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Zbl 0868.47035
Isac, G.
On an Altman type fixed point theorem on convex cones. (English)
[J] Rocky Mt. J. Math. 25, No.2, 701-714 (1995). ISSN 0035-7596

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.
[W.Kryszewski (Torún)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties
47J20 Inequalities involving nonlinear operators

Keywords: convex cone; contraction; Galerkin approximation; fixed point theory; complementarity theory; Hartmann-Stampacchia theorem

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