Geodesic
On the structure of fixed point sets of some compact maps in the Fréchet space
Mathematica Bohemica, Tome 118 (1993) no. 4, pp. 343-358

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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 $R_\delta$.
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 $R_\delta$.
DOI : 10.21136/MB.1993.126160
Classification : 46A04, 46E05, 46N20, 47H10, 47N20, 54C55
Keywords: compact map; compact $R_\delta$-set 
Kubáček, Zbyněk. On the structure of fixed point sets of some compact maps in the Fréchet space. Mathematica Bohemica, Tome 118 (1993) no. 4, pp. 343-358. doi: 10.21136/MB.1993.126160
@article{10_21136_MB_1993_126160,
     author = {Kub\'a\v{c}ek, Zbyn\v{e}k},
     title = {On the structure of fixed point sets of some compact maps in the {Fr\'echet} space},
     journal = {Mathematica Bohemica},
     pages = {343--358},
     year = {1993},
     volume = {118},
     number = {4},
     doi = {10.21136/MB.1993.126160},
     mrnumber = {1251881},
     zbl = {0839.47037},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126160/}
}
TY  - JOUR
AU  - Kubáček, Zbyněk
TI  - On the structure of fixed point sets of some compact maps in the Fréchet space
JO  - Mathematica Bohemica
PY  - 1993
SP  - 343
EP  - 358
VL  - 118
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126160/
DO  - 10.21136/MB.1993.126160
LA  - en
ID  - 10_21136_MB_1993_126160
ER  - 
%0 Journal Article
%A Kubáček, Zbyněk
%T On the structure of fixed point sets of some compact maps in the Fréchet space
%J Mathematica Bohemica
%D 1993
%P 343-358
%V 118
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1993.126160/
%R 10.21136/MB.1993.126160
%G en
%F 10_21136_MB_1993_126160

[1] N. Aronszajn: Le correspondant topologique de l'unicité dans la théorie des équations différentielles. Ann. Math. 43 (1942), 730-738. | DOI | MR | Zbl

[2] E. F. Beckenbach, R. Bellman: Inequalities. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. | MR | Zbl

[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. | DOI | MR | Zbl

[4] K. Borsuk: Theory of retracts. PWN, Warszawa, 1967. | MR | Zbl

[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. | DOI | MR

[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. | DOI | MR

[7] V. Šeda, Z. Kubáček: On the set of fixed points of a compact operator. Czech. Math. J., to appear.

[8] G. Vidossich: A fixed point theorem for function spaces. J. Math. Anal. Appl. 36 (1971), 581-587. | DOI | MR | Zbl

[9] G. Vidossich: On the structure of the set of solutions of nonlinear equations. J. Math. Anal. Appl. 34 (1971), 602-617. | DOI | MR

Cité par Sources :