A class of cone bounded quasiconvex mappings in topological vector spaces
Let \(X\) be a topological vector space, \(S\subset X\) be a nonempty convex set and \(Y\) be a topological vector space, partially ordered by a Daniell closed pointed convex cone \(K\) with nonempty interior. A mapping \(f:S\longrightarrow Y\) is said to be \(K\)-bounded quasiconvex if, for any \(x^{1},x^{2}\in S\) and \(\lambda \in \left( 0,1\right) ,\) \(K\)-\(bou\{f( x^{1}) ,f( x^{2}) \} -f( \lambda x^{1}+( 1-\lambda) x^{2}) \subset K,\) \(K\)-\(bou\{ f( x^{1}) ,f( x^{2}) \} \) denoting the set of minimal upper bounds of \(\{ f( x^{1}) ,f( x^{2}) \} .\) The authors study the fundamental properties of this notion as well as the related concepts of \(K\)-bounded strict quasiconvexity, \(K\)-bounded secondary strict quasiconvexity and \(K\)-bounded strong quasiconvexity. In particular, they prove that the concept of \(K\)-bounded quasiconvexity is not new, as it is equivalent to that of \(K\)-quasiconvexity introduced by \textit{Dinh The Luc} [``Theory of vector optimization''' (Lect. Notes Econ. Math. Sci. 319, Springer, Berlin) (1988; Zbl 0654.90082)].
scientific article; zbMATH DE number 2101827