On the discontinuous infinite-dimensional generalized quasivariational inequality problem (Q1810925)

The author solves the following generalized quasivariational inequality: \[ \begin{cases} \text{Find} (\widehat{x},\widehat{\varphi})\in X\times E^* \;\text{such that}\\ \widehat{x}\in G(\widehat{x}),\;\widehat{\varphi}\in F(\widehat{x}), \;\langle\widehat{\varphi},\widehat{x}-y\rangle\leq 0 \qquad\forall y\in G(\widehat{x}), \end{cases} \] where \(E\) is a topological vector space, \(E^*\) its topological dual, \(X\) is a nonempty subset of \(E\) and \(G\colon X\longrightarrow 2^X\), \(F\colon X\longrightarrow 2^{E^*}\) are two multifunctions. Using techniques similar to [\textit{P. Cubiotti}, Optimization Theory Appl. 92, 457-475 (1997; Zbl 0869.49006)] an existence result is obtained. The paper extends to an infinite-dimensional situation the results of [\textit{J. C. Yao} and \textit{J.-S. Guo}, J. Math. Anal. Appl. 182, 371-392 (1994; Zbl 0809.49005)] and [\textit{P. Cubiotti}, J. Optimization Theory Appl. 92, 477-495 (1997; Zbl 0879.90179)].