On the Case of Complex Roots of the Characteristic Operator Polynomial of a Linear $n$th-Order Homogeneous Differential Equation in a Banach Space

Abstract

We consider a linear homogeneous $n$th-order differential equation, $n\in \mathbb {N}$, with constant bounded operator coefficients in a Banach space. Under some conditions on the (real and complex) roots of the corresponding characteristic equation, we obtain a formula expressing the general solution of the differential equation via the operator functions given by the exponential, sine, and cosine of the roots. The case in which $n$ is even and the characteristic equation has $n/2$ pairs of complex-conjugate pure imaginary roots is investigated in detail. The case of a second-order differential equation is considered separately.