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

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(2π), x ′ (0) = x ′ (2π) and x ′′ (0) = x ′′ (2π). Our approach is based on the R-boundedness and L p-multiplier of linear operators