Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space
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...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/189814 |
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| author | Hiroko Manaka |
| author_facet | Hiroko Manaka |
| author_sort | Hiroko Manaka |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-68e8cff7a36241d1aa89e460e3409526 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-68e8cff7a36241d1aa89e460e34095262025-02-03T01:31:07ZengWileyAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/189814189814Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach SpaceHiroko Manaka0Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, Tokiwadai, Hodogayaku, Yokohama 240-8501, JapanLet 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.http://dx.doi.org/10.1155/2010/189814 |
| spellingShingle | Hiroko Manaka Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space Abstract and Applied Analysis |
| title | Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space |
| title_full | Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space |
| title_fullStr | Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space |
| title_full_unstemmed | Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space |
| title_short | Convergence Theorems for a Maximal Monotone Operator and a 𝑉-Strongly Nonexpansive Mapping in a Banach Space |
| title_sort | convergence theorems for a maximal monotone operator and a 𝑉 strongly nonexpansive mapping in a banach space |
| url | http://dx.doi.org/10.1155/2010/189814 |
| work_keys_str_mv | AT hirokomanaka convergencetheoremsforamaximalmonotoneoperatorandavstronglynonexpansivemappinginabanachspace |