Fixed point theorems in CAT(0) spaces and Ã¢Â„Â-trees - W. A. Kirk (Fixed Point Theory and Applications, 2004)
We show that if U is a bounded open set in a complete CAT(0) space X, and if f:UÃ‚Â¯Ã¢Â†Â’X is nonexpansive, then f always has a fixed point if there exists pÃ¢ÂˆÂˆU such that xÃ¢ÂˆÂ‰[p,f(x)) for all xÃ¢ÂˆÂˆÃ¢ÂˆÂ‚U. It is also shown that if K is a geodesically bounded closed convex subset of a complete Ã¢Â„Â-tree with int(K)Ã¢Â‰Â Ã¢ÂˆÂ…, and if f:KÃ¢Â†Â’X is a continuous mapping for which xÃ¢ÂˆÂ‰[p,f(x)) for some pÃ¢ÂˆÂˆint(K) and all xÃ¢ÂˆÂˆÃ¢ÂˆÂ‚K, then f has a fixed point. It is also noted that a geodesically bounded complete Ã¢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.