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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.
##### Keywords:
hyperconvex space; fixed point; nonexpansive mapping
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