Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

Let E be a real uniformly convex Banach space, and let{T-i : i is an element of I} be N nonexpansive mappings from E into itself with F = {x is an element of E : T(i)x = x, i is an element of I} not equal phi, where I = {1,2,...,N}. From an arbitrary initial point x(1) = E, hybrid iteration scheme {x(n)} is defined as follows: x(n+1) = alpha(n)x(n) + (1 - alpha(n))(T(n)x(n) - lambda(n+1)mu A(T(n)x(n))), n >= 1, where A : E -> E is an L-Lipshitzian mapping, T-n = T-i, i = n(mod N), 1 <= i <= N, mu > 0, {lambda(n)} subset of [0,1), and {alpha(n)} subset of [a, b] for some a, b is an element of (0, 1). Under some suitable conditions, the strong and weak convergence theorems of {x(n)} to a common fixed point of the mappings {T-i : i is an element of I} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007). Copyright (C) 2009 Lin Wang et al.