Convergence of minimal and approximate minimal elements of sets in partially ordered vector spaces (Q1819001)

Let \(Z\) be a real normed vector space partially ordered by a closed convex pointed cone \(C\) with nonempty interior, and let \(A\) be a nonempty set in \(Z\). In the first part of the paper the authors point out conditions of minimal character ensuring that the set \(\operatorname {Min}A= \{a\in A\mid (a-C)\cap (A\setminus\{a\})= \emptyset\}\) satisfies one of the following inclusions: \[ \begin{aligned} \operatorname {Min}A &\subseteq \text{Lim inf Min }A_n,\\ \operatorname {Min}A &\subseteq \text{Lim sup Min }A_n,\\ \text{Lim sup Min }A_n &\subseteq \operatorname {Min} A. \end{aligned} \] In the second part of the paper they associate with any \(\varepsilon\in C\) the set \(\varepsilon\)-\(S(A)= (\varepsilon+ \operatorname {Min}A-C)\cap A\) of restricted \(\varepsilon\)-minimal elements of \(A\) and the set \(\varepsilon\)-\(\widehat{S}(A)= (\varepsilon+ \operatorname {Min}A- \operatorname {int}C)\cap A\) of strict restricted \(\varepsilon\)-minimal elements of \(A\). Concerning these sets they also reveal conditions of minimal character such that one of the following inclusions holds: \[ \begin{aligned} \varepsilon\text{-} \widehat{S}(A) &\subseteq \text{Lim inf} (\varepsilon\text{-} \widehat{S}(A_n)+ C),\\ \varepsilon\text{-} \widehat{S}(A) &\subseteq \text{Lim sup} (\varepsilon\text{-} \widehat{S}(A_n)+ C),\\ \text{Lim sup } \varepsilon\text{-} S(A_n) &\subseteq \varepsilon\text{-} S(A). \end{aligned} \]