On Uniqueness of Conjugacy of Continuous and Piecewise Monotone Functions

We investigate the existence and uniqueness of solutions [inline-graphic not available: see fulltext] of the functional equation [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext], where [inline-graphic not available: see fulltext] are closed intervals, and [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext] are some continuous piecewise monotone functions. A fixed point principle plays a crucial role in the proof of our main result.