Solution of a second-order algebro-differential equation in a Banach space

This article is devoted to the study of the algebro-differential equation
\begin{equation*}
A\frac{d^2u}{dt^2}=B\frac{du}{dt}+Cu(t)+f(t),
\end{equation*}
where $A,$ $B,$ $C$ are closed linear operators acting from a Banach space $E_1$ into a Banach space $E_2$ whose domains are everywhere dense in $E_1$. $A$ is a Fredholm operator with zero index (hereinafter, Fredholm), the function $f(t)$ takes values in $E_2$; $t\in[0;T]$. The kernel of the operator $A$ is assumed to be one-dimensional. For solvability of the equation with respect to the derivative, the method of cascade splitting is applied, consisting in the stepwise splitting of the equation and conditions to the corresponding equations and conditions in subspaces of lower dimensions. One-step and two-step splitting are considered, theorems on the solvability of the equation are obtained. The theorems are used to obtain the existence conditions for a solution to the Cauchy problem. In order to illustrate the results obtained, a homogeneous Cauchy problem with given operator coefficients in the space $\mathbb{R}^2$ is solved. For this, it is considered the
second-order differential equation in the finite-dimensional space $\mathbb{C}^m$
\begin{equation*}
\frac{d^2u}{dt^2}=H\frac{du}{dt}+Ku(t).
\end{equation*}
The characteristic equation $M(\lambda):=\det(\lambda^2 I-\lambda H-K)=0$ is studied. For the polynomial $M(\lambda),$ in the cases $m=2,$ $m=3,$ the Maclaurin formulas are obtained. General
solution of the equation is defined in the case of the unit algebraic multiplicity of the characteristic equation.