Convergence Theorems for a Maximal Monotone Operator and a V‐Strongly Nonexpansive Mapping in a Banach Space
Hiroko Manaka · Abstract and Applied Analysis · 2010
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.