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↦tnf(x)+(1−tn)(1/n)∑j=1n(PT)jx. Consider also the iterative processes {yn} and {zn} generated by yn+1=αnf(yn)+(1−αn)(1/(n+1))∑j=0n(PT)jyn, n≥0, and zn+1=(1/(n+1))∑j=0nP(αnf(zn)+(1−αn)(TP)jzn),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}