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

Let EE be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure, KK a nonempty bounded closed convex subset of EE, {Ti}i=1N a finite family of asymptotically nonexpansive self-mappings on KK with common sequence {kn}n=1∞⊂[1,∞), {tn},{sn}{tn},{sn} be two sequences in (0, 1) such that sn+tn=1(n≥1)sn+tn=1(n≥1) and ff be a contraction on KK. Under suitable conditions on the sequences {sn},{tn}{sn},{tn}, we show the existence of a sequence {xn}{xn} satisfying the relation xn=(1−1kn)xn+snknf(xn)+tnknTrnnxn where n=lnN+rnn=lnN+rn for some unique integers ln≥0ln≥0 and 1≤rn≤N1≤rn≤N. Further we prove that {xn}{xn} converges strongly to a common fixed point of {Ti}i=1N, which solves some variational inequality, provided ‖xn−Tixn‖→0‖xn−Tixn‖→0 as n→∞n→∞ for i=1,2,…,Ni=1,2,…,N. As an application, we prove that the iterative process defined by z0∈K,zn+1=(1−1kn)zn+snknf(zn)+tnknTrnnzn, converges strongly to the same common fixed point of {Ti}i=1N.