Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces

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  , T:K⟶KT:K⟶K an asymptotically nonexpansive mapping with sequence {kn}n⊂[1,∞){kn}n⊂[1,∞). Let {tn}⊂(0,1){tn}⊂(0,1) be such that tn→1tn→1 as n→∞n→∞ and f be a contraction on K  . Under suitable conditions on the sequence {tn}{tn}, we show the existence of a sequence {xn}n{xn}n satisfying the relation xn=(1-tnkn)f(xn)+tnknTnxn, and prove that {xn}n{xn}n converges strongly to the fixed point of T  , which solves some variational inequality, provided ∥xn-Txn∥→0∥xn-Txn∥→0 as n→∞n→∞. As an application, we prove that the iterative process defined by z0∈K,zn+1≔(1-tnkn)f(zn)+tnknTnzn,n∈N, converges strongly to the same fixed point of T.