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Pre-vector variational inequalities. (English) Zbl 0858.49008

The author considers abstract vector-valued variational inequalities of the following form: \[ \text{Find}\quad x\in E: \langle T(x),\eta(y,x)\rangle\not<0,\quad\forall y\in E, \] where \(E\) is a (closed, convex) subset of a Banach space \(X\), \(\eta\) denotes a function with \(\eta: E\times E\to E\) and \(T\) is a mapping from \(E\) to the space of all linear, bounded operators from \(X\) to a Banach space \(Y\). Here, \(Y\) is ordered by a closed convex cone \(D\), such that \(x\not<y\) if \(y-x\not\in\text{int }D\). Choosing \(X=\mathbb{R}^n\), \(Y=\mathbb{R}\), \(D=\mathbb{R}_+\), \(\eta(y,x)=y- x\) and \(T\) as gradient of a smooth real valued function one obtains an inequality problem in \(\mathbb{R}^n\) as a simple application of this abstract problem.
After the generalization of some properties like monotone, hemicontinuous or convex to the above situation, existence statements are established for the above abstract vector variational inequality, where different assumptions on the operator \(T\) and the function \(\eta\) are considered. Furthermore, the close relation of the inequality problem to vector optimization problems of so-called \(\eta\)-invex (convex) functions is shown as an abstract generalization of the connection between the minimization of convex functions and variational inequalities.
Moreover, existence results are derived for another type of vector variational inequalities.

MSC:

49J40 Variational inequalities
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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