  The Banach-Picard iteration is widely used to find fixed points of locally
contractive (LC) maps. This paper extends the Banach-Picard iteration to
distributed settings; specifically, we assume the map of which the fixed point
is sought to be the average of individual (not necessarily LC) maps held by a
set of agents linked by a communication network. An additional difficulty is
that the LC map is not assumed to come from an underlying optimization problem,
which prevents exploiting strong global properties such as convexity or
Lipschitzianity. Yet, we propose a distributed algorithm and prove its
convergence, in fact showing that it maintains the linear rate of the standard
Banach-Picard iteration for the average LC map. As another contribution, our
proof imports tools from perturbation theory of linear operators, which, to the
best of our knowledge, had not been used before in the theory of distributed
computation.
