From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Functional equation Date: 18 Mar 1998 05:59:28 GMT In article <350AA2DE.D4@tirf.inpg.fr>, Anisse TALEB wrote: >Well, I need help on this functional equation: > > f(ax+b)=cf(x)+d x in an interval I. > >and with a,b,c,d some real constants, a and c are assumed to be nonzero. >f(x) is the unknown function. > >I think that this equation is related to iteration theory. What do you think can be said about f? Consider the case a=1, b>0: then your defining equation simply determines the behaviour of f on all of I from its behaviour on [0, b] (or some translate thereof inside I); on the other hand, you've given no restrictions at all on how f is to behave on that smaller interval. Likewise if b=0 and a>1, you may take any f on [1,a] and then use your defining equation to extend the definition of f. Up to translation, these two special cases really include all combinations of a and b except for a=-1 (which gives an unconstrained f on a half-line, extended by a kind of mirror symmetry) and of course a=1,b=0. (You already ruled out a=0.) I don't really know what "iteration theory" means -- discrete dynamical systems, I suppose -- but yes, this seems to amount to the iterates of linear transformations in R^1. Functional equations: index/39-XX.html Dynamical systems: index/58-XX.html dave