Fixed point theorems in <inline-formula><graphic file="1687-1812-2004-738084-i1.gif"/></inline-formula> spaces and <inline-formula><graphic file="1687-1812-2004-738084-i2.gif"/></inline-formula>-trees
<p/> <p>We show that if <inline-formula><graphic file="1687-1812-2004-738084-i3.gif"/></inline-formula> is a bounded open set in a complete <inline-formula><graphic file="1687-1812-2004-738084-i4.gif"/></inline-formula> space <in...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2004-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://www.fixedpointtheoryandapplications.com/content/2004/738084 |
| Summary: | <p/> <p>We show that if <inline-formula><graphic file="1687-1812-2004-738084-i3.gif"/></inline-formula> is a bounded open set in a complete <inline-formula><graphic file="1687-1812-2004-738084-i4.gif"/></inline-formula> space <inline-formula><graphic file="1687-1812-2004-738084-i5.gif"/></inline-formula>, and if <inline-formula><graphic file="1687-1812-2004-738084-i6.gif"/></inline-formula> is nonexpansive, then <inline-formula><graphic file="1687-1812-2004-738084-i7.gif"/></inline-formula> always has a fixed point if there exists <inline-formula><graphic file="1687-1812-2004-738084-i8.gif"/></inline-formula> such that <inline-formula><graphic file="1687-1812-2004-738084-i9.gif"/></inline-formula> for all <inline-formula><graphic file="1687-1812-2004-738084-i10.gif"/></inline-formula>. It is also shown that if <inline-formula><graphic file="1687-1812-2004-738084-i11.gif"/></inline-formula> is a geodesically bounded closed convex subset of a complete <inline-formula><graphic file="1687-1812-2004-738084-i12.gif"/></inline-formula>-tree with <inline-formula><graphic file="1687-1812-2004-738084-i13.gif"/></inline-formula>, and if <inline-formula><graphic file="1687-1812-2004-738084-i14.gif"/></inline-formula> is a continuous mapping for which <inline-formula><graphic file="1687-1812-2004-738084-i15.gif"/></inline-formula> for some <inline-formula><graphic file="1687-1812-2004-738084-i16.gif"/></inline-formula> and all <inline-formula><graphic file="1687-1812-2004-738084-i17.gif"/></inline-formula>, then <inline-formula><graphic file="1687-1812-2004-738084-i18.gif"/></inline-formula> has a fixed point. It is also noted that a geodesically bounded complete <inline-formula><graphic file="1687-1812-2004-738084-i19.gif"/></inline-formula>-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.</p> |
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| ISSN: | 1687-1820 1687-1812 |