Viscosity approximation methods for asymptotically nonexpansive mappings

Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, K a nonempty closed convex subset of E  , T:K→KT:K→K an asymptotically nonexpansive mapping with sequence {kn}⊂[1,∞),limn→∞kn=1{kn}⊂[1,∞),limn→∞kn=1. Let {αn}⊂(0,1){αn}⊂(0,1) be such that limn→∞αn=0,limn→∞kn-1αn=0 and f be a contraction on K  . Under suitable conditions, we show the existence of a sequence {zn}{zn} satisfying the relation zn=αnf(zn)+(1-αn)Tnznzn=αnf(zn)+(1-αn)Tnzn, and prove that {zn}{zn} converges strongly to the fixed point of T, which solves some variational inequality, provided T   is asymptotically regular. As an application, we prove that the iterative process defined by x0∈Kx0∈K, xn+1≔αnf(xn)+βnxn+γnTnxnxn+1≔αnf(xn)+βnxn+γnTnxn, converges strongly to the same fixed point of T.