Geodesic
Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem
Canadian mathematical bulletin, Tome 24 (1981) no. 3, pp. 351-357

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1981-053-5
Mots-clés : 54 H 25, 54 C 60, 90 D 99, Fixed point theory, isolated fixed points, Kakutani's theorem, multifunctions, upper and lower semicontinuity, convex sets 
Schirmer, Helga. Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem. Canadian mathematical bulletin, Tome 24 (1981) no. 3, pp. 351-357. doi: 10.4153/CMB-1981-053-5
@article{10_4153_CMB_1981_053_5,
     author = {Schirmer, Helga},
     title = {Conditions for the {Uniqueness} of the {Fixed} {Point} in {Kakutani's} {Theorem}},
     journal = {Canadian mathematical bulletin},
     pages = {351--357},
     year = {1981},
     volume = {24},
     number = {3},
     doi = {10.4153/CMB-1981-053-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-053-5/}
}
TY  - JOUR
AU  - Schirmer, Helga
TI  - Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem
JO  - Canadian mathematical bulletin
PY  - 1981
SP  - 351
EP  - 357
VL  - 24
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-053-5/
DO  - 10.4153/CMB-1981-053-5
ID  - 10_4153_CMB_1981_053_5
ER  - 
%0 Journal Article
%A Schirmer, Helga
%T Conditions for the Uniqueness of the Fixed Point in Kakutani's Theorem
%J Canadian mathematical bulletin
%D 1981
%P 351-357
%V 24
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-053-5/
%R 10.4153/CMB-1981-053-5
%F 10_4153_CMB_1981_053_5

[1] 1. Berge, C., Topological Spaces, Oliver and Boyd, Edinburgh and London, 1963. Google Scholar

[2] 2. Brown, R. F., The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, 111, 1971. Google Scholar

[3] 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.38.2.121 Google Scholar

[4] 4. Fort, M. K. Jr, Essential and non-essential fixed points, Amer. J. Math. 72 (1962), 315-322. Google Scholar

[5] 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] 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] 7. Kakutani, S., A generalization of Brouwer's fixed point theorem, Duke Math. J. 8 (1962), 457-459. Google Scholar

[8] 8. Valentine, F. A., Convex Sets, McGraw Hill, New York, San Francisco, Toronto, London, 1964. Google Scholar

Cité par Sources :