Viscosity Iterative Approximation For Nonexpansive Mapping

Approximation to fixed points for nonlinear operator is the kernel of the study of the fixed point theory. In this paper we mainly study the following two problems:At first, we study the convergence of the following modified Mann's iterative sequence, for x₀ ∈ K, {x_n} is defined by:where α_n,β_n ∈ (0,1) satisfy proper conditions. We proved {x_n} strongly converges to some fixed point x of T, and x is the unique solution to some variational inequality in F(T).Then we study the convergence of the following modified viscosity iterative sequence:In the assumption of appropriate condition, we proved that the above sequence converges strongly to a fixed point of nonexpansive mapping in Hilbert space and in a uniformly convex Banach space respectively. Our results extended and improved the corresponding ones by H.K. Xu[J. Math.Anal. Appl. 298(2004),279-291.] and Tae-Hwa Kim and Hong-Kun Xu [Nonlinear Anal. 61(2005) 51-60].