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A note on equilibrium problems with properly quasimonotone bifunctions. (English) Zbl 0985.90090

Let \(K\subset \mathbb{R}^n\) be a convex compact subset and \(F: K\times K\to \mathbb{R}\) be a given bifunction. Setting \(\inf_\Delta F:= \inf_{x\in K} F(x,x)\) and \(\sup_\Delta F:= \sup_{x\in K} F(x,x)\), the authors discuss the equilibrium problem \[ \text{fin }\overline x\in K: F(\overline x,y)\geq \inf_\Delta F\quad\forall y\in K\tag{EP} \] and the associated dual equilibrium problem \[ \text{find }\overline y\in K: F(x,\overline y)\leq \sup_\Delta F\quad \forall x\in K.\tag{DEP} \] If \(\overline x\) and \(\overline y\) are solutions of (EP) and (DEP) then of course it is \[ \inf_\Delta F\leq \inf_{y\in K} F(\overline x,y)\leq F(\overline x,\overline y)\leq \sup_{x\in K} F(x,\overline y)\leq \sup_\Delta F. \] In case of a constant behavior of \(F\) on the diagonal set, all terms are equal and hence \((\overline x,\overline y)\) is a saddlepoint of \(F\).
In the paper, the authors present sufficient conditions for the existence of solutions of both problems in the general case. The most important notion is the proper quasimonotonicity of a bifunction. In the first part, relationships between proper quasimonotonicity and other generalized monotonicity notions are pointed out. In the second part, these notions ar used to discuss the equilibrium problems. In detail, the following results are proved:
1. If \(F(x,.)\) is l.s.c. \(\forall x\in K\) and \(F(\cdot,\cdot)-\sup_\Delta F\) is properly quasimonotone then the set of solutions (DEP) is nonempty.
2. If \(F(x,\cdot)\) is quasiconvex \(\forall x\in K\) then the set of solutions of (DEP) is convex.
3. If \(\sup_\Delta F- F(\cdot,\cdot)\) is strictly pseudomonotone then the set of solutions of (DEP) is compact.
4. If \(F(x,.)\) is l.s.c. and quasiconvex \(\forall x\in K\) then \(F(\cdot,\cdot)- \sup_\Delta F\) is properly quasiconvex.
5. If \(F(x,.)\) is l.s.c. and semistrictly quasiconvex \(\forall x\in K\), \(F(.,y)\) is hemicontinuous \(\forall y\in K\), \(F(\cdot,\cdot)\) is properly quasimonotone with \(F= 0\) on the diagonal set then the set of solutions of (EP) is nonempty.

MSC:

90C47 Minimax problems in mathematical programming
49K35 Optimality conditions for minimax problems
49J35 Existence of solutions for minimax problems
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