## On some generalization of Karamardian’s theorem on the complementarity problem.(English)Zbl 0653.90076

Assume that $$<E,F>$$ is a dual system of locally convex spaces, $$K\subset E$$ is a closed convex cone, K * is the dual of K that is $$\{$$ $$y\in F:$$ $$<x,y>\geq O$$, $$x\in K\}$$ and f: $$K\to F$$ is a mapping. The complementarity problem associated with K and f is: find $$x_*\in K$$ such that $$f(x_*)\in K$$ * and $$<x_*,f(x_*)>=O$$. The paper presents a generalization of Karamardian’s theorem (concerning the existence of solutions) improving previous incorrect results. A new variant of the theorem in Hilbert space where K is a Galerkin cone, is given.
Reviewer: A.L.Dontchev

### MSC:

 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)