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.


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
Full Text: DOI


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