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Fix-finite approximation property in normed vector spaces. (English) Zbl 1054.46006

A subset A of a normed space \(X\) satisfies the Schauder condition if for any relatively compact subset \(K\) of \(A\) and for every \(\varepsilon > 0\) there exists a finite cover \(\{B(x_{i},\eta_{x_{i}})\;; x_{i} \in A, 0< \eta_{x_{i}}<\epsilon, i=1,2,\dots ,n\}\) of \(K\) such that if \(\bigcap_{j=1}^{p}B(x_{i_{j}},\eta_{x_{i_{j}}}) \cap K \neq \emptyset\) then conv\(\{x_{i_{j}},j=1,2, \ldots,n\}=A\). The pair \((D,A))\) where \(D\) and \(A\) are subsets in a metric space, satisfies the fix-finite approximation property (FFAP) for a family \(\mathcal{F}\) of maps from \(D\) to \(A\) if for every \(f \in \mathcal{F}\) and \(\varepsilon>0\) there exists \(g \in \mathcal{F}\) which is \(\varepsilon\)-close to \(f\) and has only a finite number of fixed points. The paper contains two theorems on the FFAP. The first theorem proves the FFA property for a pair \((D,A)\), \(D\) a compact subset and any \(n\)-function for \(\mathcal{F}\), and in the second, \(D\) is a path and simply connected compact subset containing \(A\) and \(\mathcal{F}\) is an \(n\)-valued continuous multifunction.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
54H25 Fixed-point and coincidence theorems (topological aspects)
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