## Abstract and Applied Analysis

### Convergence Theorems for a Maximal Monotone Operator and a $V$-Strongly Nonexpansive Mapping in a Banach Space

Hiroko Manaka

#### Abstract

Let E be a smooth Banach space with a norm $\Vert \cdot\,\!\Vert$. Let $V(x,y)={\Vert x\Vert}^{2}+{\Vert y\Vert }^{2}-2\langle x,Jy\rangle$ for any $x,y\in E$, where $\langle \cdot\,\!,\cdot\,\!\rangle$ stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction $V(\cdot\,\!,\cdot\,\!)$, a generalized nonexpansive mapping and a $V$-strongly nonexpansive mapping are defined in $E$. In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a $V$-strongly nonexpansive mapping.

#### Article information

Source
Abstr. Appl. Anal., Volume 2010 (2010), Article ID 189814, 20 pages.

Dates
First available in Project Euclid: 1 November 2010

https://projecteuclid.org/euclid.aaa/1288620762

Digital Object Identifier
doi:10.1155/2010/189814

Mathematical Reviews number (MathSciNet)
MR2680413

Zentralblatt MATH identifier
1368.47071

#### Citation

Manaka, Hiroko. Convergence Theorems for a Maximal Monotone Operator and a $V$ -Strongly Nonexpansive Mapping in a Banach Space. Abstr. Appl. Anal. 2010 (2010), Article ID 189814, 20 pages. doi:10.1155/2010/189814. https://projecteuclid.org/euclid.aaa/1288620762