# Article

 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 (print) ISSN: 2464-7136 (online) Volume: 118 Issue: 4 Year: 1993 Pages: 343-358 Summary lang: English . Category: math . 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 DOI: 10.21136/MB.1993.126160 . Date available: 2009-09-24T21:01:12Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/126160 . 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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