Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T : C →C be a nonexpansive mapping with a nonempty fixed points set. Given a point u ∈ C, the initial guess x0 ∈ C is chosen arbitrarily and given sequences {αn}∞n=0, {βn}∞n=0 and {γn}∞n=0 in (0, 1), the following conditions are satisfied: (i) ∞n=0 αn=∞; (ii) αn→0, βn→0 and 0