A selection theorem in metric trees. (English) Zbl 1102.54022

A metric space \((M,d)\) is called metric tree if any two points \(x,y\in M\) are endpoints of a metric segment (isometric image of a real segment), denoted by \([ x,y]\), such that: (a) if \(x,y,z\in M\), then \([ x,y] \cap [ x,z] = [ x,w]\) for some \(w\in M\); (b) if \(x,y,z\in M\) and \([ x,y] \cap [ y,z] =\{ y\}\), then \([ x,y] \cup [ y,z] =[ x,z].\) A subset \(C\) of a metric tree is convex if for all \(x,y\in C\) we have \([ x,y] \subset C\).
The authors prove that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Next, it is shown that a set-valued mapping \(T^*:M\rightarrow 2^M\) of a metric tree \(M\) with nonempty bounded closed values has a selection \(T:M\rightarrow M\) for which \(d(T(x), T(y))\leq d_H(T^*(x),T^*(y))\) for all \(x,y\in M\). Finally, fixed point theorems are derived from this result.


54C65 Selections in general topology
47H04 Set-valued operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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