Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

Abstract

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 → 2^X and Ф : X → 2^C, and a single-valued map ψ : $X\times C\times X\to IR$, find a pair $(\hat x,\hat z)\in X\times C$ such that $\hat x\in \Gamma(\hat x)$, $\hat z\in$ Ф $(\hat x)$ 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 Ф. 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 $\psi(x,z,y)=\langle z,x-y\rangle$).