Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let T ℑ: ={T(t): t ∈ ℝ⁺} be a strongly continuous semigroup of asymptotically nonexpansive self-mappings on K with a sequence {L_t} ⊂ [1,∞ ). Then, for a given μ₀ ∈ K and s_n ∈(0, 1), t_n > 0 there exists a sequence {u_n} ⊂ K such that u_n=(1-α_n)T(t _n)u_n+α_nu₀, for each n,∈ Ndbl; atisfying||u_n-T (t)u_n||→0 as→n ∞, for any t+, where [image omitted]. If, in addition, E is uniformly convex with uniformly G[image omitted]teaux differentiable norm, then it is proved that F(ℑ) ≠ ∅ and the sequence {u_n} converges strongly to a point of F(ℑ) under certain mild conditions on {L_t}, {t _n} and {s_n}. Moreover, it is proved that an explicit sequence {x_n} generated from x1 ∈ K by x_n+1: α _nu₀+(1-α_nn)T(t_n)x _n, n ≥1,converges to a fixed point of T under appropriate assumption imposed upon the sequence {x_n}.