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Title: On the structure of fixed point sets of some compact maps in the Fréchet space (English)
Author: Kubáček, Zbyněk
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 118
Issue: 4
Year: 1993
Pages: 343-358
Summary lang: English
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Category: math
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Summary: The aim of this note is 1. to show that some results (concerning the structure of the solution set of equations (18) and (21)) obtained by Czarnowski and Pruszko in [6] can be proved in a rather different way making use of a simle generalization of a theorem proved by Vidossich in [8]; and 2. to use a slight modification of the "main theorem" of Aronszajn from [1] applying methods analogous to the above mentioned idea of Vidossich to prove the fact that the solution set of the equation (24), (25) (studied in the paper [7]) is a compact $R_\delta$. (English)
Keyword: compact map
Keyword: compact $R_\delta$-set
MSC: 46A04
MSC: 46E05
MSC: 46N20
MSC: 47H10
MSC: 47N20
MSC: 54C55
idZBL: Zbl 0839.47037
idMR: MR1251881
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Date available: 2009-09-24T21:01:12Z
Last updated: 2015-08-30
Stable URL: http://hdl.handle.net/10338.dmlcz/126160
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Reference: [1] N. Aronszajn: Le correspondant topologique de l'unicité dans la théorie des équations différentielles.Ann. Math. 43 (1942), 730-738. Zbl 0061.17106, MR 0007195, 10.2307/1968963
Reference: [2] E. F. Beckenbach, R. Bellman: Inequalities.Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. Zbl 0186.09606, MR 0158038
Reference: [3] I. Bihari: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.Acta Math. Acad. Sci. Hung. 7 (1956), 81-94. Zbl 0070.08201, MR 0079154, 10.1007/BF02022967
Reference: [4] K. Borsuk: Theory of retracts.PWN, Warszawa, 1967. Zbl 0153.52905, MR 0216473
Reference: [5] F. F. Browder, G. P. Gupta: Topological degree and non-linear mappings of analytic type in Banach spaces.J. Math. Anal. Appl. 26 (1969), 390-402. MR 0257826, 10.1016/0022-247X(69)90162-0
Reference: [6] K. Czarnowski, T. Pruszko: On the structure of fixed point sets of compact maps in $B_0$ spaces with applications to integral and differential equations in unbounded domain.J. Math. Anal. Appl. 154 (1991), 151-163. MR 1087965, 10.1016/0022-247X(91)90077-D
Reference: [7] V. Šeda, Z. Kubáček: On the set of fixed points of a compact operator.Czech. Math. J., to appear.
Reference: [8] G. Vidossich: A fixed point theorem for function spaces.J. Math. Anal. Appl. 36 (1971), 581-587. Zbl 0194.44903, MR 0285945, 10.1016/0022-247X(71)90040-0
Reference: [9] G. Vidossich: On the structure of the set of solutions of nonlinear equations.J. Math. Anal. Appl. 34 (1971), 602-617. MR 0283645, 10.1016/0022-247X(71)90100-4
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