  We in this paper study the nonexpansive operators equipped with arbitrary
metric and investigate the connections between firm nonexpansiveness,
cocoerciveness and averagedness. The convergence of the associated fixed-point
iterations is discussed with particular focus on the case of degenerate metric,
since the degeneracy is often encountered when reformulating many existing
first-order operator splitting algorithms as a metric resolvent. This work
paves a way for analyzing the generalized proximal point algorithm with a
non-trivial relaxation step and degenerate metric.
