STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*, and K be a nonempty closed convex subset of E. Suppose that T is nonexpansive mapping from K into itself such that F = F(T) not equal 0. For arbitrary initial value x(0) is an element of K and fixed point u is an element of K, define iteratively a sequence{x(n)), as follows: x(n+1) = alpha(n)u + beta(n)x(n) + gamma(n)Tx(n) , n >= 0, where {alpha(n)}, {beta(n)}, {gamma(n)} subset of (0, 1) satisfy proper conditions. We prove that {x(n)} converges strongly to P(F)u as n ->infinity no, where P-F is a unique sunny nonexpansive retraction of K onto F. Also we prove that the same conclusions still hold in in a uniformly convex Banach space with uniformly Geteaux differentiable norm or uniformly smooth Banach spaces.