Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces
We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn:n∈ℕ} be an infinite family of commuting nonexpansive mapp...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2006-06-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/FPTA/2006/59692 |
| _version_ | 1818727295391629312 |
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| author | Tomonari Suzuki |
| author_facet | Tomonari Suzuki |
| author_sort | Tomonari Suzuki |
| collection | DOAJ |
| description | We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn:n∈ℕ} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0,1/2) satisfying limntn=limnαn/tnℓ=0 for ℓ∈ℕ. Fix u∈C and define a sequence {un} in C by un=(1−αn)((1−∑k=1ntnk)T1un+∑k=1ntnkTk+1un)+αnu for n∈ℕ. Then {un} converges strongly to Pu, where P is the unique sunny nonexpansive retraction from C onto ∩n=1∞F(Tn). |
| first_indexed | 2024-12-17T22:11:50Z |
| format | Article |
| id | doaj.art-a028e1ca7c0445d3b0b30d3372a4c537 |
| institution | Directory Open Access Journal |
| issn | 1687-1820 1687-1812 |
| language | English |
| last_indexed | 2024-12-17T22:11:50Z |
| publishDate | 2006-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Applications |
| spelling | doaj.art-a028e1ca7c0445d3b0b30d3372a4c5372022-12-21T21:30:43ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122006-06-01200610.1155/FPTA/2006/59692Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spacesTomonari SuzukiWe prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn:n∈ℕ} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0,1/2) satisfying limntn=limnαn/tnℓ=0 for ℓ∈ℕ. Fix u∈C and define a sequence {un} in C by un=(1−αn)((1−∑k=1ntnk)T1un+∑k=1ntnkTk+1un)+αnu for n∈ℕ. Then {un} converges strongly to Pu, where P is the unique sunny nonexpansive retraction from C onto ∩n=1∞F(Tn).http://dx.doi.org/10.1155/FPTA/2006/59692 |
| spellingShingle | Tomonari Suzuki Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces Fixed Point Theory and Applications |
| title | Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces |
| title_full | Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces |
| title_fullStr | Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces |
| title_full_unstemmed | Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces |
| title_short | Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces |
| title_sort | browder s type strong convergence theorems for infinite families of nonexpansive mappings in banach spaces |
| url | http://dx.doi.org/10.1155/FPTA/2006/59692 |
| work_keys_str_mv | AT tomonarisuzuki browderstypestrongconvergencetheoremsforinfinitefamiliesofnonexpansivemappingsinbanachspaces |