Discontinuous implicit generalized quasi-variational inequalities in Banach spaces

摘要

We consider the following implicit quasi-variational inequality problem: given two topological vector spaces E and F, two nonempty sets X is contained in E and C is contained in F, two multifunctions Γ: X → 2~X and Φ: X → 2~C, and a single-valued map ψ: X × C × X → IR, find a pair (x,z) ∈ X × C such that x ∈ Γ(x), z ∈ Φ(x) and ψ (x, z, y) ≤ 0 for all y ∈ Γ(x). We prove an existence theorem in the setting of Banach spaces where no continuity or monotonicity assumption is required on the multifunction Φ. 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 ψ(x, z,y) = ).