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Variational inequalities with operator solutions. (English) Zbl 1009.47064

The authors consider the following operator variational inequality (OVI) problem: find \(f_0\in K\) such that \(\langle f-f_0,T(f_0)\rangle\notin C(f_0),\) for every \(f\in K,\) where \(X,Y\) are Hausdorff topological vector spaces, \((X,Y)^*\) denotes the space of linear and continuous operators from \(X\) into \(Y\), \(K\subset (X,Y)^*\) is a nonempty, convex set, \(T:K\to X\) is a mapping and \(C:K\to 2^Y\) is a set-valued mapping with convex, cone values such that \(0\notin C(f)\) for all \(f\in K.\)
The authors study the solvability of the problem (OVI) using the \(C\)-pseudomonotonicity assumption, improving a previous result by Yu and Yao. As a particular case of (OVI) the authors study the solvability of scalar variational inequalities on Hausdorff topological vector spaces. Finally, they generalize the notion of pseudomonotonicity introduced by Brézis defining the \(B\)-pseudomonotonicity and prove, under this assumption, the solvability of (OVI).
Reviewer: Rita Pini (Milano)

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J53 Set-valued and variational analysis
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