Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices

Using a theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings.Also, using a theorem of Mil’man and Mil’man, we show that for the above classes of Banach spaces, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings.Further, we show that a cascading nonexpansive mapping on a closed, bounded, convex set in a Banach space always has an approximate fixed point sequence.