A qualitative result for higher-order discontinuous implicit differential equations

Let $n,kin{f N}$, and let $T>0$, $Ysubseteq{f R}^n$ and $\xi=(\xi_0,\xi_1,ldots,\xi_{k-1})in({f R}^n)^k$. Given a function $f:[0,T] imes({f R}^n)^k imes Y o{f R}$, we consider the Cauchy problem $f(t,u,u^prime,ldots,u^{(k)})=0$ in $[0,T]$, $u^{(i)}(0)=\xi_i$ for every $i=0,1,ldots,k-1$. We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, we prove that, under suitable assumptions, the solution set $mathcal{S}^f_T(\xi)$ of the above problem is nonempty, and the multifunction $\xiin({f R}^n)^k o mathcal{S}^f_T(\xi)$ admits an upper semicontinuous multivalued selection, with nonempty, compact and connected values. The assumptions of our result do not require any kind of continuity for the function $f(cdot,cdot, y)$. In particular, a function $f$ satisfying our assumptions could be discontinuous, with respect to the second variable, even at all points $\xiin({f R}^n)^k$.