Common Fixed Points of Commutative Semigroups of Nonexpansive Mappings

Let C be a closed convex subset of a Banach space E. A mapping T on C is called a nonexpansive mapping if ‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We denote by F (T ) the set of fixed points of T . Kirk [21] proved that F (T ) is nonempty in the case that C is weakly compact and has normal structure. See also [3, 4, 5, 14] and others. If C is weakly compact and E has the Opial property, then C has normal structure; see [15]. Thus, F (T ) is nonempty in the case that C is weakly compact and E has the Opial property. Let (S,+) be a commutative semigroup, i.e., (i) s+ t ∈ S for s, t ∈ S; (ii) (s+ t) + u = s+ (t+ u) for s, t, u ∈ S; and (iii) s+ t = t+ s for s, t ∈ S. Then a family {T (t) : t ∈ S} of mappings on C is called a commutative semigroup of nonexpansive mappings on C (a nonexpansive semigroup on C, for short) if the following are satisfied: (sg 1) for each t ∈ S, T (t) is a nonexpansive mapping on C; and (sg 2) T (s+ t) = T (s) ◦ T (t) for all s, t ∈ S. We put F (S) = ⋂