A General Iterative Method Based on the Hybrid Steepest Descent Scheme 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, G is a L-Lipschitzian mapping on H with coefficient L > 0, and F : H → H is a k- Lipschitzian and η-strongly monotone operator with k > 0, η > O. Let 0 <; μ <; 2η/k², 0 <; γ <; μ(η - μk²/2)/L = τ/L. We pointed out the relationship between Yamada´s method and viscosity iteration and proved that the sequence {x_η} generated by the iterative method x_η+1 = α_nγG(x_n) + (I - μα_nF)Tx_n converges strongly to a fixed point x̃ ∈ F_ix(T), which solves the variational inequality ((γG - μF)x̃, x-x̃) ≤ 0, for x ∈ F_ix(T).