UNIFORM NORMAL STRUCTURE AND STRONG 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 {epsilon(n)}, and asymptotically pseudocontractive with sequence {k(n)} where {k(n)} and {epsilon(n)} satisfy certain mild conditions. Let a sequence {x(n)} be generated from x(1) is an element of K by x(n+1) := (1 - lambda(n))x(n) + lambda(n)T(n)x(n) - lambda(n)theta(n)(x(n) - x(1)) for all integers n >= 1 where {lambda(n)} and {theta(n)} are real sequences satisfying appropriate conditions, then parallel to x(n) - Tx(n)parallel to -> 0 as n -> infinity. Moreover if E has uniform normal structure with coefficient N(E), L < N(E)(1/2) as well as has a uniformly Gateaux differentiable norm and T satisfies an additional mild condition, then {x(n)} also converges strongly to a fixed point of T. The results presented in this paper are improvements, extension and complement of some earlier and recent ones in the literature.