Common fixed points of one-parameter nonexpansive semigroups in strictly convex 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 ∈ ℠+ } be a strongly continuous semigroup of nonexpansive mappings on C . Fix u ∈ C and t 1 , t 2 ∈ ℠+ with t 1 < t 2 . Define a sequence { x n } in C by x n = ( 1 − α n ) / ( t 2 − t 1 ) ∫ t 1 t 2 T ( s ) x n d s + α n u for n ∈ ℕ , where { α n } is a sequence in ( 0 , 1 ) converging to 0 . Then { x n } converges strongly to a common fixed point of { T ( t ) : t ∈ ℠+ } .