Iterative Methods for Pseudocontractive Mappings in Banach Spaces

Abstract

Let $E$ a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let $C$ be a nonempty closed convex subset of $E$, $T:C\to C$ a continuous pseudocontractive mapping with $F(T)\ne \varnothing$, and $A:C\to C$ a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant $k\in (\mathrm{0,1})$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be sequences in $(\mathrm{0,1})$ satisfying suitable conditions and for arbitrary initial value $x_0\in C$, let the sequence $\{x_n\}$ be generated by $x_n={\alpha }_{n}A{x}_n+{\beta }_n{x}_{n-1}+(1-{\alpha }_n-{\beta }_n)T{x}_n,n\ge 1.$ If either every weakly compact convex subset of $E$ has the fixed point property for nonexpansive mappings or $E$ is strictly convex, then $\{x_n\}$ converges strongly to a fixed point of $T$, which solves a certain variational inequality related to $A$.