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 |
| _version_ | 1818384358656966656 |
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| author | W. A. Kirk |
| author_facet | W. A. Kirk |
| author_sort | W. A. Kirk |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-14T03:21:00Z |
| format | Article |
| id | doaj.art-9cb6c4daa6d24de19138a38c1ddcadd2 |
| institution | Directory Open Access Journal |
| issn | 1687-1820 1687-1812 |
| language | English |
| last_indexed | 2024-12-14T03:21:00Z |
| publishDate | 2010-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Applications |
| spelling | doaj.art-9cb6c4daa6d24de19138a38c1ddcadd22022-12-21T23:19:02ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122010-01-01201010.1155/2010/393470A Note on Geodesically Bounded ℝ-TreesW. A. KirkIt 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.http://dx.doi.org/10.1155/2010/393470 |
| spellingShingle | W. A. Kirk A Note on Geodesically Bounded ℝ-Trees Fixed Point Theory and Applications |
| title | A Note on Geodesically Bounded ℝ-Trees |
| title_full | A Note on Geodesically Bounded ℝ-Trees |
| title_fullStr | A Note on Geodesically Bounded ℝ-Trees |
| title_full_unstemmed | A Note on Geodesically Bounded ℝ-Trees |
| title_short | A Note on Geodesically Bounded ℝ-Trees |
| title_sort | note on geodesically bounded x0211d trees |
| url | http://dx.doi.org/10.1155/2010/393470 |
| work_keys_str_mv | AT wakirk anoteongeodesicallyboundedx0211dtrees AT wakirk noteongeodesicallyboundedx0211dtrees |