Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces

Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H , P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H . For a contraction f on C and {tn} ⊆ (0,1), let xn be the unique fixed point of the contraction x → tn f (x) + (1− tn)(1/n) ∑n j=1(PT) x. Consider also the iterative processes {yn} and {zn} generated by yn+1 = αn f (yn) + (1− αn)(1/(n+ 1)) ∑n j=0(PT) j yn,n ≥ 0, and zn+1 = (1/(n+ 1)) ∑n j=0P(αn f (zn) + (1− αn)(TP) j zn),n ≥ 0, where y0,z0 ∈ C,{αn} is a real sequence in an interval [0,1]. Strong convergence of the sequences {xn},{yn}, and {zn} to a fixed point of T which solves some variational inequalities is obtained under certain appropriate conditions on the real sequences {αn} and {tn}.