Numerical Stability of Fixed-Point Interation

The fixed-point iteration $x_{n+1} = phi(x_n)$ for some Lipschitz-continuous function $phi$ with Lipschitz-constant $L<1$ is one of the methods in numerical analysis to obtain a solution $x^*$ of the nonlinear system $y = f(x^*)$. How would I determine whether the fixed-point iteration is numerically stable? Or, in other words, how would I determine the stability in the sense of forward analysis, that is, the smallest number $sigma$ which satisfies $$Vert tilde g(x) – tilde g(tilde x)Vertleq kappasigmaepsilon$$, where $tilde g$ is an algorithm that solves the problem (i.e. in that case the fixed-point iteration), $kappa$ the relative condition of the problem (i.e. $frac{Vert x^*Vert}{Vert f(x^*)Vert}Vertmathrm D f(x^*)^{-1}Vert$), and $epsilon$ is the machine precision.

Unfortunately, I am completely lost. One of the major confusions arises from the fact that I find very differently looking definitions of numerical stability…