On upper semi-continuity of simultaneous operators (Q1073289)

Let (X,d) be a metric space. Given a nonempty closed subset K of X and a nonempty bounded subset E of X, we denote by D(K,F) the number \(\sup_{y\in F}\inf_{x\in K}d(x,y)\) (resp. denote by \(\delta\) (F,K) the number \(\inf_{x\in K}\sup_{y\in F}d(y,x)).\) A set U is said to be nearly compact (resp. approximatively compact) with respect to a set V if from any sequence \(\{x_ n\}\) in U such that \(\inf_{y\in V}d(x_ n,y)\to D(V,U)\) (resp. \(\sup_{y\in V}d(x_ n,y)\to \delta (Y,U))\) one may extract a subsequence which converges to some point in U. Let \(C(B,X)=\{A\) is a nonempty closed subset of \(X\}\). Then Hausdorff metric H on CB(X) is defined as follows: For A,B\(\in CB(X)\), \[ H(A,B):=\max \{\sup_{a\in A}d(a,B),\sup_{b\in B}d(b,A)\}. \] For any \(K\in CB(X)\), let \(Q_ F(K)\) denote the set \(\{\) \(y\in F:\inf_{x\in K}d(x,y)=D(V,F)\}\) (resp. \(P_ K(F)\) denote the set \(\{\) \(x\in K:\) \(\sup_{y\in F}d(y,x)=\delta (F,K)\})\). Then the operator \(Q_ F\) (resp. \(P_ K)\) is called the simultaneous farthest operator (resp. best simultaneous approximation operator) supported on F (resp. K). In this paper, the author proves the following: (1) If F is nearly compact with respect to each member K of CB(X), then the operator \(Q_ F\) is upper semi-continuous. (2) If K is approximatively compact with respect to each member F of CB(X), then the operator \(P_ K\) is upper semi-continuous.