On the distributed computation of a common fixed point of a family of paracontractions

This paper is concerned with a distributed algorithm for computing a common fixed point of a family of m > 1 nonlinear maps Mi : Rn → Rn, assuming that each map is a paracontraction and that at least one such common fixed point exists. Each agent i knows the paracontraction Mi and receives only a subset of the entries of the state vectors of its current neighbors at each time t. Using only this information, each agent recursively updates its own estimate of a common fixed point. Neighbor relations are characterized by a collection of time-varying directed neighbor graphs Nk(t), one for each entry k ∊ {1,…, n} of the state vector. It is shown under certain technical conditions that the algorithm causes all agent estimates to converge to the same common fixed point.