Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gateaux differentiable and T : C -> C be a continuous pseudo-contraction with a nonempty fixed point set F(T). For arbitrary given element u is an element of C and for t is an element of (0, 1), let {y(t)} be the unique continuous path such that y(t) = (1 - t)Ty(t) + tu. Assume that y(t) -> p is an element of F(T) as t -> 0. Let {alpha(n)}, {beta(n)} and {gamma(n)} be three real sequences in (0, 1) satisfying the following conditions: (i) alpha(n) + beta(n) + gamma(n) = 1; (ii) lim(n ->infinity) alpha(n) = lim(n ->infinity) beta(n) = 0; (iii) lim(n ->infinity) beta(n)/1 - gamma(n) = 0; or (iii)' Sigma(infinity)(n=0) alpha(n)/1 - gamma(n) = infinity. Let {epsilon(n)} be a summable sequence of positive numbers. For arbitrary initial datum x(0) = x(0)(0) is an element of C and a fixed n >= 0, construct elements {x(n)(m)} as follows: x(n)(m + 1) = alpha(u)(n) + beta(n)x(n) + gamma(n)Tx(n)(m), m = 0, 1, 2, .... Suppose that there exists a least positive integer N(n) satisfying the following condition: parallel to Tx(n)(N(n) + 1) - Tx(n)(N(n))parallel to <= gamma(-1)(n) (1 - gamma(n))epsilon(n). Define iteratively a sequence {x(n)} in an explicit manner as follows: x(n + 1) = x(n + 1)(0) = x(n)(N(n) + 1) = alpha(n)u + beta(n)x(n) + gamma(n)Tx(n)(N(n)), n >= 0. Then {x(n)} converges strongly to a fixed point of T. For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x(n), this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311-3317]. (C) 2008 Elsevier Ltd. All rights reserved.