  In this paper, we prove the following theorem: Let {T(t) : t >= 0} be a
one-parameter continuous semigroup of mappings on a subset C of a Banach space
E. The set of fixed points of T(t) is denoted by F(T(t)) for each t >= 0. Then
cap_{t >= 0} F(T(t)) = F(T(1)) cap F(T(sqrt 2)) holds. Using this theorem, we
discuss convergence theorems to a common fixed point of {T(t) : t >= 0}.
