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 {εn}, and asymptotically pseudocontractive with constant {kn}, where {kn} and {εn} satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by xn+1 := (1 − λn)xn + λnT nxn − λnθn(xn − x1), for all integers n 1, where {λn} and {θn} are real sequences satisfying appropriate conditions, then xn − T xn →0 as n→∞. Moreover, if E is reflexive, and has uniform normal structure with coefficient N(E) andL