Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let T-1, T-2,..., T-m : K -> E be asymptotically nonexpansive mappings of K into E with sequences (respectively) {k(in)}(n=1)(infinity) satisfying k(in) -> 1 as n -> infinity, i = 1, 2,..., m, and Sigma(infinity)(n=1) (k(in) - 1) < infinity. Let {alpha(in)}(n=1)(infinity) be a sequence in [epsilon, 1 - epsilon], epsilon is an element of (0, 1), for each i is an element of {1, 2,..., m} (respectively). Let {x(n)} be a sequence generated for m >= 2 by x1 is an element of K, x(n+1) = P[(1 - alpha(1n))x(n) + alpha T-1n(1)(PT1)(n-1)y(n+m-2)], y(n+m-2) = P[(1 - alpha(2n))x(n) + alpha T-2n(2)(PT2)(n-1)y(n+m-3)], . . . y(n) = P[(1 - alpha(mn))x(n) + alpha T-mn(m)(PTm)(n-1)x(n)], Let boolean AND(m)(i=1) F(T-i) not equal 0. Stron and weak convergence of the sequence {x(n)} to a common fixed point of the 9 family {Ti}(i=1)(m), are proved. Furthermore, if T-1, T-2,...,T-m are nonexpansive mappings and the dual E* of E satisfies the Kadec-Klee property, weak convergence theorem is also proved. (c) 2006 Elsevier Inc. All rights reserved.