## Abstract and Applied Analysis

### Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces

#### Abstract

Let $X$ be a real reflexive Banach space with a weakly continuous duality mapping ${J}_{\phi }$. Let $C$ be a nonempty weakly closed star-shaped (with respect to $u$) subset of $X$. Let $\scr F$ = $\{T(t):t\in [0,+\infty )\}$ be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of $C$, which is uniformly continuous at zero. We will show that the implicit iteration scheme: ${y}_{n}={\alpha }_{n}u+(1-{\alpha }_{n})T({t}_{n}){y}_{n}$, for all $n\in \Bbb N$, converges strongly to a common fixed point of the semigroup $\scr F$ for some suitably chosen parameters $\{{\alpha }_{n}\}$ and $\{{t}_{n}\}$. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 202095, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393442740

Digital Object Identifier
doi:10.1155/2013/202095

Mathematical Reviews number (MathSciNet)
MR3049365

Zentralblatt MATH identifier
06209208

#### Citation

Sahu, D. R.; Wong, Ngai-Ching; Yao, Jen-Chih. Strong Convergence Theorems for Semigroups of Asymptotically Nonexpansive Mappings in Banach Spaces. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 202095, 8 pages. doi:10.1155/2013/202095. https://projecteuclid.org/euclid.aaa/1393442740