Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces

Let E be a uniformly convex Banach space having a uniformly Gateaux differentiable norm, D a nonempty closed convex subset of E, and T : D -> K(E) a nonself multimap such that F(T) not equal phi and P-T is nonexpansive, where F(T) is the fixed point set of T, K(E) is the family of nonempty compact subsets of E and P-T(x) = {u(x) is an element of Tx : parallel to x - u(x)parallel to = d(x, Tx)}. Suppose that D is a nonexpansive retract of E and that for each v is an element of D and t is an element of (0, 1), the contraction S-t defined by S(t)x = tP(T)x + (1 - t)v has a fixed point x(t) is an element of D. Let {alpha(n)}, {beta(n)} and {gamma(n)} be three real sequences in (0, 1) satisfying approximate conditions. Then for fixed u is an element of D and arbitrary x(0) is an element of D, the sequence {x(n)} generated by x(n) is an element of alpha(n)u + beta(n)x(n-1) + gamma P-n(T)(x(n)), for all n >= 0, converges strongly to a fixed point of T. (C) 2009 Elsevier Ltd. All rights reserved.