Article
MSC:
46A04,
46E05,
46N20,
47H10,
47N20,
54C55 | MR 1251881 | Zbl 0839.47037 | DOI: 10.21136/MB.1993.126160
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Keywords:
compact map; compact $R_\delta$-set
compact map; compact $R_\delta$-set
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 $R_\delta$.
References:
[1] N. Aronszajn: Le correspondant topologique de l'unicité dans la théorie des équations différentielles. Ann. Math. 43 (1942), 730-738. DOI 10.2307/1968963 | MR 0007195 | Zbl 0061.17106
[2] E. F. Beckenbach, R. Bellman: Inequalities. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0158038 | Zbl 0186.09606
[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 10.1007/BF02022967 | MR 0079154 | Zbl 0070.08201
[4] K. Borsuk: Theory of retracts. PWN, Warszawa, 1967. MR 0216473 | Zbl 0153.52905
[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 10.1016/0022-247X(69)90162-0 | MR 0257826
[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 10.1016/0022-247X(91)90077-D | MR 1087965
[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 10.1016/0022-247X(71)90040-0 | MR 0285945 | Zbl 0194.44903
[9] G. Vidossich: On the structure of the set of solutions of nonlinear equations. J. Math. Anal. Appl. 34 (1971), 602-617. DOI 10.1016/0022-247X(71)90100-4 | MR 0283645
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