On the structure of fixed point sets of some compact maps in the Fréchet space. (English) Zbl 0839.47037

Summary: The aim of this note is
1. to show that some results obtained by K. Czarnowski and T. Pruszko in [J. Math. Anal. Appl. 154, No. 1, 151-163 (1991; Zbl 0729.47054)] can be proved in a rather different way making use of a simple generalization of a theorem proved by G. Vidossich in [ibid. 36, 581-587 (1971; Zbl 0219.47054)]; and
2. to use a slight modification of the “main theorem” of N. Aronszajn from [Ann. Math., II. Ser. 43, 730-738 (1942; Zbl 0061.17106)] applying methods analogous to the above-mentioned idea of Vidossich to prove the fact that the solution set of the equation \[ x'(t)= f(t, x_t),\quad t\in [b, \infty),\quad x_b= \psi \] is a compact \(R_\delta\).


47H10 Fixed-point theorems
46E05 Lattices of continuous, differentiable or analytic functions
46N20 Applications of functional analysis to differential and integral equations
46A04 Locally convex Fréchet spaces and (DF)-spaces
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