## Paolo Cubiotti, On the unified approach between upper and lower semicontinuous differential inclusions

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DOI: 10.23952/jnva.5.2021.1.03

Volume 5, Issue 1, 1 February 2021, Pages 23-42

Abstract. In this paper, we consider a $k$-th order differential inclusion with a multifunction $F$ such that its restriction to the complement of a suitable null-measure set is lower semicontinuous and bounded. We prove that there exists an upper semicontinuous multifunction $G$ such that each generalized solution of the corresponding $k$-th order differential inclusion is also a generalized solution of the original differential inclusion. As an application, we prove an existence and qualitative result for the Cauchy problem associated to a class of $k$-th order differential inclusions. In particular, we give sufficient conditions under which the solution multifuncion admits an upper semicontinuous multivalued selection with nonempty compact connected values. Finally, as a further application, we prove an analogous existence and qualitative result for the generalized solutions of the Cauchy problem associated to a class of $k$-th order implicit discontinuous differential equations.