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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.

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