Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces
Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C→E a non-expansive nonself-mapping with F(T)≠∅. In this paper, we st...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2007-10-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/2007/59262 |
| _version_ | 1831717998137704448 |
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| author | Rabian Wangkeeree |
| author_facet | Rabian Wangkeeree |
| author_sort | Rabian Wangkeeree |
| collection | DOAJ |
| description | Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C→E a non-expansive nonself-mapping with F(T)≠∅. In this paper, we study the strong convergence of two sequences generated by xn+1=αnx+(1−αn)(1/n+1)∑j=0n(PT)jxn and yn+1=(1/n+1)∑j=0nP(αny+(1−αn)(TP)jyn) for all n≥0, where x,x0,y,y0∈C, {αn} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n→∞, where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others. |
| first_indexed | 2024-12-21T01:09:02Z |
| format | Article |
| id | doaj.art-81d6b02735084c6481a1b28a2b774483 |
| institution | Directory Open Access Journal |
| issn | 1687-1820 1687-1812 |
| language | English |
| last_indexed | 2024-12-21T01:09:02Z |
| publishDate | 2007-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Applications |
| spelling | doaj.art-81d6b02735084c6481a1b28a2b7744832022-12-21T19:20:58ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122007-10-01200710.1155/2007/59262Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach SpacesRabian WangkeereeLet E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C→E a non-expansive nonself-mapping with F(T)≠∅. In this paper, we study the strong convergence of two sequences generated by xn+1=αnx+(1−αn)(1/n+1)∑j=0n(PT)jxn and yn+1=(1/n+1)∑j=0nP(αny+(1−αn)(TP)jyn) for all n≥0, where x,x0,y,y0∈C, {αn} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n→∞, where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.http://dx.doi.org/10.1155/2007/59262 |
| spellingShingle | Rabian Wangkeeree Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces Fixed Point Theory and Applications |
| title | Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces |
| title_full | Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces |
| title_fullStr | Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces |
| title_full_unstemmed | Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces |
| title_short | Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces |
| title_sort | strong convergence of cesafa ro mean iterations for nonexpansive nonself mappings in banach spaces |
| url | http://dx.doi.org/10.1155/2007/59262 |
| work_keys_str_mv | AT rabianwangkeeree strongconvergenceofcesaƒaromeaniterationsfornonexpansivenonselfmappingsinbanachspaces |