Generalized \(g\)-quasivariational inequality (Q2368460)

The authors introduce the following generalized \(g\)-quasivariational inequality. Suppose that \(X\) is nonempty subset of a metric space \(E\) and \(Y\) is nonempty subset of a topological vector space \(F\). Let \(g: X \to Y\) and \(\psi: X \times Y \to \mathbb{R}\) be two functions and let \(S: X \to 2^Y\) and \(T: Y \to 2^{F^*}\) be two maps. Then the generalized \(g\)-quasivariational inequality problem (GgQVI) is to find a point \(\overline {x} \in X\) and a point \(f \in T(g(\overline {x} ))\) such that \(g(\overline {x} ) \in S(\overline {x} )\) and \(\sup_{y \in S(\overline {x})} \{ \Re \langle f,y -g(\overline {x} ) \rangle +\psi(\overline {x},y) \} =\psi(\overline {x}, g(\overline {x}))\). The authors prove sufficient conditions for the existence of a solution of the aforementioned problem (GgQVI).