A Note on Geodesically Bounded ℝ-Trees
It is proved that a complete geodesically bounded R-tree is the closed convex hull of the set of its extreme points. It is also noted that if X is a closed convex geodesically bounded subset of a complete R-tree Y, and if a nonexpansive mapping T:X→Y satisfies inf⁡{d(x,T(x)):x...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2010-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/2010/393470 |
| Summary: | It is proved that a complete geodesically bounded R-tree is the closed convex hull of the set of its extreme points. It is also noted that if X is a closed convex geodesically bounded subset of a complete R-tree Y, and if a nonexpansive mapping T:X→Y satisfies inf⁡{d(x,T(x)):x∈X}=0, then T has a fixed point. The latter result fails if T is only continuous. |
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| ISSN: | 1687-1820 1687-1812 |