Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

Abstract

Let $K$ be a nonempty closed convex subset of a real Banach space $E$, let $S:K\to K$ be nonexpansive, and let $T:K\to K$ be Lipschitz strongly pseudocontractive mappings such that $p\in F\left(S\right)\cap F\left(T\right)=\left\{x\in K:Sx=Tx=x\right\}$ and $∥x-Sy∥\le ∥Sx-Sy∥ \text{and} ∥x-Ty∥\le ∥Tx-Ty∥$ for all $x, y\in K$. Let $\left\{{\beta }_n\right\}$ be a sequence in $\left[0, 1\right]$ satisfying (i) ${\sum }_{n=1}^{\infty }{\beta }_n=\infty$; (ii) ${\text{lim}}_{n\to \infty }{\beta }_n=0.$ For arbitrary $x_0\in K$, let $\left\{x_n\right\}$ be a sequence iteratively defined by $x_n=S{y}_n, y_n=\left(1-{\beta }_n\right)x_{n-1}+{\beta }_{n}T{x}_n, n\ge 1.$ Then the sequence $\left\{x_n\right\}$ converges strongly to a common fixed point $p$ of $S$ and $T$.