A Distributed Algorithm for Computing a Common Fixed Point of a Family of Paracontractions

A distributed algorithm is described for finding a common fixed point of a family of $m>1$ nonlinear maps $M_i : \mathbb{R}^n \rightarrow \mathbb{R}^n$ assuming that each map is a paracontraction and that such a common fixed point exists. The common fixed point is simultaneously computed by $m$ agents assuming each agent $i$ knows only $M_i$, the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of the fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph $\mathbb{N}(t)$ whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any family of paracontractions $M_i, i \in \{1,2,\ldots,m\}$ which has at least one common fixed point, and any sequence of strongly connected neighbor graphs $\mathbb{N}(t)$, $t=1,2,\ldots$, the algorithm causes all agent estimates to converge to a common fixed point.