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

Abstract EN

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.