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Extensions of two fixed point theorems of Ky Fan. (English) Zbl 0551.47024

Let E, F be Hausdorff topological vector spaces, where F has sufficiently many continuous linear functionals, \(X\subset E\) be a nonempty compact convex subset. The following two results, each extends a well-known fixed point theorem of Ky Fan, are proved. Let F be a upper semi-continuous set-valued map on X whose values are nonempty closed convex subsets of F, g:X\(\to F\) be a continuous map satisfying \((a)\quad f(x)\cap g(X)\neq \emptyset\) for all \(x\in X\); (b) \(g^{-1}(C)\) is convex (or empty) for any closed convex set C in F. Then there exists a point \(x_ 0\in X\) such that \(g(x_ 0)\in f(x_ 0).\)
Theorem 3. Let f,g:X\(\to F\) be two continuous maps, where g satisfies (c) g(X) is convex and \(g^{-1}(y)\) is convex for any \(y\in g(X)\). Then either there exists a point \(x_ 0\in X\) such that \(g(x_ 0)=f(x_ 0),\) or there exist a point \(x_ 0\in X\) and a continuous seminorm p on F such that for all \(y\in g(X)\), \(0<p(g(x_ 0)-f(x_ 0))\leq p(y-f(x_ 0)).\)

MSC:

47H10 Fixed-point theorems
46A55 Convex sets in topological linear spaces; Choquet theory

References:

[1] Fan, K.: Fixed-point and minimas theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. USA38, 121-126 (1952) · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121
[2] Fan, K.: A generalization of Tychonoffs fixed point theorem. Math. Ann.142, 305-310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421
[3] Fan, K.: Sur un théorème minimax. C.R. Acad. Sci. Paris, Groupe I.259, 3925-3928 (1964) · Zbl 0138.37304
[4] Fan, K.: Extensions of two fixed point theorems of F.E. Browder Math. Z.112, 234-240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225
[5] Ha, C.-W. Minimax and fixed point theorems. Math. Ann.248, 73-77 (1980) · doi:10.1007/BF01349255
[6] Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J.8, 457-459 (1941) · Zbl 0061.40304 · doi:10.1215/S0012-7094-41-00838-4
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