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.