Solutions of the inhomogeneous Cauchy functional equation and applications on the stability of additive functionsKnown results and open problemsMaximal families of commuting automorphisms and the differential equation of Aczél-JabotinskyLinear functional equations and iteration theory

Linear functional equations and iteration theory

The linear functional equation for the function q, of order ordq³1, which is analytic in z=0, is
q(p(z))=a(z)q(z)+b(z)  (L)
where a,b,p are given functions which are analytic in z=0 and which fulfill orda=0, ordb³1, p(z)=a1 z +a2 z2+... and 0<|a1|<1. It is well known that the solution of (L) is given by the "Schröder" series. (Cf. [23][22].) In the general case we want to find an embedding of (L) together with an analytic iteration group (p(t,z))tÎC of p(z) into a family (Lt)tÎC of linear equations
q(p(t,z))=[~a] (t,z)q(z)+[~b] (t,z)  (Lt)
which is "covariant" with respect to the iteration group, such that
{
[~a] (0,z)=1 [~b] (0,z)=0
[~a] (1,z)=a(z) [~b] (1,z)=b(z)
.
and
{
[~a] (t+s,z)=[~a] (t,p(s,z))[~a] (s,z)
[~b] (t+s,z)=[~a] (t,p(s,z))[~b] (s,z)+ [~b] (t,p(s,z))
.
are fulfilled for all t,sÎC. This approach would lead e. g. to new descriptions of solutions q(z) in form of integrals or limits of expressions in [~a] (t,z) and [~b] (t,z), which generalize the Schröder series. The following problems must be solved:
  1. Prove the existence of functions [~a] (t,z), [~b] (t,z) and describe the solutions in the case that the given series a(z) and b(z) are convergent.
  2. Application of this approach to non linear equations
    q(p(z))=F(z,q(z)).
  3. Application of this method to multi dimensional situations
    q(p(z))=a(z)q(z)+b(z),
    where z=t(z1,...,zn), q(z)=t(q1(z),...,qn(z)), a(z) is an (n,n)-matrix of holomorphic functions (or formal series) and b(z)=t(b1(z),...,bn(z)), and where p is a contracting automorphism, which is iterable.
This problem posed by L. Reich in [38] was also considered in the real case by G. Guzik [17] and in an abstract situation by Z. Moszner [26].
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001

Solutions of the inhomogeneous Cauchy functional equation and applications on the stability of additive functionsKnown results and open problemsMaximal families of commuting automorphisms and the differential equation of Aczél-JabotinskyLinear functional equations and iteration theory