  We show that a variant of a function defined by Cholamjiak and Suantai can be
used to characterize 2-uniform smoothness. We then obtain greatly simplified
proofs of a convergence theorem of Marino and Xu and of one of Zhou using a
generalization of a lemma of Browder and Petryshyn and the aforementioned
characterization. We also obtain more easily rates of asymptotic regularity
corresponding to the studied iterations. Finally, we derive a way to relate two
constants which are characteristic to 2-uniformly smooth spaces.
