## Abstract and Applied Analysis

### 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 $p\in F(S)\cap F(T)=\{x\in K:Sx=Tx=x\}$ and ${\|}x-Sy{\|}\le {\|}Sx-Sy{\|} \text{and} {\|}x-Ty{\|}\le {\|}Tx-Ty{\|}$ for all $x, y\in K$. Let $\{{\beta }_{n}\}$ be a sequence in $[0, 1]$ 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 $\{{x}_{n}\}$ be a sequence iteratively defined by ${x}_{n}=S{y}_{n}, {y}_{n}=(1-{\beta }_{n}){x}_{n-1}+{\beta }_{n}T{x}_{n}, n\ge 1.$ Then the sequence $\{{x}_{n}\}$ converges strongly to a common fixed point $p$ of $S$ and $T$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 735673, 5 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606012

Digital Object Identifier
doi:10.1155/2014/735673

Mathematical Reviews number (MathSciNet)
MR3232859

Zentralblatt MATH identifier
07022976

#### Citation

Kang, Shin Min; Rafiq, Arif; Ali, Faisal; Kwun, Young Chel. Strong Convergence for Hybrid Implicit S -Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 735673, 5 pages. doi:10.1155/2014/735673. https://projecteuclid.org/euclid.aaa/1412606012

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