Abstract
We show that if is a bounded open set in a complete
space
, and if
is nonexpansive, then
always has a fixed point if there exists
such that
for all
. It is also shown that if
is a geodesically bounded closed convex subset of a complete
-tree with
, and if
is a continuous mapping for which
for some
and all
, then
has a fixed point. It is also noted that a geodesically bounded complete
-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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Kirk, W. Fixed point theorems in spaces and
-trees.
Fixed Point Theory Appl 2004, 738084 (2004). https://doi.org/10.1155/S1687182004406081
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DOI: https://doi.org/10.1155/S1687182004406081