A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let HH be a real Hilbert space. Suppose that TT is a nonexpansive mapping on HH with a fixed point, ff is a contraction on HH with coefficient 0<α<10<α<1, and F:H→HF:H→H is a kk-Lipschitzian and ηη-strongly monotone operator with k>0,η>0k>0,η>0. Let 0<μ<2η/k2,0<γ<μ(η−μk22)/α=τ/α. We proved that the sequence {xn}{xn} generated by the iterative method xn+1=αnγf(xn)+(I−μαnF)Txnxn+1=αnγf(xn)+(I−μαnF)Txn converges strongly to a fixed point x̃∈Fix(T), which solves the variational inequality 〈(γf−μF)x̃,x−x̃〉≤0, for x∈Fix(T)x∈Fix(T).