Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions

Let C be a closed convex subset of a real Hilbert space H and assume that T is an asymptotically κ-strict pseudo-contraction on C with a fixed point, for some 0 ≤ κ < 1. Given an initial guess x0 ∈ C and given also a real sequence {αn} in (0, 1), the modified Mann's algorithm generates a sequence {xn} via the formula: xn + 1 = αn xn + (1 - αn) Tn xn, n ≥ 0. It is proved that if the control sequence {αn} is chosen so that κ + δ < αn < 1 - δ for some δ ∈ (0, 1), then {xn} converges weakly to a fixed point of T. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence. © 2007 Elsevier Ltd. All rights reserved.