Let f be a continuous
real valued function defined on the real line. If f has a periodic point of
period k, does f have to have a periodic point of some other period m? A
Russian mathematician, A. N. Sharkovsky obtained a complete answer to
this question. Sharkovsky’s result is elegant, however, his proof is difficult.
Recently, P. D. Straffin attempted to give a simple proof of the sufficient part of
Sharkovsky’s theorem by means of directed graphs. However, his proof contains a
gap. In this paper, the authors fill in the gap in Straffin’s work. They also
give a proof of the necessary part of the theorem, which is also based on
directed graphs, and thus, obtain a complete simple proof of Sharkovsky’s
theorem.
