Let I be an interval of the real line and be a
continuous function from I onto itself. Apply it repeatedly and the process is
kind of a feedback loop. x results in an output y=f(x).
After a single period, the new value of x is f(x), after 2 periods it's , and on and on.
The sequence: is called the orbit of x.
Let's see. Let I be the set of non negative reals and define:
for every x in I. We have the iterations:
The orbit of 1:
This sequence is increasing since every term after the second is gotten from
the previous one by replacing the final b with
Therefore, the orbit(bounded and
monotonic), is convergent.
If we let its limit be L, then:
shows that, as ,
Therefore, L is a fixed point of f. A solution of x=f(x).
With some basic algebra we see
I hope I didn't delve into too much. I just wanted to explain best I could.
I wanted to show more about its convergence, but I'm tired of fighting with this contrary LaTex.