A fixed point theorem for multifunctions in a locally convex space

Sehgal, V. M.; Morrison, Evelyn

doi:10.1090/S0002-9939-1973-0312344-6

A fixed point theorem for multifunctions in a locally convex space
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by V. M. Sehgal and Evelyn Morrison PDF

Proc. Amer. Math. Soc. 38 (1973), 643-646 Request permission

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.

References

Felix E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301. MR 229101, DOI 10.1007/BF01350721
Ky. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126. MR 47317, DOI 10.1073/pnas.38.2.121
Ky Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1960/61), 305–310. MR 131268, DOI 10.1007/BF01353421
I. L. Glicksberg, A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170–174. MR 46638, DOI 10.1090/S0002-9939-1952-0046638-5
C. J. Himmelberg, Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972), 205–207. MR 303368, DOI 10.1016/0022-247X(72)90128-X
Shizuo Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457–459. MR 4776
A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (1935), no. 1, 767–776 (German). MR 1513031, DOI 10.1007/BF01472256