Convergence of fixed-point iteration despite projection to hypercube

Question

I have a column vector $a(k) \in R^{n \times1}$, which satisfies $a(0) = 0$ and converges to $a^*$ using the algorithm $a(k+1)=S\cdot a(k)+T$ for $S \in R^{n \times n}, T \in R^{n \times 1}$. The matrix $S$ includes some variables, and I have found the values that make the iteration convergent, that is the absolute value of every eigenvalue of $S$ is less than 1, $|\text{eig}(S)|<1$. I'm trying to find the additional conditions (if any) to guarantee convergence of the iteration if I need $a_i(k) \in [0,1]$. Can anyone point to a direction to find how projection affects the convergence?