A metric approach to asymptotic analysis.
Having a subset \(E\) of the normed vector space \(X\), the \(\varepsilon\)-enlargement of \(E\) is the set \(C_\varepsilon(E):=\{x\in X\mid d(x,E)<\varepsilon\| x\| \}\cup\{0\}\). The subsets \(E,F\) of \(X\) are apart if there is some \(\varepsilon>0\) such that \(C_\varepsilon(E)\cap C_\varepsilon(F)\) is bounded. Moreover, the subset \(C\) of \(X\) is said to be a firm (outer) asymptotic approximation of \(E\subset X\) if for every \(\varepsilon>0\) there exists \(r>0\) such that \(E\setminus rU_X\subset C_\varepsilon(C)\); a cone \(C\) is a firm asymptotic cone of \(E\) if \(C\) is a firm asymptotic approximation of \(E\). In the present paper the author accomplishes a comprehensive study of these notions and gives applications to boundedness properties of multifunctions, closedness of the sum of two sets, existence of minimum points and of fixed points.
scientific article; zbMATH DE number 2017286
scientific article