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 γ¯ > 0. Let 0 < γ < γ¯ /α. It is proved that the sequence {xn} generated by the iterative method xn+1 = (I − αnA)T xn + αnγf (xn) converges strongly to a fixed point ˜x ∈ Fix(T ) which solves the variational inequality (γf −A) ˜ x,x − ˜x 0 for x ∈ Fix(T ). © 2005 Elsevier Inc. All rights reserved.