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On preservation of common fixed points and coincidences under a homotopy of mapping families of ordered sets. (English) Zbl 1412.54048
Some results concerning preservation of common fixed point properties for families of maps under order homotopy are stated. The following is the main result of this exposition.
Theorem. Let \((X,\preceq)\) be an ordered set; and \(\mathcal{F}=\{f_\alpha: \alpha\in A\}\), \(\tilde{\mathcal{F}}=\{\tilde{f}_\alpha: \alpha\in A\}\) be two families of selfmaps. Then, for each \(\alpha\in A\), let \(H_\alpha=\{H_{i,\alpha}: 1\le i\le n\}\) be a homotopy between \(f_\alpha\) and \(\tilde{f}_\alpha\), with \(f_\alpha=H_{0,\alpha}\preceq H_{1,\alpha}\succeq \dots\preceq H_{n,\alpha}=\tilde{f}_\alpha\), \(\alpha\in A\) (where \(n\) does not depend on \(\alpha\)). Suppose, in addition, that the following conditions hold
i)
the family \(\mathcal{H}_i=\{H_{i,\alpha}: \alpha\in A\}\) is concordantly isotone for \(i\in \{1,\dots,n\}\)
ii)
there exists a common fixed point \(x_0\) of \(\mathcal{F}=\mathcal{H}_0\)
iii)
for each even \(s\in \{1,\dots,n\}\) and every chain \(S\in \mathcal{C}_1(\mathcal{H}_s,\preceq)\), there exists an element \(\xi\in X\) such that \(H_{s,\alpha}(\xi)\preceq \xi\) and \(H_{s,\alpha}(\xi)\) is a lower bound of \(H_{s,\alpha}(S)\) for all \(\alpha\in A\)
iv)
for each odd \(s\in \{1,\dots,n\}\) and every chain \(S\in \mathcal{C}_1^*(\mathcal{H}_s,\preceq)\), there exists an element \(\xi\in X\) such that \(H_{s,\alpha}(\xi)\succeq \xi\) and \(H_{s,\alpha}(\xi)\) is an upper bound of \(H_{s,\alpha}(S)\) for all \(\alpha\in A\)
Then, there exists a fence \(x_0\preceq x_1\succeq \dots\preceq x_n\), such that
a)
for each odd \(k\in \{1,\dots,n\}\), the point \(x_k\in \mathrm{Comfix}(\mathcal{H}_k)\cap \mathcal{O}_X^*(x_{k-1})\) is a maximal element of the set \(\mathrm{Comfix}(\mathcal{H}_k)\cap \mathcal{O}_X^*(x_{k-1})\)
b)
for each even \(k\in \{1,\dots,n\}\), the point \(x_k\in \mathrm{Comfix}(\mathcal{H}_k)\cap \mathcal{O}_X^*(x_{k-1})\) is a minimal element of the set \(\mathrm{Comfix}(\mathcal{H}_k)\cap \mathcal{O}_X^*(x_{k-1})\).
Further aspects occasioned by these developments are also discussed.
Reviewers remark: The concepts of non-increasing and/or non-decreasing chain do not seem to be appropriate at this abstract level.
MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
06A06 Partial orders, general
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