If the following is too specific, my question can be simplified as fixed point iteration
$${\mathbf X}_{k+1}={\mathbf G}f({\mathbf X}_k)+h({\mathbf X}_k)$$
and given that the spectrum radius of $\bf G$ less than $1$, any known conditions for the above fixed point iteration to converge? Any ideas or reference will be very helpful. Thanks!
Detailed information:
Consider the following fixed point iteration, where the exact form of the $$f=(f_1,f_2,...,f_N)'$$ is unknown, and each $$f_i:\mathbb R^N\to R^N$$ but we can assume certain conditions hold for $f$ when needed. Below is the fixed point iteration: $$\begin{pmatrix} X_{1}^{k + 1^{'}} \\ \vdots \\ X_{p}^{k + 1^{'}} \\ \end{pmatrix} = C\begin{pmatrix} G^{\left( 1 \right)} & & & & \\ & \ddots & & & \\ & & G^{\left( s \right)} & & \\ & & & \ddots & \\ & & & & G^{\left( p \right)} \\ \end{pmatrix}f\begin{pmatrix} X_{1}^{k + 1^{'}} \\ \vdots \\ X_{p}^{k + 1^{'}} \\ \end{pmatrix} + \begin{pmatrix} h_{A,X_{2}^{k}\left( :,1 \right)}\left( X_{1}^{k} \right)^{'} \\ \vdots \\ h_{X_{s - 1}^{k}\left( :,N \right),X_{s + 1}^{k}\left( :,1 \right)}\left( X_{s}^{k} \right)^{'} \\ \vdots \\ h_{X_{p - 1}^{k}\left( :,N \right),B}\left( X_{p}^{k - 1} \right)^{'} \\ \end{pmatrix}$$
Denote $$G = \begin{pmatrix} G^{\left( 1 \right)} & & & & \\ & \ddots & & & \\ & & G^{\left( s \right)} & & \\ & & & \ddots & \\ & & & & G^{\left( p \right)} \\ \end{pmatrix}_{N\times N}$$ with each $G^{(s)}$ is a symmetric block matrix of size $\frac{N}{p}\times\frac{N}{p}$.
and $$h_{A,B}(t)=\frac{c_1B-c_2A+(A-B)t}{c_1-c_2}$$
So briefly the iteration can be viewed as $$\mathbf{X}_{k+1} = G\mathbf{f}(\mathbf{X}_k)+h(\mathbf{X}_k)$$ Suppose we know the spectrum radius of the matrix $G$, which is a symmetric matrix, is bounded by $$\frac{(\frac{N}{p}+1)}{(N+1)^2(\frac{N}{p}-1)(cos(\frac{k\pi}{\frac{N}{p}+1})-1)}$$
My question: How to show the fixed point iteration converges? Of course we can assume $f$ satisfy certain constraints, and we can make certain choices of $p$ depends on $N$.
It is very complicated to me, I think of linearize $f$ by considering $Jf$, but that will not only give errors but also resulted in multiplication of $G$ and $Jf$, and I do not know how to find the eigenvalues of multiplications of a symmetric matrix and an arbitrary matrix. Any ideas?
Thanks for any help and suggestions!