Sharkovskii's Theorem Under Set-Valued Functions

Sharkovskii's remarkable theorem from 1964 demonstrated significant results about periodic orbits of continuous functions on the real line. His work produced the Sharkovskii ordering. If m >> n in the Sharkovskii ordering and if f has a periodic orbit of period m, it must also have a periodic orbit of period n. While Sharkovskii worked with classical continuous functions, this paper expands Sharkovskii's theorem to a class of set-valued functions. In particular, we show that the ordering holds for upper semicontinuous set-valued functions with the strong intermediate value property.