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 ds+ α n u for 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∈ + } .