Prelinear equations
Let lÎR and let j,w be functions from
R to R, then the Schröder functional
equation (cf. [23][22])
j(w(x))=lj(x) (xÎR) (S)
is a functional equation for the unknown function j.
Gy. Targonski [44] realized that from this equation under
certain assumptions the equation
(j(w))n=jn-1(j(wn)) (Sn)
follows for all n>1, where wn is the n-th iterate of w.
Conversely there is the problem whether the system ((Sn),n³2) or even a
subsystem of it imply the equation (S). (Cf. [27].)
We want to study a similar problem for the linear functional equation (for the
unknown function j)
j(w(x))=m(x)j(x). (L)
The same way as the Schröder equation (S) leads to (Sn) the linear
equation (L) leads to a system of functional equations
Qn(j(x),j(w(x)),j(wn(x)))=0 (Ln)
with universal polynomials Qn for n³2.
It should be determined whether the validity of the system ((Ln),n³2)
or of a subsystem implies the validity of (L). Which relations hold between
different members of ((Ln),n³2)? (Cf. [40].)
harald.fripertinger@kfunigraz.ac.at,
last changed: February 9, 2001
