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

Abstract

Let E be a smooth Banach space with a norm $\Vert \cdot \Vert$. Let $V\left(x,y\right)={\Vert x\Vert }^2+{\Vert y\Vert }^2-2\langle x,Jy\rangle$ 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 $V\left(\cdot ,\cdot \right)$, 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.