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Zbl 1080.90086
Huang, Nan Jing; Li, Jun; Thompson, H.B.
Implicit vector equilibrium problems with applications.
(English)
[J] Math. Comput. Modelling 37, No. 12-13, 1343-1356 (2003). ISSN 0895-7177

Summary: Let $X$ and $Y$ be Hausdorff topological vector spaces, $K$ a nonempty, closed, and convex subset of $X, C : K \to 2^Y$ a point-to-set mapping such that for any $x \in K, C(x)$ is a pointed, closed, and convex cone in $Y$ and int $C(x) \ne \emptyset$. Given a mapping $g : K \to K$ and a vector valued bifunction $f : K \times K \to Y$, we consider the implicit vector equilibrium problem (IVEP) of finding $x^{\ast} \in K$ such that $f(g(x^{\ast}), y) \notin-\operatorname{int}C(x)$ for all $y\in K$. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak $C$-pseudomonotonicity, $C$-pseudomonotonicity, and strict $C$-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.
MSC 2000:
*90C47 Minimax problems
49J40 Variational methods including variational inequalities
90C29 Multi-objective programming, etc.
90C33 Complementarity problems

Keywords: Implicit vector equilibrium problems; Vector variational inequality; Weak $C$-pseudo monotonicity; $C$-convex; Duality

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