# R-boundedness Approach to linear third differential equations in a UMD Space

## TL;DR

This paper investigates the existence of periodic solutions for third-order differential equations in UMD spaces using R-boundedness and $L^{p}$-multiplier techniques, advancing the theoretical understanding of such equations.

## Contribution

It introduces an R-boundedness approach combined with $L^{p}$-multiplier theory to establish periodic solutions for third-order differential equations in UMD spaces.

## Key findings

- Established conditions for existence of periodic solutions
- Applied R-boundedness to third-order differential equations
- Extended operator theory in UMD spaces

## Abstract

The aim of this work is to study the existence of a periodic solutions of third order differential equations $z'''(t) = Az(t) + f(t)$ with the periodic condition $x(0) = x(2\pi), x'(0) = x'(2\pi)$ and $x''(0) = x''(2\pi)$. Our approach is based on the R-boundedness and $L^{p}$-multiplier of linear operators.