A general iterative method for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0 < α < 1, and a strongly positive linear bounded operator A with coefficient γ̄ > 0. Let 0 < γ < γ̄/α. It is proved that the sequence {xn} generated by the iterative method xn+1 = (I - αnA)T xn + αn γ f (xn) converges strongly to a fixed point x̃ ∈ Fix(T) which solves the variational inequality 〈(γf - A)x̃, x - x ̃〉 ≤ 0 for x ∈ Fix(T). © 2005 Elsevier Inc. All rights reserved.