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