Suppose I have a fixed point iteration of the form
Suppose further that after some initial testing, I found that it does not converge to an a priori known fixed point $x^*$. I recall from optimization class that often when the standard newton's method does not converge, one can implement a global convergence strategy to ensure convergence from any initial point (e.g. line search with backtracking). Does anything similar exist for general fixed point iterations? Is there strategy that I can implement to ensure that the sequence of iterations does converge?