Known results and open problemsSolutions of the inhomogeneous Cauchy functional equation and applications on the stability of additive functionsAsymptotic formulas for the iterates of a function

Asymptotic formulas for the iterates of a function

Based on an asymptotic solution of the translation equation by L. Berg (cf. [5]) D. Gronau proved in [15] a similar formula for the iterates of a function F:UÍR -> R which has a fixed point in 0, where UÍR is an open neighborhood of 0. This asymptotic formula of the N-th iterate FN of F is given by
FN((x)/(N)) = åi=1r (1)/(Ni) fi(x)+o((1)/(Nr)) for N -> ¥. (A)
The fi are real functions defined in a neighborhood of 0 and rÎN is a fixed integer. Whenever real functions fi for i=1,...,r exist such that (A) is fulfilled in a neighborhood of 0, then these fi are uniquely defined. In this case the functions fi fulfill a recursive system of functional equations which could be solved under some additional regularity assumptions up to the index r=4 (cf. [16] and [7]).

In [15] it was shown that for a differentiable function F an expansion of the form (A) exists if and only if F'(0)=1. If FÎC2 then (A) exists for r=1. If F is analytic in a neighborhood of 0 then (A) exists for any rÎN. In this case it is not relevant whether F can be embedded in the sense of (E) or not.

For special cases F(x)=(1-x)2, F(x)=exp(x)-1 the functions fi were computed for i=1,...,3. Numerical computations showing the quality of this approximation can be found in [16] and in [7].

The research work in connection with asymptotic development of the form (A) are not finished yet. Further cooperation together with L. Berg (Rostock) (cf. [3][4]) is planned:

  1. It is an open problem to prove the following conjecture: The asymptotic development (A) exists, if fÎCr+1 and f'(0)=1.
  2. Further investigations of the recursive system of functional equations for fi. In this context we can only expect some results about the structure of the solutions fi.
  3. Numerical computations and estimations about the quality of the approximations.
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001

Known results and open problemsSolutions of the inhomogeneous Cauchy functional equation and applications on the stability of additive functionsAsymptotic formulas for the iterates of a function