On some differential-operator equations of arbitrary order

Yu. A. Dubinskii

Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 1-21

Résumé

On the half-line $(0,+\infty)$ we investigate the following equation in a Banach space: \begin{equation} \sum^s_{j=0}Aj\frac{d^ju(t)}{dt^j}=h(t),\quad s\geqslant1, \end{equation} where $A_0,\dots,A_s$ are closed operators which commute with $\frac d{dt}$. We consider the following classes of equations: parabolic, inverse parabolic, hyperbolic, quasi-elliptic, and quasi-hyperbolic. We present boundary value problems for these classes and prove that they are well-posed. The proofs are based on a solvability theorem for the operator equation $\sum^s_{j=0}A_jB^ju=h $, where $B$ is a closed operator. Bibliography: 20 titles.