Abstract and Applied Analysis
Volume 2010 (2010), Article ID 189814, 20 pages
doi:10.1155/2010/189814
Research Article

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

Hiroko Manaka

Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama 240-8501, Japan

Received 25 November 2009; Revised 31 March 2010; Accepted 16 July 2010

Academic Editor: S. Reich

Copyright © 2010 Hiroko Manaka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let E be a smooth Banach space with a norm ‖⋅‖. Let V(x,y)=‖x‖2+‖y‖2−2〈x,Jy〉 for any x,y∈E, where 〈⋅,⋅〉 stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction V(⋅,⋅), 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.