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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.

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