Hukuhara differentiability of continuous sine and cosine families of linear set-valued functions

We give necessary and sufficient conditions for a regular sine family of continuous linear set-valued functions associated with a regular cosine family of continuous linear set-valued functions, to be continuous. Then we show that the continuity and Hukuhara differentiability of regular sine and cosine families are equivalent. As an application, we prove the existence and uniqueness of a solution of a second order differential problem. Our results are a much stronger version of results in [5], [6] and [10] that are valid for Banach spaces.