Common Fixed Points of One-parameter Nonexpansive Semigroups in Strictlyconvex Banach Spaces

One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: let C be a bounded closed convex subset of a Hilbert space E, and let {T(t) : t ∈ R+} be a strongly continuous semigroup of nonexpansive mappings on C. Fix u ∈ C and t1, t2 ∈ R+ with t1 < t2. Define a sequence {xn} in C by xn = (1− αn)/ (t2 − t1) ∫ t2 t1 T(s)xnds+ αnu for n ∈N, where {αn} is a sequence in (0,1) converging to 0. Then {xn} converges strongly to a common fixed point of {T(t) : t ∈R+}.