A general composite 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 α∈(0,1)α∈(0,1), F:H→HF:H→H is a kk-Lipschitzian and ηη-strongly monotone operator with k>0,η>0k>0,η>0, and A:H→HA:H→H is a strongly positive bounded linear operator with coefficient γ̄∈(1,2). Let 0<μ<2η/k2,0<γ<μ(η−μk22)/α=τ/α. It is shown that the sequence {xn}{xn} generated by the following general composite iterative method: {yn=(I−αnμF)Txn+αnγf(xn),xn+1=(I−βnA)Txn+βnyn,∀n≥0, where {αn}⊂[0,1]{αn}⊂[0,1] and {βn}⊂(0,1]{βn}⊂(0,1], converges strongly to a fixed point x̄∈Fix(T), which solves the variational inequality 〈(I−A)x̄,x−x̄〉≤0,∀x∈Fix(T).