On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

Abstract

We investigate the structure of solutions of an equation $y″(t) = By(t)$, where $B$ is a weakly positive operator in a Banach space B, on the interval $(0, \infty)$ and establish the existence of their limit values as $t → 0$ in a broader locally convex space containing $B$ as a dense set. The analyticity of these solutions on $(0, \infty)$ is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions.