### A generalization of N. Aronszajn's theorem on connectedness of the fixed point set of a compact mapping

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It is proved that under some conditions the set of all solutions of an initial value problem for $n$-th order functional differential system on an unbounded interval 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)...

For a generalized pendulum equation we estimate the number of periodic solutions from below using lower and upper solutions and from above using a complex equation and Jensen’s inequality.

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