Multivalued second order differential problem

Let $K$ be a closed convex cone with nonempty interior in a real Banach space and let $F,G,Hcolon K o cc(K)$ be three given continuous additive set-valued functions. We study the existence and uniqueness of a solution of the second order differential problem $$ D^2Phi (t,x) = Phi (t,H(x)) $$ $$ Phi (0,x) = F(x),quad DPhi (t,x)|_{t=0} = G(x) $$ for $tgeq 0$ and $xin K$, where $DPhi (t,x)$ and $D^2Phi (t,x)$ denote the Hukuhara derivative and the second Hukuhara derivative of $Phi (t,x)$ with respect to $t$.