A strong convergence theorem for contraction semigroups in Banach spaces

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t) : t> 0} is a contraction semigroup on C such that Fix(F) 6 = ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined by xn = αnu+ (1 − αn)T (tn)xn for all n> 1 converges strongly to a member of Fix(F). 1