Higher order abstract Cauchy problems: their existence and uniqueness families.

*(English)*Zbl 1073.34072The authors deal with the abstract Cauchy problem for higher-order linear differential equations

$${u}^{\left(n\right)}\left(t\right)+\sum _{k=0}^{n-1}{A}_{k}{u}^{\left(k\right)}\left(t\right)=0,\phantom{\rule{4pt}{0ex}}t\ge 0,\phantom{\rule{1.em}{0ex}}{u}^{\left(k\right)}\left(0\right)={u}_{k},\phantom{\rule{4pt}{0ex}}0\le k\le n-1,\phantom{\rule{2.em}{0ex}}\left(1\right)$$

and its inhomogeneous version, where ${A}_{0},\cdots ,{A}_{n-1}$ are linear operators in a Banach space $X$. The authors introduce a new operator family of bounded linear operators from a Banach space $Y$ into $X$, called an existence family for (1), so that the existence and continuous dependence on initial data can be studied and some basic results in a quite general setting can be obtained. Necessary and sufficient conditions, ensuring (1) to possess an exponentially bounded existence family, are presented in terms of Laplace transforms. As applications, two concrete initial value problems for partial differential equations are studied.

Reviewer: Messoud A. Efendiev (Berlin)