A general composite 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 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 < 2 η / k 2 , 0 < γ < μ ( η − μ k 2 2 ) / α = τ / α . It is shown that the sequence { x n } generated by the following general composite iterative method: { y n = ( I − α n μ F ) T x n + α n γ f ( x n ) , x n + 1 = ( I − β n A ) T x n + β n y n , ∀ n ≥ 0 , where { α n } ⊂ [ 0 , 1 ] and { β n } ⊂ ( 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 ) .