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

Abstract

The aim of this work is to study the existence of a periodic solutions of third order differential equations $z^{\prime \prime \prime }(t)=Az(t)+f(t)$ with the periodic condition $x(0)=x(2\pi ),x^{\prime }(0)=x^{\prime }(2\pi )$ and $x^{\prime \prime }(0)=x^{\prime \prime }(2\pi )$. Our approach is based on the R-boundedness and $L^p$-multiplier of linear operators.