# Existence and uniqueness of functional differential equations with n delay

## TL;DR

This paper establishes necessary and sufficient conditions for the existence and uniqueness of periodic solutions in functional differential equations with multiple delays, using operator-valued Fourier multipliers.

## Contribution

It introduces a novel framework based on R-boundedness of Fourier multipliers to analyze solutions of delayed differential equations.

## Key findings

- Derived conditions for existence and uniqueness of solutions.
- Connected solution properties to R-boundedness of Fourier multipliers.
- Provided a mathematical foundation for analyzing delayed differential equations.

## Abstract

In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained in terms of R-boundedness of operator valued Fourier multipliers.