## Abstract and Applied Analysis

### Iterative Methods for Pseudocontractive Mappings in Banach Spaces

Jong Soo Jung

#### 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 $k\in (0,1)$. Let $\{{\alpha }_{n}\}$ and $\{{\beta }_{n}\}$ be sequences in $(0,1)$ 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 $A$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 643602, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393444403

Digital Object Identifier
doi:10.1155/2013/643602

Mathematical Reviews number (MathSciNet)
MR3039129

Zentralblatt MATH identifier
06209386

#### Citation

Jung, Jong Soo. Iterative Methods for Pseudocontractive Mappings in Banach Spaces. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 643602, 7 pages. doi:10.1155/2013/643602. https://projecteuclid.org/euclid.aaa/1393444403