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This paper addresses the problem of seeking a common fixed point for acollection of nonexpansive operators over time-varying multi-agent networks inreal Hilbert spaces, where each operator is only privately and approximatelyknown to each individual agent, and all agents need to cooperate to solve thisproblem by propagating their own information to their neighbors through localcommunications over time-varying networks. To handle this problem, inspired bythe centralized inexact Krasnosel'ski\u{\i}-Mann (IKM) iteration, we propose adistributed algorithm, called distributed inexact Krasnosel'ski\u{\i}-Mann(D-IKM) iteration. It is shown that the D-IKM iteration can converge weakly toa common fixed point of the family of nonexpansive operators. Moreover, underthe assumption that all operators and their own fixed point sets are(boundedly) linearly regular, it is proved that the D-IKM iteration convergeswith a rate $O(1/k^{\ln(1/\xi)})$ for some constant $\xi\in(0,1)$, where $k$ isthe iteration number. To reduce computational complexity and burden of storageand transmission, a scenario, where only a random part of coordinates for eachagent is updated at each iteration, is further considered, and a correspondingalgorithm, named distributed inexact block-coordinate Krasnosel'ski\u{\i}-Mann(D-IBKM) iteration, is developed. The algorithm is proved to be weaklyconvergent to a common fixed point of the group of considered operators, and,with the extra assumption of (bounded) linear regularity, it is convergent witha rate $O(1/k^{\ln(1/\xi)})$. Furthermore, it is shown that the convergencerate $O(1/k^{\ln(1/\xi)})$ can still be guaranteed under a more relaxed(bounded) power regularity condition.

Distributed Algorithms for Computing a Common Fixed Point of a Group of Nonexpansive Operators