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Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems. (English) Zbl 1047.49014
In recent years, the so-called exceptional families of elements (EFE) have been frequently used to study the existence of solutions for the complementarity problem; see for instance [{\it G. Isac}, {\it V. Bulavski} and {\it V. Kalashnikov}, J. Glob. Optim. 10, No. 2, 207--225 (1997; Zbl 0880.90127)]. The same method has been generalized very recently to variational inequality problems. The present paper introduces a further generalization of EFE to nonlinear projection equations (NPE), i.e., to equations of the form $h(x)=\Pi_{K}\left( g(x)-f(x)\right) $ where $f,g,h$ are functions from $\Bbb{R}^{n}$ to itself, and $\Pi_{K}$ is the orthogonal projection to a subset $K$ of $\Bbb{R}^{n}$. It is shown that, under suitable assumptions, either NPE has a solution or there exists an EFE. Applications are then given to special cases of NPE such as the generalized complementarity problem. In case the functions involved are continuous, these results generalize considerably previously known ones.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C48Programming in abstract spaces
Full Text: DOI
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