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 < infinity), 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) not equal 0, then under certain appropriate assumptions made on the sequences {alpha(n)} and {t(n)} of the parameters, we show that the sequence {x(n)} implicitly defined by x(n) = alpha(n)u + (1 - alpha(n))T(t(n))x(n) for all n >= 1 converges strongly to a member of Fix(F).