  We present necessary conditions for monotonicity, in one form or another, of
fixed point iterations of mappings that violate the usual nonexpansive
property. We show that most reasonable notions of linear-type monotonicity of
fixed point sequences imply {\em metric subregularity}. This is specialized to
the alternating projections iteration where the metric subregularity property
takes on a distinct geometric characterization of sets at points of
intersection called {\em subtransversality}. Our more general results for fixed
point iterations are specialized to establish the necessity of
subtransversality for consistent feasibility with a number of reasonable types
of sequential monotonicity, under varying degrees of assumptions on the
regularity of the sets.
