On the unified approach between upper and lower semicontinuous differential inclusions
Journal of Nonlinear and Variational Analysis · 2021
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.