Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups

We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let CC be a closed convex subset of a Hilbert space EE. Let {T(t):t≥0}{T(t):t≥0} be a strongly continuous semigroup of nonexpansive mappings on CC. The set of all fixed points of T(t)T(t) is denoted by F(T(t))F(T(t)) for each t≥0t≥0. Let ττ be a nonnegative real number and let {tn}{tn} be a sequence in RR satisfying τ+tn≥0τ+tn≥0 and tn≠0tn≠0 for n∈Nn∈N, and limntn=0limntn=0. Then {F(T(τ+tn))}{F(T(τ+tn))} converges to ⋂t≥0F(T(t))⋂t≥0F(T(t)) in the sense of Mosco.