Hagopian, Charles L. A fixed-point theorem for tree-like continua. (English) Zbl 0786.54043 Topol. Proc. 16, 57-62 (1991). 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 \(t>0\). Reviewer: J.J.Charatonik (Wrocław) Cited in 2 Documents MSC: 54H25 Fixed-point and coincidence theorems (topological aspects) 54F15 Continua and generalizations 54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites Keywords:tree-like continuum without the fixed point property; homotopy Citations:Zbl 0447.54039; Zbl 0770.54043 PDFBibTeX XMLCite \textit{C. L. Hagopian}, Topol. Proc. 16, 57--62 (1991; Zbl 0786.54043)