How to find the starting values in a Fixed-point iteration

Question

Hi I have to find which of the following equations and for which starting values converges towards a fixed point.

The equations are:

$ a) x = e^x−sin(x)+x$

$b) x = sin(x) − e^x + x$

$c) x = arcsin(e^x), x < 0$

$d) x = ln(sin(x)), x ∈ (0, π)$

I know how to find out if a fixed point iteration converges towards a fixed point but the problem is that I really don't know how to find the starting values.

Can anyone help me?

Answer 1

The fixed point theorem involves an interval $[a,b]$. Once you are able to establish the conditions of the theorem in a given interval, you can just pick any initial condition in that interval.

Obviously, this strategy will give admissible starting points but will not rule out other starting points outside the interval $[a,b]$.