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, a finite family of asymptotically nonexpansive self-mappings on K with common sequence , {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 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 , 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 , converges strongly to the same common fixed point of .