Semicontinuity properties of the strong best co-approximation operator (Q1056850)

Let E be a normed linear space, G a linear subspace of E and \(R_{S,G}^ a \)set-valued operator defined on the subset \(\{\) \(x\in E: R_{S,G}(x)\neq \emptyset \}\) of E by \(R_{S,G}(x)=\{g_ 0\in G: \| g_ 0-g\| +r\| x-g_ 0\| \leq \| x-g\|,\quad \forall g\in G\},\) where \(0<r\leq 1\). The present paper studies the semicontinuity properties of the operator \(R_{S,G}\), for the first time, analogous to the investigations made by many authors in the case of the semicontinuity properties of the set-valued metric projection. Certain necessary and sufficient conditions for \(R_{S,G}\) to be upper semicontinuous (lower semicontinuous) and upper (K)-semicontinuous (lower (K)-semicontinuous) are obtained. Some sufficient conditions under which \(R_{S,G}\) is upper semicontinuous (lower semicontinuous) are also provided. A preliminary version has been presented by the authors in Constructive function theory, Proc. int. Conf., Varna/Bulg. 1981, 495-498 (1983; Zbl 0547.41022).