An existence result for the generalized vector equilibrium problem
Let \(X\) and \(Y\) be two topological vector spaces, \(K\) a nonempty convex subset of \(X\), and \(F : K \times K \to 2^Y\) a multifunction. Let \(C : K\to 2^Y\) be a multifunction such that for each \(x\in K\), \(C(x)\) is a closed convex cone with \(\text{int }C(x)\neq 0\), where \(\text{int } C(x)\) denotes the interior of \(C(x)\). Then, the authors consider the following generalized vector equilibrium problem which consists of finding \[ F(\overline x, y)\nsubseteq -\text{int }C(\overline x)\quad \text{for all } y\in K. \] By using the Fan-Browder type fixed-point theorem due to \textit{E. Tarafdar} [J. Math. Anal. Appl. 128, 475-479 (1987; Zbl 0644.47050)], they present an existence result for the above problem, extending the result in \textit{E. Tarafdar} and \textit{G. X.-Z. Yuan} [Nonlinear Anal., Theory Methods Appl. 30, No.~7, 4171-4181 (1997; Zbl 0912.49004)]. It is very interesting for us.
scientific article; zbMATH DE number 1428528