## 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.

### MSC:

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

### Keywords:

metric trees; selection theorems; hyperconvex spaces
Full Text: