Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

• Authors: Paolo Cubiotti;Jen-Chih Yao
• Venue: Journal of Global Optimization
• Year: 2007

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X $$\sqsubseteq$$ E and C $$\sqsubseteq$$ F, two multifunctions Γ : X 驴 2 X and 驴 : X 驴 2 C , and a single-valued map 驴 : $$X\times C\times X\to IR$$ , find a pair $$(\hat x,\hat z)\in X\times C$$ such that $$\hat x\in \Gamma(\hat x)$$ , $$\hat z\in$$ 驴 $$(\hat x)$$ and $$\psi(\hat x,\hat z,y)\le 0$$ for all $$y\in\Gamma(\hat x)$$ . We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction 驴. Our result extends to non-compact and infinite-dimensional setting a previous results of the authors (Theorem 3.2 of Cubbiotti and Yao [15] Math. Methods Oper. Res. 46, 213---228 (1997)). It also extends to the above problem a recent existence result established for the explicit case (C = E * and $$\psi(x,z,y)=\langle z,x-y\rangle$$ ).