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

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  • Hiroko Manaka

Abstract

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.

Suggested Citation

  • Hiroko Manaka, 2010. "Convergence Theorems for a Maximal Monotone Operator and a V‐Strongly Nonexpansive Mapping in a Banach Space," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    • Handle: RePEc:wly:jnlaaa:v:2010:y:2010:i:1:n:189814
    DOI: 10.1155/2010/189814
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    References listed on IDEAS

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    1. Takanori Ibaraki & Yasunori Kimura & Wataru Takahashi, 2003. "Convergence theorems for generalized projections and maximal monotone operators in Banach spaces," Abstract and Applied Analysis, Hindawi, vol. 2003, pages 1-9, January.
    2. Takanori Ibaraki & Yasunori Kimura & Wataru Takahashi, 2003. "Convergence theorems for generalized projections and maximal monotone operators in Banach spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2003(10), pages 621-629.
    3. Dan Butnariu & Elena Resmerita, 2006. "Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2006(1).
    4. Dan Butnariu & Elena Resmerita, 2006. "Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces," Abstract and Applied Analysis, Hindawi, vol. 2006, pages 1-39, February.
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