On the notion of derivo-periodicity

A De Blasi-like differentiable multivalued function is shown to have a periodic derivative (i.e., to be derivo-periodic) if and only if it is a sum of a function of a continuous (single-valued) periodic function, linear function and a bounded interval (a multivalued constant). At the same time, the single-valued part is derivo-periodic a.e. in the usual sense. In the single-valued case, a characterization of a more general class of derivo-periodic ACG∗-functions is given. Derivo-periodicity in terms of the Clarke subdifferentials and an impossibility of an almost-periodic analogy are also discussed. The obtained results are finally applied to differential equations and inclusions.