Let p>0 and , where all the square roots are positive. Design a fixed point iteration with some F which has x as a fixed point. We prove that the fixed point iteration converges for all choices of initial guesses greater than -p+1/4.
Let so x is a fixed point for F since F(x)=x.
Now let . We have .
I can see that for , we have that g'(x) <1.
From there I am not sure how to proceed to obtain convergence for .