Fixed point theorems in CAT(0) spaces and â„Â-trees
We show that if U is a bounded open set in a complete CAT(0) space X, and if f:U¯→X is nonexpansive, then f always has a fixed point if there exists p∈U such that x∉[p,f(x)) for all x∈∂U. It is also shown that if K is a geodesically bounded closed convex subset of a compl...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2004-12-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://dx.doi.org/10.1155/S1687182004406081 |
| Summary: | We show that if U is a bounded open set in a complete CAT(0) space X, and if f:U¯→X is nonexpansive, then f always has a fixed point if there exists p∈U such that x∉[p,f(x)) for all x∈∂U. It is also shown that if K is a geodesically bounded closed convex subset of a complete â„Â-tree with int(K)≠∅, and if f:K→X is a continuous mapping for which x∉[p,f(x)) for some p∈int(K) and all x∈∂K, then f has a fixed point. It is also noted that a geodesically bounded complete â„Â-tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory. |
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| ISSN: | 1687-1820 1687-1812 |