I am reading a journal article (Berry, 1994. The RAND Journal of Economics, Vol.25, No. 2, P242-262). I tried to go through the proof of the existence and uniqueness of a fixed point for a system of nonlinear equations: $\mathbf\delta = \mathbf r(\delta)$, where $\delta = (\delta_1, \ldots , \delta_N)$ and $\mathbf r = (r_1, \ldots, r_N)$ is a vector function that maps $\delta$ onto itself in a compact convex space.
I do not understand something in the proof: The proof says "A well-known sufficient condition for uniqueness is $\sum_k |\partial r_j/\delta_k| <1$," where $k = 1, \ldots, N$. In other words, the sufficient condition for uniqueness of the fixed point is that the row sums of the absolute values of the Jacobian elements of $\mathbf r$ with respect to $\delta$ are less than one. I don't know where this sufficient condition comes from. I have searched everywhere, but could not find the source for such a condition. Can someone explain to me why "$\sum_k |\partial r_j/\delta_k| <1$" is a sufficient condition for the uniqueness of the fixed point?