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

Let X,M,Λ be Hausdorff topological spaces, Y be a topological vector space and CY be closed and such that intC. Given two multifunctions K:X×Λ2 X and F:X×X×M2 Y , the “parametric vector quasiequilibrium problem” consists in finding, given λΛ and μM, some x ¯clK(x ¯,λ) such that F(x ¯,y,μ)(Y-intC) for every yK(x ¯,λ).

Assuming that the solution set S 1 (λ,μ) is nonempty in a neighborhood of (λ 0 ,μ 0 )Λ×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 (λ,μ). These results generalize and sometimes improve previously known results on quasivariational inequalities.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
49J45Optimal control problems involving semicontinuity and convergence; relaxation