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On a class of nonconvex equilibrium problems. (English) Zbl 1070.65053

The author considers an equilibrium problem of the form: Find \(u \in K\) such that \[ F(g(u), g(v)) \geq 0 \quad \forall v \in K, \] where \(K\) is a closed set in a Hilbert space, \(g : K \rightarrow K\) is a mapping, and \(F : K \times K \rightarrow R\) is a bifunction. Moreover, the set \( g^{-1}(K)\) is supposed to be convex. Extensions of several known results from the usual equilibrium problems such as the Minty lemma and the auxiliary problem based iterative algorithms are given. A similar extension of the well-posedness result from variational inequalities is also described and substantiated.

MSC:

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
49J27 Existence theories for problems in abstract spaces
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