Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

Let {tn}⊂(0,1) be such that tn → 1 as n → ∞, let α and β be two positive numbers such that α + β = 1, and let f be a contraction. If T be a continuous asymptotically pseudocontractive self‐mapping of a nonempty bounded closed convex subset K of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {tn}, we show the existence of a sequence {xn} n satisfying the relation xn = (1 − tn/kn)f(xn) + (tn/kn)Tnxn and prove that {xn} converges strongly to the fixed point of T, which solves some variational inequality provided T is uniformly asymptotically regular. As an application, if T be an asymptotically nonexpansive self‐mapping of a nonempty bounded closed convex subset K of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by z0 ∈ K, zn+1 = (1 − tn/kn)f(zn) + (αtn/kn)Tnzn + (βtn/kn)zn converges strongly to the fixed point of T.