## 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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