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 ${F_{t}: tgeq 0}$ is a regular cosine family of continuous linear set-valued functions $F_{t}colon Klongrightarrow cc(K)$, $xin F_{t}(x)$ for $tgeq 0$, $xin K$ 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 $xin K$ and $tgeq 0$, 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 $t=0$.