Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space - Hiroko Manaka (Abstract and Applied Analysis)
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.