General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space

Let H be a Hilbert space and f a fixed contractive mapping with coefficient 0 < alpha < 1, A a strongly positive linear bounded operator with coefficient (gamma) over bar > 0. Consider two iterative methods that generate the sequences {x(n)} {y(n)} by the algorithm, respectively. x(n) = (i - alpha(n)A) 1/t(n) integral(in)(0) T(s)x(n)ds + alpha(n)gamma f(x(n)) (I) Yn+1 = (i - alpha(n)A) 1/t(n) integral(in)(0) T(s)y(n)ds + alpha(n)gamma f(y(n)) (II) where {alpha(n)} and {t(n)} are two sequences satisfying certain conditions, and J = {T(s) : s >= 0} is a one-parameter nonexpansive semigroup on H. It is proved that the sequences {x(n)}, {y(n)} generated by the iterative method (I) and (II), respectively, converge strongly to a common fixed point x* e F(J) which solves the variational inequality <(A - gamma f)x*, x* - z > <= 0 Z is an element of F(J). (c) 2008 Elsevier Ltd. All rights reserved.