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 |
| Summary: | 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). |
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| ISSN: | 1687-1820 1687-1812 |