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A fixed-point theorem for tree-like continua. (English) Zbl 0786.54043

D. P. Bellamy [Houston J. Math. 6, 1-13 (1980; Zbl 0447.54039)] constructed a tree-like continuum without the fixed point property. P. Minc [Topology Appl. 46, No. 2, 99-106 (1992; Zbl 0770.54043)] used Bellamy’s example to construct a tree-like continuum that admits fixed point free mappings arbitrarily close to the identity. The author shows that it is impossible to define a homotopy on a tree-like continuum that is made up of such mappings: given a tree-like continuum \(M\) and a mapping \(H\) of \(M\times [0,1]\) onto \(M\) such that \(H(x,0)= x\) for each point \(x\) of \(M\), if \(f_ t: M\to M\) is defined by \(f_ t(x)= H(x,t)\), then \(f_ t\) has a fixed point for some \(t>0\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54F15 Continua and generalizations
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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