Second Hukuhara derivative and cosine family of linear set-valued functions

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 $K$. If $\{F_{t}: t\geq 0\}$ is a regular cosine family of continuous linear set-valued
functions $F_{t}\colon K\longrightarrow cc(K)$, $x\in F_{t}(x)$ for $t\geq 0$, $x\in K$ and $F_{t}\circ F_{s}=F_{s}\circ F_{t}$ for $s,t \geq 0$, then
\[
D^{2}F_{t}(x)=F_{t}(H(x))
\]
for $x\in K$ and $t\geq 0$, where $D^{2}F_{t}(x)$ denotes the second Hukuhara derivative of $F_{t}(x)$ with respect to $t$ and $H(x)$ is the second Hukuhara derivative of this multifunction at $t=0$.