Convergence theorems for strongly continuous semi-groups of asymptotically nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E. Let T {colon equals} {T (t) : t ∈ R⁺} be a strongly continuous semi-group of asymptotically nonexpansive mappings from K into K with a sequence {L_t} ⊂ [1, ∞). Suppose F (T) ≠ 0{combining long solidus overlay}. Then, for a given u₀ ∈ K and t_n > 0 there exists a sequence {u_n} ⊂ K such that u_n = (1 - α_n) T (t_n) u_n + α_n u₀, for n ∈ N such that {α_n} ⊂ (0, 1) and L_tn - 1 < α_n, where t_n ∈ R⁺. Suppose, in addition, that E is reflexive strictly convex with a uniformly Gâteaux differentiable norm and that lim_{n → ∞} t_n = ∞, lim_{n → ∞} α_n = lim_{n → ∞} frac(L_tn - 1, α_n) = 0. Then the sequence {u_n} converges strongly to a point of F (T). Moreover, it is proved that an explicit sequence {x_n} generated from x₁ ∈ K by x_{n + 1} {colon equals} α_n u + (1 - α_n) T (t_n) x_n, n ≥ 1, converges to a fixed point of T.