# Article

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Keywords:
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:
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[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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