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 ). Keywords: Nonexpansive mapping; Iterative method; Variational inequality; Fixed point; Projection; Viscosity approximation