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→CT:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈Cu∈C, the initial guess x0∈Cx0∈C is chosen arbitrarily and given sequences {αn}n=0∞, {βn}n=0∞ and {γn}n=0∞ in (0,1)(0,1), the following conditions are satisfied:(i)∑n=0∞αn=∞;(ii)αn→0αn→0, βn→0βn→0 and 0