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, l p 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 ) ≠ ∅, then under certain appropriate assumptions made on the sequences {α n } and { t n } of the parameters, we show that the sequence { x n } implicitly defined by for all n ≥ 1 converges strongly to a member of Fix( F ).