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An existence result for the generalized vector equilibrium problem. (English) Zbl 1014.49008

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 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 E. Tarafdar and G. X.-Z. Yuan [Nonlinear Anal., Theory Methods Appl. 30, No. 7, 4171-4181 (1997; Zbl 0912.49004)]. It is very interesting for us.
Reviewer: Kim Jong-Kyu

MSC:

49J40 Variational inequalities
47H10 Fixed-point theorems
49J53 Set-valued and variational analysis
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References:

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