# Remarks and improvement regarding the theorem of continuous dependence of solutions of F.D.E., using some new results for continuous convergences

## TL;DR

This paper enhances the theorem on continuous dependence of solutions for functional differential equations by removing the need for uniform boundedness assumptions and correcting previous proof inconsistencies.

## Contribution

It introduces new results on continuous convergences to improve and correct the existing theorem without requiring uniform boundedness of the sequence.

## Key findings

- The theorem now holds without the uniform boundedness assumption.
- Corrected and clarified proof of the continuous dependence theorem.
- Extended applicability of the theorem to broader classes of functional differential equations.

## Abstract

We improve the theorem on continuous dependence of solutions of functional differential equations (see J. Hale, Functional differential equations, theorem 5.1), using some new results on continuous convergences. Namely, we prove this theorem without the assumption of uniform boundeness for the sequence (f_k). Also we restore some inconsistencies regarding the proof of this theorem.