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 $\{F_t : t ≥ 0\}$ is a regular cosine family of continuous linear set-valued functions $F_t:K → cc(K), x ∈ F_t(x)$ for $t ≥ 0, x ∈ K$ and $F_t ᴑ F_s = F_s ᴑ F_t$ for $s; t ≥ 0$, then \[D^2F_t(x) = F_t(H(x))\] for $x ∈ K$ and $t ≥ 0$, where $D^2F_t(x)$ 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 $t = 0$.