A class of operator equilibrium problems. (English) Zbl 1072.47061

The authors study a class of operator equilibrium problems defined as follows: find \(f\in K\) such that \[ F(f,g)\notin -C(f),\qquad \forall g\in K, \] where \(K\subset L(X,Y),\) \(X,Y\) are Hausdorff topological vector spaces, \(F:K\times K\to Y,\) and \(C:K\to 2^Y\) has solid open convex cone images, with \(0\notin C(f)\), for every \(f\in K.\) Under the assumptions on the bi-operator \(F\) of \(C(f)\)-pseudo monotonicity and hemicontinuity in the first argument, and natural quasi \(P\)-convexity in the second one, a Minty-type lemma is proved, providing the equivalence between the set of the Stampacchia and Minty solutions. An existence theorem is then established, using quite standard techniques. Other existence results are proved under \(B\)-\(C(f)\)-pseudo monotonicity assumptions, and in topological (not necessarily Hausdorff) vector spaces.
Reviewer: Rita Pini (Milano)


47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
90C47 Minimax problems in mathematical programming
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
91B50 General equilibrium theory
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