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Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem

Published online by Cambridge University Press:  20 November 2018

Helga Schirmer
Helga Schirmer*
Affiliation:
Carleton University, Ottawa, Canada
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Abstract

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Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.

Keywords

54 H 2554 C 6090 D 99Fixed point theoryisolated fixed pointsKakutani's theoremmultifunctionsupper and lower semicontinuityconvex sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 3 , 01 September 1981 , pp. 351 - 357
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Berge, C., Topological Spaces, Oliver and Boyd, Edinburgh and London, 1963.Google Scholar
2. Brown, R. F., The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, 111, 1971.Google Scholar
3. Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S. 38 (1962), 121-126.10.1073/pnas.38.2.121Google Scholar
4. Fort, M. K. Jr, Essential and non-essential fixed points, Amer. J. Math. 72 (1962), 315-322.Google Scholar
5. Glicksberg, I. L., A further generalization of the Kakutani fixed point theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1962), 170-174.Google Scholar
6. Hamilton, O. H., A fixed point theorem for upper semicontinuous transformations on n-cells for which the image of points are non-acylic continua, Duke Math. J. 14 (1962), 689-693. Corrections, ibid. 24 (1962), 59.Google Scholar
7. Kakutani, S., A generalization of Brouwer's fixed point theorem, Duke Math. J. 8 (1962), 457-459.Google Scholar
8. Valentine, F. A., Convex Sets, McGraw Hill, New York, San Francisco, Toronto, London, 1964.Google Scholar