A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems
Assume \(B\), \(\Lambda\) are metric spaces and \(Y\) is a metric vector space with closed convex cone \(C\) such that \(\text{int} C\neq \emptyset\). The authors consider for \(F:B\times B\times \Lambda \rightarrow Y\) and a set valued map \(A:\Lambda \rightarrow 2^B\) the mapping \(S:\Lambda\rightarrow B\) of efficient solutions of a parametric vector equilibrium problem given by \[ S(\mu):=\{x\in A(\mu) \mid F(x,y,\mu)\notin -C\setminus\{0\}, \forall y\in A(\mu)\}. \] Theorem 3.1: If \(A\) and \(F\) are continuous mappings, \(A\) is compact valued and the metric regularity condition (the authors use more restrictively \(D=1\)) \[ \exists D>0~ \forall \mu\in \Lambda~ \forall x\in A(\mu)\setminus S(\mu) ~\exists y\in S(\mu): d_B(x,y)\leq D \inf_{w\in Y\setminus (-\text{int}\,C)}d_Y(F(x,y,\mu),w) \] holds, where \(d_B\) resp. \(d_Y\) are the metrics on \(B\) and \(Y\), then the set valued mapping \(S\) is lower semicontinuous in the sense of Kuratowski. Under the same assumptions (Theorem 3.2) the lower semicontinuity is valid for the weak efficient solution set mapping [replace in \(S\) the set \(-C\setminus\{0\}\) by \(-\text{int}\,C\)]. They illustrate their results by two finite dimensional examples.
scientific article; zbMATH DE number 6142818