A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings
Let H be a real Hilbert space and C⊂H a closed convex subset. Let T:C→C be a nonexpansive mapping with the nonempty set of fixed points Fix(T). Kim and Xu (2005) introduced a modified Mann iteration x0=x∈C, yn=αnxn+(1−αn)Txn, xn+1=βnu+(1−βn)yn, where u∈C is an arbitrary (but fixed) element, and {αn}...
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/768595 |
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| author | Songnian He Wenlong Zhu |
| author_facet | Songnian He Wenlong Zhu |
| author_sort | Songnian He |
| collection | DOAJ |
| description | Let H be a real Hilbert space and C⊂H a closed convex subset. Let T:C→C be a nonexpansive mapping with the nonempty set of fixed points Fix(T). Kim and Xu (2005) introduced a modified Mann iteration x0=x∈C, yn=αnxn+(1−αn)Txn, xn+1=βnu+(1−βn)yn, where u∈C is an arbitrary (but fixed) element, and {αn} and {βn} are two sequences in (0,1). In the case where 0∈C, the minimum-norm fixed point of T can be obtained by taking u=0. But in the case where 0∉C, this iteration process becomes invalid because xn may not belong to C. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of T and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection PC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others. |
| format | Article |
| id | doaj-art-503b339c31b9498fb1cf5a737d579d96 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
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| series | Abstract and Applied Analysis |
| spelling | doaj-art-503b339c31b9498fb1cf5a737d579d962025-02-03T06:42:20ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/768595768595A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive MappingsSongnian He0Wenlong Zhu1College of Science, Civil Aviation University of China, Tianjin 300300, ChinaCollege of Science, Civil Aviation University of China, Tianjin 300300, ChinaLet H be a real Hilbert space and C⊂H a closed convex subset. Let T:C→C be a nonexpansive mapping with the nonempty set of fixed points Fix(T). Kim and Xu (2005) introduced a modified Mann iteration x0=x∈C, yn=αnxn+(1−αn)Txn, xn+1=βnu+(1−βn)yn, where u∈C is an arbitrary (but fixed) element, and {αn} and {βn} are two sequences in (0,1). In the case where 0∈C, the minimum-norm fixed point of T can be obtained by taking u=0. But in the case where 0∉C, this iteration process becomes invalid because xn may not belong to C. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of T and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection PC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.http://dx.doi.org/10.1155/2013/768595 |
| spellingShingle | Songnian He Wenlong Zhu A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings Abstract and Applied Analysis |
| title | A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings |
| title_full | A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings |
| title_fullStr | A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings |
| title_full_unstemmed | A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings |
| title_short | A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings |
| title_sort | modified mann iteration by boundary point method for finding minimum norm fixed point of nonexpansive mappings |
| url | http://dx.doi.org/10.1155/2013/768595 |
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