schröder's equation is a functional equation. let's assume A is a subset of the real numbers and g maps A to itself. the goal is to find a nonzero (invertible, if possible) function f and a real number r such that [tex]f\circ g=rf[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

motivation: if there is an invertible f, then the nth iterate of g is given by [tex]g^{n}=f^{-1}\left( r^{n}f\right) [/tex]. this can lead to understanding the dynamics of g better.

let V be the set of functions from A to R, the set of real numbers.

let S(r,g) denote [tex]\left\{ f\in V:f\circ g=rf\right\} [/tex]. S(r,g) is a subspace of V.

define a map k(r,g) from V to V by the following:

[tex]k\left( r,g\right) \left( f\right) =f\circ g-rf[/tex].

k(r,g) is a linear operator on V with kernel S(r,g).

putting in this context, are there any theorems (with more assumptions perhaps) that

1. give one at least a sense of when the kernel of a map is not {0}

2. can find a basis for the kernel of a map explicitly

3. can approximate solutions to k(r,g)(f)=0?

any thoughts would be helpful. if you want to get real specific, assume A=[0,1] and g is either 2^(x-1) or x^2.

as an example, [tex]x+\frac{b}{a-1}\in S\left( a,ax+b\right) [/tex] , abusing the notation a little to let ax+b represent the function g such that g(x)=ax+b.

i know that S(r,g)=fix(k(r-1,g)), where fix(h) is the set of fixed points of h. this opens me up to all the fixed point theorems in functional analysis except that i already know it has a fixed point, namely the zero function. what will be interesting is uniqueness fixed point theorems to show that 0 is the only fixed point. however, under one or two more assumptions on g and A, for example, i can turn my attention to bounded functions and use the sup norm and perhaps talk about contractions. all i can do in that case, and here it's critical that g maps A to A, is show that the norm of k(r,g) is at most 1+|r| which doesn't prove it's a contraction. it would be extremely helpful if anyone were able to tell me what the norm of this operator k(r,g) is. in other words, the sup of ||k(r,g)(f)|| where ||f||=1. then i can tell for which r k is a contraction.

also, is it true that if an operator is NOT a contraction then it doesn't have a unique fixed point or could it not be a contraction and still have a unique fixed point?

any thoughts would be helpful.

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# Schröder's equation and functional analysis

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