Given the following function:
$$g(z)=C*\begin{pmatrix} x^2+y^2-2 \\ x^2-y^2-1 \\ \end{pmatrix}+z, \; \; \;z=(x,y)\in [0.93,1.52]\times [0.41,1]$$
Prove that $g $ is a contraction for $C=\begin{pmatrix} c & c \\ c & -c \\ \end{pmatrix}$
I looked at the Jacobian matrix of $g$ which is : $J=C*\begin{pmatrix} 2x & 2y \\ 2x & -2y \\ \end{pmatrix}+\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$
However, I didn't find any norm such that $||J||\lt1$ which would prove that $g$ is in fact a contraction.
Would appreciate some help.
Assuming your $*$ is matrix multiplication $$J = \pmatrix{4cx+1 & 0\cr 0 & 4cy+1\cr}$$ so this will be a contraction with the Euclidean norm in any convex region where $|4cx+1|<1$ and $|4cy+1|<1$. For this to be true in your rectangle $(x,y) \in [0.93, 1.52] \times [0.41, 1]$ you need $-25/76 < c < 0$.