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 Gâteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E, {Ti}i=1 a finite family of asymptotically nonexpansive self-mappings on K with common sequence {kn}∞n=1 ⊂ [1,∞), {tn}, {sn} be two sequences in (0, 1) such that sn + tn = 1 (n ≥ 1) and f be a contraction on K . Under suitable conditions on the sequences {sn}, {tn}, we show the existence of a sequence {xn} satisfying the relation xn = (1− 1 kn )xn+ sn kn f (xn)+ tn kn T n rnxn where n = lnN + rn for some unique integers ln ≥ 0 and 1 ≤ rn ≤ N . Further we prove that {xn} converges strongly to a common fixed point of {Ti}i=1, which solves some variational inequality, provided ‖xn − Tixn‖ → 0 as n → ∞ for i = 1, 2, . . . ,N . As an application, we prove that the iterative process defined by z0 ∈ K , zn+1 = (1 − 1 kn )zn + sn kn f (zn)+ tn kn T n rnzn, converges strongly to the same common fixed point of {Ti} N i=1. © 2008 Elsevier Ltd. All rights reserved.