Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
floor functional equation
#1
let {x} = x - floor(x)
let f(x) be a nonlinear real-analytic function and satisfy for x > 0

f(f({x}/e)) = {x}/e

seems outside the books not ?
Reply
#2
(05/26/2011, 12:27 PM)tommy1729 Wrote: let {x} = x - floor(x)
let f(x) be a nonlinear real-analytic function and satisfy for x > 0

f(f({x}/e)) = {x}/e

seems outside the books not ?

doesnt any half-iterate of f pay the bill?
Reply
#3
euh no.

how do you arrive at half-iterates ??

the half-iterate of a polynomial or the half-iterate of an exponential does not satisfy f(f({x}/e)) = {x}/e

we are searching for a solution to f(x) in f(f({x}/e)) = {x}/e.

not its half-iterate ?
Reply
#4
(05/27/2011, 11:59 AM)tommy1729 Wrote: euh no.

how do you arrive at half-iterates ??

the half-iterate of a polynomial or the half-iterate of an exponential does not satisfy f(f({x}/e)) = {x}/e

we are searching for a solution to f(x) in f(f({x}/e)) = {x}/e.

not its half-iterate ?

Sorry, I meant a half-iterate of x not of f. We discussed that somewhere on the forum already. f(f(x))=x hence f(f({x}/e))={x}/e.
Reply
#5
yes that is true.

for those confused :

0 < x f(f({x}/e)) = {x}/e.

reduces to

0 < x < e f(f(x)) = x

in fact i noticed i made a mistake. ( when i had no computer in the neighbourhood )

if f(x) is real-analytic we get a contradiction since f(f(x)) is then also real-analytic and the equation f(f(x)) = x leads to f(f(x)) - x = 0 where f(f(x)) - x is also real-analytic.

but f(f(x)) - x = 0 for 0 < x < e

so on the interval [0,e] we simply have a constant 0 function but another function elsewhere ; this clearly is not real-analytic.

so the question reduces to finding :

non-linear Coo f(x) that satisfies for 0 < x < e => f(f(x)) = x

that should have been the OP.
Reply
#6
the easy f^(-1) ( 1 - f(z) ) does wonders
Reply
#7
The easiest is perhaps f(x)=1/x, or f(x)=-x, if you dont need strict increase.
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
Big Grin Logical tetration equation Xorter 0 1,171 02/01/2018, 10:29 PM
Last Post: Xorter
  Functional power Xorter 0 1,196 03/11/2017, 10:22 AM
Last Post: Xorter
  Some sort equation ... tommy1729 0 1,442 10/15/2015, 12:12 PM
Last Post: tommy1729
  An intresting equation ? Taking squares by equation. tommy1729 0 1,542 05/08/2015, 11:37 PM
Last Post: tommy1729
  Conservation of functional equation ? tommy1729 0 1,580 05/01/2015, 10:03 PM
Last Post: tommy1729
  Is this THE equation for parabolic fix ? tommy1729 5 5,712 04/16/2015, 10:01 PM
Last Post: tommy1729
  Mick's differential equation tommy1729 3 3,034 04/10/2015, 04:14 PM
Last Post: tommy1729
  tommy equation tommy1729 3 3,682 03/18/2015, 08:52 AM
Last Post: sheldonison
  A system of functional equations for slog(x) ? tommy1729 3 4,162 07/28/2014, 09:16 PM
Last Post: tommy1729
  [2014] Inconsistant equation ? tommy1729 0 1,679 07/27/2014, 02:38 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)