Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings

Let K be a nonempty closed convex and bounded subset of a real Banach space E and T : K → K be uniformly L-Lipschitzian, uniformly asymptotically regular with sequence {ε_n}, and asymptotically pseudocontractive with constant {k_n}, where {k_n} and {ε_n} satisfy certain mild conditions. Let a sequence {x_n} be generated from x₁ ∈ K by x_n+1 := (1 - λ_n)x_n + λ_nTⁿx_n - λ_nθ_n(x_n - x₁), for all integers n ≥ 1, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, then ∥x_n - Tx_n∥ → 0 as n → ∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) and L < N(E)^1/2 and has a uniformly Gâteaux differentiable norm, and T satisfies an additional mild condition, then {x_n} also converges strongly to a fixed point of T.