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A fixed point theorem for multifunctions in a locally convex space


Authors: V. M. Sehgal and Evelyn Morrison
Journal: Proc. Amer. Math. Soc. 38 (1973), 643-646
MSC: Primary 47H10; Secondary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1973-0312344-6
MathSciNet review: 0312344
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Abstract: Let $ S$ be a convex subset of a locally convex space $ E$ and $ K$ a compact subset of $ S$. Let $ f:S \to F$ (a topological space) and $ g:K \to F$ be multifunctions. In this paper sufficient conditions are given for the existence of an $ x \in K$ such that $ f(x) \cap g(x) \ne \emptyset $. The result generalizes a recent theorem of Himmelberg (J. Math. Anal. Appl. 38 (1972), 205-207) and in special cases extends some of the other well-known results.


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  • [1] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. MR 37 #4679. MR 0229101 (37:4679)
  • [2] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121-126. MR 13, 858. MR 0047317 (13:858d)
  • [3] -, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1960/61), 305-310. MR 24 #A1120. MR 0131268 (24:A1120)
  • [4] I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170-174. MR 13, 764. MR 0046638 (13:764g)
  • [5] C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205-207. MR 0303368 (46:2505)
  • [6] S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J. 8 (1941), 457-459. MR 3, 60. MR 0004776 (3:60c)
  • [7] A. Tychonoff, Ein fixpunktsatz, Math. Ann. 111 (1935), 767-776. MR 1513031

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0312344-6
Keywords: Multifunctions, upper semicontinuity, fixed point
Article copyright: © Copyright 1973 American Mathematical Society

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