On vector quasi-variational inequality problems in \(H\)-spaces (Q2720374)

Let \((X,\{\Gamma_A\})\) be a compact \(H\)-space and \((Y,\{\Gamma_B\})\) an \(H\)-space. Let \(E\) be a real topological vector space with a body cone \(K\), i.e., the interior of \(K\) is nonempty. Suppose that: (i) \(S:X\to 2^X\) is a continuous set-valued mapping with nonempty compact \(H\)-convex values and \(S^{-1}(x)\) is open for any \(x\in X\); (ii) \(T:X\to 2^Y\) is a set-valued mapping with nonempty \(H\)-convex values and \(T^{-1}:Y\to 2^X\) is transfer open valued; (iii) \(\varphi:X\times Y\times X\to E\) is a continuous function such that \(\varphi(x,y,x)\notin \text{int}K\) for all \(x\in X\) and \(y\in T(x)\), and the function \(z\mapsto \varphi(x,y,z)\) is \(H^*\)-convex. The authors prove that there exists \(x^*\in S(x^*)\) and \(y^*\in T(x^*)\) such that \(\varphi(x^*,y^*,x)\not\in -\text{ int}K\) for all \(x\in S(x^*)\). They also prove another similar result.