)
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.