Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings
Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T:C→C is a Lipschitzian mapping, and x*∈C is a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by xn+1=(1-αn)xn+αnTxn,n≥0, where {αn} is a sequence in [0, 1], ∑...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/327878 |
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| _version_ | 1832550102699343872 |
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| author | Chang-He Xiang Jiang-Hua Zhang Zhe Chen |
| author_facet | Chang-He Xiang Jiang-Hua Zhang Zhe Chen |
| author_sort | Chang-He Xiang |
| collection | DOAJ |
| description | Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T:C→C is a Lipschitzian mapping, and x*∈C is a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by xn+1=(1-αn)xn+αnTxn,n≥0, where {αn} is a sequence in [0, 1], ∑n=0∞αn2<∞, ∑n=0∞αn=∞. We prove that the sequence {xn} strongly converges to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that limsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0. |
| format | Article |
| id | doaj-art-d1df681da22c4aca844beb3a51fc328e |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-d1df681da22c4aca844beb3a51fc328e2025-02-03T06:07:46ZengWileyJournal of Applied Mathematics1110-757X1687-00422012-01-01201210.1155/2012/327878327878Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian MappingsChang-He Xiang0Jiang-Hua Zhang1Zhe Chen2College of Mathematics, Chongqing Normal University, Chongqing 400047, ChinaSchool of Management, Shandong University, Shandong Jinan 250100, ChinaCollege of Mathematics, Chongqing Normal University, Chongqing 400047, ChinaSuppose that E is a real normed linear space, C is a nonempty convex subset of E, T:C→C is a Lipschitzian mapping, and x*∈C is a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by xn+1=(1-αn)xn+αnTxn,n≥0, where {αn} is a sequence in [0, 1], ∑n=0∞αn2<∞, ∑n=0∞αn=∞. We prove that the sequence {xn} strongly converges to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that limsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.http://dx.doi.org/10.1155/2012/327878 |
| spellingShingle | Chang-He Xiang Jiang-Hua Zhang Zhe Chen Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings Journal of Applied Mathematics |
| title | Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings |
| title_full | Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings |
| title_fullStr | Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings |
| title_full_unstemmed | Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings |
| title_short | Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings |
| title_sort | necessary and sufficient condition for mann iteration converges to a fixed point of lipschitzian mappings |
| url | http://dx.doi.org/10.1155/2012/327878 |
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