The solvability of $$f(p(x)) = q(f(x))$$ for given strictly monotonous continuous real functions p, q

Abstract

We investigate the functional equation $$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) where p and q are given real functions. In the paper “On solvability of$$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494”, we solved the problem of the solvability of $$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output ’solvable/not solvable’.