We consider continuous solutions f: R+→ R+=(0,∞) of the functional equation f(xf(x)) =φ(f(x)) where φ is a given continuous map R+ → R+. If φ is an increasing homeomorphism the solutions are completely described, if not there are only partial results. In this paper we bring some necessary conditions upon a possible range Rf. In particular, if φ|Rf has no periodic points except for fixed points then there are at most two fixed points in Rf, and all possible types of Rf and all possible types of behavior of f can be described. The paper contains techniques which essentially simplify the description of the class of all solutions.
2000 Mathematics Subject Classification.
Primary 39B12; 39B22; 37E05. Secondary 26A18.
Key words: Iterative functional
equation; invariant curves; real solutions; periodic orbits.