Strong Convergence Theorems for Asymptotically Nonexpansive Mappings and Asymptotically Nonexpansive Semigroups

A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T ; that is, F(T)= {x ∈ C : Tx = x}. Also, recall that a family S= {T(s) | 0≤ s <∞} of mappings from C into itself is called an asymptotically nonexpansive semigroup on C if it satisfies the following conditions: (i) T(0)x = x for all x ∈ C; (ii) T(s+ t)= T(s)T(t) for all s, t ≥ 0; (iii) there exists a positive valued function L : [0,∞)→ [1,∞) such that lims→∞Ls = 1 and ‖T(s)x−T(s)y‖ ≤ Ls‖x− y‖ for all x, y ∈ C and s≥ 0; (iv) for all x ∈ C, s → T(s)x is continuous.