Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems (Q596800)

Let \(X,M,\Lambda\) be Hausdorff topological spaces, \(Y\) be a topological vector space and \(C\subseteq Y\) be closed and such that \(\text{int}C\neq\emptyset\). Given two multifunctions \(K:X\times\Lambda\rightarrow2^{X}\) and \(F:X\times X\times M\rightarrow2^{Y}\), the ``parametric vector quasiequilibrium problem'' consists in finding, given \(\lambda\in\Lambda\) and \(\mu\in M\), some \(\bar{x}\in clK(\bar{x},\lambda)\) such that \(F(\bar{x} ,y,\mu)\cap(Y\backslash-\text{int}C)\neq\emptyset\) for every \(y\in K(\bar{x} ,\lambda)\). Assuming that the solution set \(S_{1}(\lambda,\mu)\) is nonempty in a neighborhood of \((\lambda_{0},\mu_{0})\in\Lambda\times M\), the present paper gives necessary conditions for the multifunction \(S_{1}\) to be lower semicontinuous, or upper semicontinuous. Also, a ``strong'' version of the quasiequilibrium problem is investigated, and sufficient conditions are given for its solution set to be equal to \(S_{1}(\lambda,\mu)\). These results generalize and sometimes improve previously known results on quasivariational inequalities.