Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex subset of a real uniformly convex Banach space E , which is also a nonexpansive retract of E with nonexpansive retraction P . Let {Ti : i ∈ I } be N nonself asymptotically nonexpansive mappings from K to E such that F = {x ∈ K : Ti x = x, i ∈ I } 6= φ, where I = {1, 2, . . . , N }. From arbitrary x0 ∈ K , {xn} is defined by xn = P((1− αn)xn−1 + αnTn(PTn) xn−1), n ≥ 1 where n = (m−1)N+i , Tn = Tn(mod N ) = Ti , i ∈ I , the mod N function takes values in I , {αn} is a real sequence in [δ, 1−δ] for some δ ∈ (0, 1). Some strong and weak convergence theorems of {xn} to some q ∈ F are obtained under some suitable conditions in real uniformly convex Banach spaces. c © 2007 Elsevier Ltd. All rights reserved.