Variational inequalities given by semi-pseudomonotone mappings (Q2702961)

In this paper, the authors consider the following problem:NEWLINENEWLINENEWLINELet \(X\) be a real Banach space with \(X^*\) its dual space, \(C\subseteq X^*\) a nonempty convex set and let \(T: C\times C\to 2^X\) be a set-valued mapping. Find an element \(\overline u\in C\) such that NEWLINE\[NEWLINE\sup_{x\in T(\overline u,\overline u)}\langle x,u-\overline u\rangle\geq 0,\quad\forall u\in C.\tag{P}NEWLINE\]NEWLINE As usual, they use the well-known intersection theorem of \textit{K. Fan} [Math. Ann. 142, 305-310 (1961; Zbl 0093.36701)] to prove the existence of a solution of problem (P) under pseudomonotonicity (in the sense of Karamardian) in the first variable and certain kind of continuity in the second variable of the operator \(T\). As a result, they deduce the well-known theorem of \textit{H. Brézis} [Ann. Inst. Fourier 18, No. 1, 115-275 (1968; Zbl 0169.18602)] concerning variational inequalities.