Existence of solutions for second-order differential inclusions associated with highly non-semicontinuous multifunctions

In this paper we deal with the existence of generalized solutions for the Cauchy problem associated with a second-order differential inclusion, both in explicit and in implicit form. We firstly prove an existence result for an inclusion of the type u^∈ F(t,u, u⁾, where F:[0,T]× Rⁿ× Rⁿ →2^ Rⁿ is a given closed-valued multifunction. The main peculiarity of this latter result is as follows: our assumptions do not imply any kind of semicontinuity for the multifunction F(t,·\,,·\,). That is, a multifunction F can satisfy all the assumptions and, at the same time, for every t∈[0,T] the multifunction F(t,·\,,·\,) can be neither upper nor lower semicontinuous even at each point (x,z)∈ Rⁿ× Rⁿ. A viable version of this result is also proved. Furtherly, as an application, an analogous result is proved for an inclusion of the type u^∈ Q(t,u, u⁾+S(t,u, u⁾, where Q:[0,T]× Rⁿ× Rⁿ → Rⁿ has convex values, and S:[0,T]× Rⁿ× Rⁿ → Rⁿ has closed values. Again, our assumptions do not imply any kind of semicontinuity for the multifunctions Q(t,·\,,·\,) and S(t,·\,,·\,). Then we consider an application to the implicit differential inclusion ψ(u^)∈ F(t,u, u⁾+G(t,u, u⁾, where F is convex-valued and G is closed-valued. As regards the function ψ, we only assume that it is continuous and locally nonconstant. Finally, we present a further application to the Cauchy problem associated with a Sturm-Liouville type differential inclusion.