On a limit formula for regular iterations

Abstract Let f be an increasing homeomorphism of [ 0 , ∞ ) onto itself with no nonzero fixed point such that d : = f ′ ( 0 ) exists and 0 d 1 . We investigate the limit f α , ∞ ( x ) : = lim n → ∞ ⁡ f − n ( α f n ( x ) ) ( x ≥ 0 , α > 0 ) and its properties. We show that the Schroder equation σ ( f ( x ) ) = d σ ( x ) has a regularly varying solution if and only if for some a > 0 the limit f α , ∞ ( a ) exists for α in a dense set A in R + and the map A ∋ α → f α , ∞ ( a ) is injective.