A general 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 0 < alpha < 1, and F : H -> H is a k-Lipschitzian and eta-strongly monotone operator with k > 0, eta > 0. Let 0 < mu < 2 eta/k(2), 0 < gamma < mu(eta-mu k(2)/2)/alpha = tau/alpha. We proved that the sequence {x(n)} generated by the iterative method x(n+1) = alpha(n)gamma f(x(n)) + (I -mu alpha(n)F)Tx(n) converges strongly to a fixed point (x) over tilde is an element of F(ix) (T), which solves the variational inequality <(gamma f - mu F)(x) over tilde, x-(x) over tilde > <= 0, for x is an element of F(ix)(T). (C) 2010 Elsevier Ltd. All rights reserved.