  The common fixed points problem requires finding a point in the intersection
of fixed points sets of a finite collection of operators. Quickly solving
problems of this sort is of great practical importance for engineering and
scientific tasks (e.g., for computed tomography). Iterative methods for solving
these problems often employ a Krasnosel'ski\u{\i}-Mann type iteration. We
present an Asynchronous Sequential Inertial (ASI) algorithmic framework in a
Hilbert space to solve common fixed points problems with a collection of
nonexpansive operators. Our scheme allows use of out-of-date iterates when
generating updates, thereby enabling processing nodes to work simultaneously
and without synchronization. This method also includes inertial type
extrapolation terms to increase the speed of convergence. In particular, we
extend the application of the recent ARock algorithm" [Peng, Z. et al, SIAM J.
on Scientific Computing 38, A2851-2879, (2016)] in the context of convex
feasibility problems. Convergence of the ASI algorithm is proven with no
assumption on the distribution of delays, except that they be uniformly
bounded. Discussion is provided along with a computational example showing the
performance of the ASI algorithm applied in conjunction with a diagonally
relaxed orthogonal projections (DROP) algorithm for estimating solutions to
large linear systems.
