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Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. (English) Zbl 1048.49004
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.

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
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