On the solution existence of generalized quasivariational inequalities with discontinuous multifunctions

Authors: Bùi Trọng Kiên, Ngai-Ching Wong, Jen-Chih Yao,

https://doi.org/10.1007/s10957-007-9239-4

Publisher, magazine: ,

Publication year: 2007

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Abstract

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E *, a multifunction Φ:X→2E∗ and a multifunction Γ:X→2X, find a point (x^,z^)∈X×E∗ such that x^∈Γ(x^),z^∈Φ(x^),⟨z^,x^−y⟩≤0 , ∀y∈Γ(x^) . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous.

Tags: Generalized quasivariational inequalities, Lower semicontinuity, Hausdorff upper semicontinuity, Hausdorff lower semicontinuity, Multifunctions, Closed graphs, Open graphs.