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On fixed point theorems of Leray–Schauder type. (English) Zbl 1136.54026
A metric space \((X,d)\) is called hyperconvex if every family \((\overline{B}(x_i;r_i))_{i\in I}\) of closed balls has nonempty intersection whenever \(d(x_i,x_j)\leq r_i+r_j\) for \(i,j\in I\). A pair \((E,e)\) is called a hyperconvex hull of \((X,d)\) if \(E\) is hyperconvex, \(e:X\to E\) is an isometric embedding, and there is no hyperconvex subset of \(E\) which properly includes \(e(X)\). It is known that hyperconvex hulls always exist. Denote the set of hyperconvex hulls of \(A\subset X\) by \(\mathcal{H}(A)\). The authors prove the following result. Let \(X\) be a hyperconvex metric space, let \(\Omega\subset X\) be open and nonempty, and let \(H:[0,1]\times X\to X\) be a homotopy. Assume that \(H(0,\cdot)\) has a subadditive modulus of continuity and that \(H(\{0\}\times\overline{\Omega)})\) is contained in a compact hyperconvex subset \(V\) of \(\overline{\Omega}\) and that there are no fixed points of \(H(\lambda,\cdot)\) on \(\partial\Omega\) for each \(\lambda\in[0,1]\). Assume further that, for each \(C\subset\Omega\) such that \(C=\Omega\cap P\) for some \(P\in\mathcal{H}(H([0,1]\times C)\cup V)\), we have that \(C\) is relatively compact. The conclusion is that \(H(1,\cdot)\) has a fixed point in \(\bar{\Omega}\). Finally, the authors point out a minor slip in an article by N. Aronszajn and P. Panitchpakdi [Pac. J. Math. 6, 405–439 (1956; Zbl 0074.17802)] which, however, was dealt with in a corrigendum [ibid. 7, 1729 (1957; Zbl 0074.17802)].

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54E35 Metric spaces, metrizability
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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