 
Summary: BROWDER'S CONVERGENCE FOR ONEPARAMETER
NONEXPANSIVE SEMIGROUPS
SHIGEKI AKIYAMA AND TOMONARI SUZUKI
Abstract. We give the sufficient and necessary condition of Browder's convergence
theorem for oneparameter nonexpansive semigroups which was proved in [T. Suzuki,
Browder's type convergence theorems for oneparameter semigroups of nonexpansive
mappings in Banach spaces, Israel J. Math., 157 (2007), 239257]. We also discuss the
perfect kernels of topological spaces.
1. Introduction
Let C be a closed convex subset of a Banach space E. A family of mappings {T(t) : t
0} is called a oneparameter strongly continuous semigroup of nonexpansive mappings
(oneparameter nonexpansive semigroup, for short) on C if the following are satisfied:
(i) For each t 0, T(t) is a nonexpansive mapping on C, that is,
T(t)x  T(t)y x  y
holds for all x, y C.
(ii) T(s + t) = T(s) T(t) for all s, t 0.
(iii) For each x C, the mapping t T(t)x from [0, ) into C is strongly continu
ous.
There are six papers concerning the existence of common fixed points of {T(t) : t 0};
see [1, 2, 4, 5, 9, 11]. Recently, Suzuki [11] proved that t0 F T(t) is nonempty
