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 $F_i(\cdot): \mathbb{R}^n\to\mathbb{R}$.