Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities
Let {𝑡𝑛}⊂(0,1) be such that 𝑡𝑛→1 as 𝑛→∞, let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differen...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/453452 |
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| author | Xionghua Wu Yeong-Cheng Liou Zhitao Wu Pei-Xia Yang |
| author_facet | Xionghua Wu Yeong-Cheng Liou Zhitao Wu Pei-Xia Yang |
| author_sort | Xionghua Wu |
| collection | DOAJ |
| description | Let {𝑡𝑛}⊂(0,1) be such that 𝑡𝑛→1 as 𝑛→∞, let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {𝑡𝑛}, we show the existence of a sequence {𝑥𝑛}𝑛 satisfying the relation 𝑥𝑛=(1−𝑡𝑛/𝑘𝑛)𝑓(𝑥𝑛)+(𝑡𝑛/𝑘𝑛)𝑇𝑛𝑥𝑛 and prove that {𝑥𝑛} converges strongly to the fixed point of 𝑇, which solves some variational inequality provided 𝑇 is uniformly asymptotically regular. As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧0∈𝐾,𝑧𝑛+1=(1−𝑡𝑛/𝑘𝑛)𝑓(𝑧𝑛)+(𝛼𝑡𝑛/𝑘𝑛)𝑇𝑛𝑧𝑛+(𝛽𝑡𝑛/𝑘𝑛)𝑧𝑛 converges strongly to the fixed point of 𝑇. |
| format | Article |
| id | doaj-art-4b0ee75a7a9448a4858fd3930f42ddea |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-4b0ee75a7a9448a4858fd3930f42ddea2025-02-03T05:46:56ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/453452453452Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational InequalitiesXionghua Wu0Yeong-Cheng Liou1Zhitao Wu2Pei-Xia Yang3Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, ChinaDepartment of Information Management, Cheng Shiu University, Kaohsiung 833, TaiwanDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300387, ChinaDepartment of Mathematics, Tianjin Polytechnic University, Tianjin 300387, ChinaLet {𝑡𝑛}⊂(0,1) be such that 𝑡𝑛→1 as 𝑛→∞, let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {𝑡𝑛}, we show the existence of a sequence {𝑥𝑛}𝑛 satisfying the relation 𝑥𝑛=(1−𝑡𝑛/𝑘𝑛)𝑓(𝑥𝑛)+(𝑡𝑛/𝑘𝑛)𝑇𝑛𝑥𝑛 and prove that {𝑥𝑛} converges strongly to the fixed point of 𝑇, which solves some variational inequality provided 𝑇 is uniformly asymptotically regular. As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧0∈𝐾,𝑧𝑛+1=(1−𝑡𝑛/𝑘𝑛)𝑓(𝑧𝑛)+(𝛼𝑡𝑛/𝑘𝑛)𝑇𝑛𝑧𝑛+(𝛽𝑡𝑛/𝑘𝑛)𝑧𝑛 converges strongly to the fixed point of 𝑇.http://dx.doi.org/10.1155/2012/453452 |
| spellingShingle | Xionghua Wu Yeong-Cheng Liou Zhitao Wu Pei-Xia Yang Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities Abstract and Applied Analysis |
| title | Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities |
| title_full | Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities |
| title_fullStr | Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities |
| title_full_unstemmed | Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities |
| title_short | Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities |
| title_sort | viscosity methods of asymptotically pseudocontractive and asymptotically nonexpansive mappings for variational inequalities |
| url | http://dx.doi.org/10.1155/2012/453452 |
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