  This paper investigates the problem of finding a fixed point for a global
nonexpansive operator under time-varying communication graphs in real Hilbert
spaces, where the global operator is separable and composed of an aggregate sum
of local nonexpansive operators. Each local operator is only privately
accessible to each agent, and all agents constitute a network. To seek a fixed
point of the global operator, it is indispensable for agents to exchange local
information and update their solution cooperatively. To solve the problem, two
algorithms are developed, called distributed Krasnosel'ski\u{\i}-Mann (D-KM)
and distributed block-coordinate Krasnosel'ski\u{\i}-Mann (D-BKM) iterations,
for which the D-BKM iteration is a block-coordinate version of the D-KM
iteration in the sense of randomly choosing and computing only one
block-coordinate of local operators at each time for each agent. It is shown
that the proposed two algorithms can both converge weakly to a fixed point of
the global operator. Meanwhile, the designed algorithms are applied to recover
the classical distributed gradient descent (DGD) algorithm, devise a new
block-coordinate DGD algorithm, handle a distributed shortest distance problem
in the Hilbert space for the first time, and solve linear algebraic equations
in a novel distributed approach. Finally, the theoretical results are
corroborated by a few numerical examples.
