On a multivalued second order differential problem with Hukuhara derivative

Abstract

Let $K$ be a closed convex cone with the nonempty interior in a real Banach space and let $cc(K)$ denote the family of all nonempty convex compact subsets of $K$. Assume that continuous linear multifunctions $H,\Psi : K \to cc(K)$ are given. We consider the following problem $\begin{aligned}D^2\Phi(t,x) =& \Phi(t,H(x)), D\Phi(t,x)|_{t=0} =& {0}, \Phi(0,x) =& \Psi(x)\end{aligned}$ for $t \geq 0$ and $x \in K$, where $D\Phi(t,x)$ denotes the Hukuhara derivative of $\Phi(t,x)$ with respect to $t$.