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Zbl 1048.49004
Anh, L.Q.; Khanh, P.Q.
(Lam Quoc Anh; Phan Quoc Khanh)
Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems.
(English)
[J] J. Math. Anal. Appl. 294, No. 2, 699-711 (2004). ISSN 0022-247X

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 $y\in K(\bar{x} ,\lambda)$. \par 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 2000:
*49J40 Variational methods including variational inequalities
47J20 Inequalities involving nonlinear operators
49J45 Optimal control problems inv. semicontinuity and convergence

Keywords: quasiequilibrium problem; lower semicontinuity; upper semicontinuity; variational inequality; multifunction

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