A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Abstract

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 0 < α < 1 , and F : H → H is a k -Lipschitzian and η -strongly monotone operator with k > 0 , η > 0 . Let 0 < μ < 2 η / k 2 , 0 < γ < μ ( η − μ k 2 2 ) / α = τ / α . We proved that the sequence { x n } generated by the iterative method x n + 1 = α n γ f ( x n ) + ( I − μ α n F ) T x n converges strongly to a fixed point x ̃ ∈ F i x ( T ) , which solves the variational inequality 〈 ( γ f − μ F ) x ̃ , x − x ̃ 〉 ≤ 0 , for x ∈ F i x ( T ) .