A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient α∈(0,1), F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0, and A:H→H is a strongly positive bounded linear operator with coefficient γ∈(1,2). Let 0<μ<2η ^k2,0<γ<μ(η-μ^k22)α= τα. It is shown that the sequence ^xn generated by the following general composite iterative method: ^yn=(I- ^αnμF)T^xn+^αnγf( ^xn),xn+₁=(I-^βnA)T^xn+ ^βnyn,∀n<0, where ^αn⊂ [0,1] and ^βn⊂(0,1], converges strongly to a fixed point x∈Fix(T), which solves the variational inequality 〈(I-A)x,x- x〉≤0,∀x∈Fix(T).