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

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  • Hiroko Manaka

Abstract

Let E be a smooth Banach space with a norm ‖ â‹… ‖ . Let 𠑉 ( ð ‘¥ , 𠑦 ) = ‖ ð ‘¥ ‖ 2 + ‖ 𠑦 ‖ 2 − 2 ⟨ ð ‘¥ , ð ½ ð ‘¦ ⟩ for any ð ‘¥ , 𠑦 ∈ ð ¸ , where ⟨ â‹… , â‹… ⟩ stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction 𠑉 ( â‹… , â‹… ) , a generalized nonexpansive mapping and a 𠑉 -strongly nonexpansive mapping are defined in ð ¸ . 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 𠑉 -strongly nonexpansive mapping.

Suggested Citation

  • Hiroko Manaka, 2010. "Convergence Theorems for a Maximal Monotone Operator and a 𠑉 -Strongly Nonexpansive Mapping in a Banach Space," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-20, September.
    • Handle: RePEc:hin:jnlaaa:189814
    DOI: 10.1155/2010/189814
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