Fixed point theorems in -trees with applications to graph theory

It is proved that a commutative family of nonexpansive mappings of a complete R -tree X into itself always has a nonempty common fixed point set if X does not contain a geodesic ray. As a consequence of this, we show that any commuting family of edge preserving mappings of a connected reflexive graph G that contains no cycles or infinite paths always has at least one common fixed edge. This approach provides a new proof of the classical fixed edge theorem of Nowakowski and Rival. Several related results are also obtained.