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

Abstract

Let $\{t_n\}\subset (\mathrm{0,1})$ be such that $t_n\to 1$ as $n\to \infty$, let $\alpha$ and $\beta$ be two positive numbers such that $\alpha +\beta =1$, 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 $\{t_n\}$, we show the existence of a sequence $\{x_n{\}}_n$ satisfying the relation $x_n=(1-t_n/k_n)f(x_n)+{(t}_n/k_n)T^n{x}_n$ and prove that $\{x_n\}$ 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 $z_0\in K,\hspace{0.17em}{z}_{n+1}=(1-t_n/k_n)f(z_n)+(\alpha t_n/k_n)T^n{z}_n+(\beta t_n/k_n)z_n$ converges strongly to the fixed point of $T$.