Sandwich with periodicity

Sandwich type results for p-sub-periodic, and p-super-periodic functions are presented. Given functions f, g: R→ R, we obtain necessary and sufficient conditions for the existence of a minimal p-periodic function F: R→ R and a maximal p-periodic function G: R→ R such that f(x) ≤ F(x) ≤ G(x) ≤ g(x). Moreover, the formulas for F and G are given, their semicontinuity, subadditivity and superadditivity as well as the lower semicontinuity of the set-valued function Φ(x):=[F(x),G(x)]are discussed. Finally, using Michael’s selection theorem, conditions ensuring the existence of a continuous p-periodic selection of the function Φ are also given.