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

## TL;DR

This paper presents a distributed algorithm enabling multiple agents to collaboratively find a common fixed point of a family of nonlinear paracontraction maps, even with limited local information and dynamic neighbor relations.

## Contribution

It introduces a novel distributed method for computing common fixed points of paracontractions with convergence guarantees under time-varying network conditions.

## Key findings

- All agents' estimates converge to a common fixed point.
- The algorithm works under general conditions on the neighbor graph.
- It applies to nonlinear maps that are paracontractions.

## Abstract

A distributed algorithm is described for finding a common fixed point of a family of m>1 nonlinear maps M_i : R^n -> R^n assuming that each map is a paracontraction and that at least one such 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 a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agents estimates to converge to the same common fixed point of the m nonlinear maps.