A fix-finite approximation theorem
Let \(E\) be a metrizable locally convex vector space whose topology is defined by a translation invariant metric \(d\) given by \(d(x,y)=\sum^\infty_{h=1}\frac1{2^n}\;\frac{p_n(x-y)}{1+p_n(x-y)}\).Let \(C(E)\) be the set of non-empty compact subsets of \(E\). For \(A,B\in C(E)\), the Hausdorff distance is defined as \(d_H(A,B)=\max\{P(A,B),P(B,A)\}\), \(P(A,B)=\sup\{d(x),B:x\in A\}\). A multifunction \(F:E\to E\) is a map from \(E\) to the set of nonempty subsets of \(E\). \(F\) is said to be (i) \(n\)-valued if \(F(x)\) consists of \(n\) points for all \(x\), (ii) continuous at \(x_0\in E\) if for every \(\varepsilon>0\), there exists \(B>0\) such that \(d(x_0,x)<\beta\Rightarrow d_H(F(x_0),F(x))<\varepsilon\), (iii) continuous on \(E\) if it is continuous at every point of \(E\), (iv) compact if it is continuous and the closure of its range \(\overline{F(E)}\) is a compact subset of \(E\). An element \(x\) of \(E\) is said to be a fixed point of \(F\) if \(x\in F(x)\). For two compact multifunctions \(F\) and \(G\) from \(E\) to \(E\), the Hausdorff distance between \(F\) and \(G\) is given by \(d_H(F,G)=\sup\{d_H(F(x),G(x)): x\in E\}\). \(F\) and \(G\) are said to be \(\varepsilon\)-near if \(d_H(F,G)<\varepsilon\). In this paper, the author proves that if \(c_i\) is a nonempty convex comapct subset of metrizable locally convex vector space \(E\) for \(i=1,2,\dots,m\) such that \(\bigcap^m_{i=1}c_i\neq \emptyset\) or \(c_i\cap c_j=\emptyset\) for \(i\neq j\), then for every \(\varepsilon >0\) and for every \(n\)-valued continuous multifunction \(F:\bigcup^m_{i=1}C_i\to \bigcup^m_{i=1}C_i\) there exists an \(n\)-valued continuous multifunction \(G:\bigcup^m_{i=1}c_i\to \bigcup^m_{i=1}c_i\) which is \(\varepsilon\)-near to \(F\) and has only a finite number of fixed points.
scientific article; zbMATH DE number 2143240