Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption

Let E be a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E, {T-i}(i=1)(N), a finite family of asymptotically nonexpansive self-mappings on K with common sequence {kn)(n=1)(infinity) subset of [1. infinity), {t(n)}, (s(n)} be two sequences in (0, 1) such that s(n) + t(n) = 1 (n >= 1) and f be a contraction on K. Under suitable conditions on the sequences {s(n)}, {t(n)}, we show the existence of a sequence {x(n)} satisfying the relation x(n) = (1-1/k(n))x(n) + s(n)/k(n) f(x(n)) + t(n)/k(n)T(rn)(n)x(n) where n = l(n)N + r(n) for some unique integers l(n) >= 0 and 1 <= r(n) <= N. Further we prove that {x(n)} converges strongly to a common fixed point of {T-i}(i=1)(N), which solves some variational inequality, provided parallel to x(n) - T(i)x(n)parallel to -> 0 as n -> infinity for i = 1, 2, N. As an application, we prove that the iterative process defined by z(0) is an element of K, z(n+1) = (1 - 1/k(n))z(n) + s(n)/k(n)f(z(n)) + t(n)/k(n)T(rn)(n)z(n), converges strongly to the same common fixed point of {T-i}(i=1)(N). (C) 2008 Elsevier Ltd. All rights reserved.