On a limit formula for regular iterations

Let f   be an increasing homeomorphism of [0,∞)[0,∞) onto itself with no nonzero fixed point such that d:=f′(0)d:=f′(0) exists and 00α>0) and its properties. We show that the Schröder equation σ(f(x))=dσ(x)σ(f(x))=dσ(x) has a regularly varying solution if and only if for some a>0a>0 the limit fα,∞(a)fα,∞(a) exists for α in a dense set A   in R+R+ and the map A∋α→fα,∞(a)A∋α→fα,∞(a) is injective.