On a generalization of Horn's fixed point theorem
The authors present the following theorem: Let \(X\) be a locally convex topological vector space, \(S_ 0\subset S_ 1\subset S_ 2\subset X\), with \(S_ 0\) and \(S_ 2\) compact convex, and \(S_ 1\) open relative to \(S_ 2\). Let \(f: S_ 2\to X\) be a continuous mapping such that for some positive integer \(m\), \(f^ j(S_ 1)\subset S_ 2\), \(1\leq j\leq m-1\); \(f^ j(S_ 1)\subset S_ 0\), \(m\leq j\leq 2m-1\). Then \(f\) has a fixed point in \(S_ 0\). This result is a generalization of \textit{W. A. Horn}'s fixed point theorem [Trans. Am. Math. Soc. 149, 391-404 (1970; Zbl 0201.462), Theorem 6].
scientific article; zbMATH DE number 60580