Analytical method for solving linear abstract differential equations of order n

In this paper, the concept of point-wise independent set of closed operators is introduced. The new concept together with the use of Hahn–Banach Theorem enable us to reduce an inhomogeneous linear abstract differential equation of order n namely, Anu(n)(t)+An−1u(n−1)(t)+⋯+A1u′(t)+A∘u(t)=f(t), into an inhomogeneous linear ordinary differential equation with constant coefficients of order n. The involved operators in the main abstract equation are assumed to be closed and linear on a given Banach space while the unknown function is assumed to be n-times differentiable.