On the general solution of a linear \(n\)th-order differential equation with constant bounded operator coefficients in a Banach space (Q2577328)

Let \(E\) be a Banach space. The author considers the equation \[ u^{(n)}+A_1u^{(n-1)} + \ldots + A_{n-1}u^\prime +A_nu = f(t), \quad 0 \leq t < \infty,\eqno(1) \] where \(A_i \in L(E), 1 \leq i \leq n\) and \(f(t) \in C([0, \infty); E)\) and associates the operator characteristic equation \[ \Lambda^n +A_1 \Lambda^{n-1} + \ldots + A_{n-1}\Lambda + A_n =0 \] with \(n\) distinct roots \(\Lambda_1, \ldots, \Lambda_n \in L(E)\) such that \(\Lambda_i\Lambda_j= \Lambda_j\Lambda_i\), for all \(1 \leq i, j \leq n\), and there exists \((\Lambda_i-\Lambda_j)^{-1} \in L(E)\), for all \(1 \leq j <i \leq n\). One of the main results of the paper states that if \(u_*\) is a particular solution of (1), then the general solution of (1) has the form \(u = u_{g.o}+u_*\), where \[ u_{g.o} = \sum_{k=1}^n\exp(\Lambda_k t)x_k, \quad x_k \in E, k=1, \ldots n. \]