| 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 |
| . |
