Multivalued second order differential problem

Abstract

Let $K$ be a closed convex cone with nonempty interior in a real Banach space and let $F,G,H: K → cc(K)$ be three given continuous additive set-valued functions. We study the existence and uniqueness of a solution of the second order differential problem

\[D^2Φ(t,x) = Φ(t,H(x)), Φ(0,x) = F(x), DΦ(t,x)|_{t=0} = G(x)\]

for $t ≥ 0$ and $x ∈ K$, where $DΦ(t,x)$ and $D^2Φ(t,x)$ denote the Hukuhara derivative and the second Hukuhara derivative of $Φ(t,x)$ with respect to $t$.