On the nonautonomous higher-order Cauchy problems

Abstract

We study the existence and uniqueness of classical solutions to the following nonautonomous higher-order Cauchy problem, \begin{equation*} \begin{cases} u^{(n+1)}(t)= A(t)u^{(n)}(t)+B_1(t)u^{(n-1)}(t) \\ \hspace{45pt} + \cdots +B_n(t)u(t)+f(t), \ \ \ \ 0 \le s \le t \le T, \\ u^{(i)}(0)=x_{i} \in E , \hspace{4pt} i=0,1,\dots, n, \end{cases} \end{equation*} by using operator matrices. The results cover some of the known results about the existence and uniqueness of the higher-order Cauchy problem. An example and applications are also given.