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A variational-like inequality problem. (English) Zbl 0649.49007

Summary: Given a closed and convex set K in \({\mathbb{R}}^ n\) and two continuous maps \(F: K\to {\mathbb{R}}^ n\) and \(\eta: K\times K\to {\mathbb{R}}^ n,\) the problem considered here is to find \(\bar x\in K\) such that \(<F(\bar x)\), \(\eta(x,\bar x)>\geq 0\) for all \(x\in k\). We call it a variational-like inequalityproblem, and develop a theory for the existence of a solution. We also show the relationship between the variational-like inequality problem and some mathematical programming problems.

MSC:

49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J45 Methods involving semicontinuity and convergence; relaxation
90C25 Convex programming
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems
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