An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

Abstract

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let $f:K\to K$ a contractive mapping and $T:K\to K$ be a uniformly continuous pseudocontractive mapping with $F(T)\ne \varnothing$. Let $\left\{{\lambda }_n\right\}\subset (0,1/2)$ be a sequence satisfying the following conditions: (i) ${\lim }_{n\to \infty }{\lambda }_n=0$; (ii) ${\displaystyle {\sum }_{n=0}^{\infty }{\lambda }_n=\infty }$. Define the sequence $\left\{x_n\right\}$ in K by $x_0\in K$, $x_{n+1}={\lambda }_{n}f(x_n)+(1-2{\lambda }_n)x_n+{\lambda }_{n}T{x}_n$, for all $n\ge 0$. Under some appropriate assumptions, we prove that the sequence $\left\{x_n\right\}$ converges strongly to a fixed point $p\in F(T)$ which is the unique solution of the following variational inequality: $\langle f(p)-p,j(z-p)\rangle \le 0$, for all $z\in F(T)$.