I need to prove the fixed point convergence of a vector scheme which is as follows $$\mathbf{A}\mathbf{X} = b \max\{\mathbf{J}(\mathbf{X}+\mathbf{a}),0\}$$
where the max-operator operates elementwise; $\mathbf{A} = \pmatrix{B&0\\0&B}$, $\mathbf{J}=\pmatrix{-I & I \\ I & -I}$, $\mathbf{a} = \pmatrix{a\\0}$. Furthermore $b \in R$, $a \in R^n$ and $B, I \in R^{nxn}$ subsequently $\mathbf{X}, \mathbf{a} \in R^{2n}$ and $\mathbf{A}, \mathbf{J} \in R^{2nx2n}$.
I know $B^{-1}$ its operator/induced norm is bounded w.r.t to the infinity norm under $R^n$; in fact $||B^{-1}|| <= 1/(b+r)$ where b is from the problem and $r \in R^+$.
I rewrote the scheme as follows $-\mathbf{A}\mathbf{X} = b \min\{-\mathbf{J}(\mathbf{X}+\mathbf{a}),0\} = b(\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\}-\mathbf{J}\mathbf{X})$ then $\mathbf{X} = -\mathbf{A^{-1}}b(\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\}-\mathbf{J}\mathbf{X})$.
I would now compute $d(T(\mathbf{X}), T(\mathbf{Y}))$ and show $d(T(\mathbf{X}), T(\mathbf{Y})) <= qd(\mathbf{X}, \mathbf{Y})$ for $q \in [0,1)$ i.e. a contraction. This gives $$d(T(\mathbf{X}), T(\mathbf{Y})) = ||b(-\mathbf{A})^{-1}(\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\} -\min\{\mathbf{J}\mathbf{Y}, -\mathbf{J}\mathbf{a}\} -\mathbf{J}(\mathbf{X}-\mathbf{Y}))||$$
I use my result on the bound of $||B^{-1}||$ which implies $||\mathbf{A^{-1}}|| <= 1/(b+r)$ given the way it's constructed from $B$.
This gives $||b(-\mathbf{A})^{-1}(\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\}-\min\{\mathbf{J}\mathbf{Y}, -\mathbf{J}\mathbf{a}\} -\mathbf{J}(\mathbf{X}-\mathbf{Y}))|| <= \\||b(-\mathbf{A})^{-1}||*||\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\}-\min\{\mathbf{J}\mathbf{Y}, -\mathbf{J}\mathbf{a}\} -\mathbf{J}(\mathbf{X}-\mathbf{Y})||$
Where the later equals $(b/(b+r))*||\min\{\mathbf{J}\mathbf{X}, -\mathbf{J}\mathbf{a}\} -\min\{\mathbf{J}\mathbf{Y}, -\mathbf{J}\mathbf{a}\}-\mathbf{J}(\mathbf{X}-\mathbf{Y})||$.
However at this point I am somewhat stuck as I am unsure on how to deal with the elementwise min(or max) operator, is it a contraction by definition? In fact this question is a follow up from the answer(in comments) I got here: Fixed point convergence of a coupled iterative vector scheme Here I decided to take $\mathbf{J}$ out, compute its norm as $2$ and argue for the min-operator to contract. However this seems to be wrong regardless... if someone could give a heads up on how to proceed to prove the contraction it would be much appreciated.