On an Altman type fixed point theorem on convex cones (Q1898287)

The paper recalls the existing connection of the fixed point theory with the so-called complementarity theory. Given a continuous map \(f:K\to H\) of a closed convex cone \(K\) in a Hilbert space \(H\), the complementarity problem concerns the existence of a point \(u\in K\) such that \(f(u)\) is contained in the dual cone \(K^*\) and \(\langle u,f(u)\rangle=0\). It appears that the solvability of the complementarity problem is equivalent to the existence of zeros of \(f\) (i.e. fixed points of \(T=I-f\)). Thus, results concerning the solvability of the complementarity problem (related to the well-known Hartmann-Stampacchia theorem) yield fixed points. For instance, the author shows that given a contraction \(S\) and a compact map \(T\), the map \(f=S+T: K\to H\) has a fixed point provided \(f(K)\subset K\) and \(K,I-f\) satisfy some auxiliary natural conditions. Some other, even more general results are proved and discussed.