Today the 17 October 2019 I discussed a very remarkable fixed point theorem discovered by the Ukrainian mathematician Oleksandr Micholayovych Sharkovsky.
We recall that a periodic point of period for a function is a point such that . With this definition, a periodic point of period is also periodic of period for every which is a multiple of . If but for every from 1 to , we say that is the least period of .
Theorem 1. (Sharkovsky’s “little” theorem) Let be an interval and let be a continuous function su. If has a point of least period 3, then it has points of arbitrary least period; in particular, it has a fixed point.