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 $u^{\prime\prime}\in F(t,u, u^\prime)$, where $F:[0,T]\times{\bf R}^n\times {\bf R}^n \to2^{{\bf R}^n}$ 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 $F(t,\cdot\,,\cdot\,)$. That is, a multifunction $F$ can satisfy all the assumptions and, at the same time, for every $t\in[0,T]$ the multifunction $F(t,\cdot\,,\cdot\,)$ can be neither upper nor lower semicontinuous even at each point $(x,z)\in{\bf R}^n\times{\bf R}^n$. 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^{\prime\prime}\in Q(t,u, u^\prime)+S(t,u, u^\prime) $, where $Q:[0,T]\times{\bf R}^n\times {\bf R}^n \to{\bf R}^n$ has convex values, and $S:[0,T]\times{\bf R}^n\times {\bf R}^n \to{\bf R}^n$ has closed values. Again, our assumptions do not imply any kind of semicontinuity for the multifunctions $Q(t,\cdot\,,\cdot\,)$ and $S(t,\cdot\,,\cdot\,)$. Then we consider an application to the implicit differential inclusion $\psi(u^{\prime\prime})\in F(t,u, u^\prime)+G(t,u, u^\prime)$, where $F$ is convex-valued and $G$ is closed-valued. As regards the function $\psi$, 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.