  The Picard iteration is widely used to find fixed points of locally
contractive (LC) maps. This paper extends the 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 sparse 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
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.
