Comparing Jacobi and Gauss-Seidel methods for nonlinear iterations

It is well known that for certain linear systems Jacobi and Gauss-Seidel iterative methods have the same convergence behavior, e.g. Stein-Rosenberg Theorem. I am wondering if similar results exist for nonlinear iterations, where at step $k$ the Jacobi iterations on, say $mathbb{R}^n$, are defined as

$$x_i^{k+1}=F_i(x_1^{k+1},cdots,x_{i-1}^{k+1},x_i^k,cdots,x_n^k),$$

and the Gauss-Seidel iterations are defined as

$$x_i^{k+1}=F_i(x_1^k,cdots,x_n^k)$$

for $i=1,cdots,n$ and we have a set of nonlinear functions $F_i(cdot): mathbb{R}^ntomathbb{R}$.