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

Necessary conditions for linear convergence of iterated expansive, set-valued mappings with application to alternating projections