Previous |  Up |  Next


compact map; compact $R_\delta$-set
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 $R_\delta$.
[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
Partner of
EuDML logo