On Solvability of Boundary Value Problems for Kinetic Operator-Differential Equations

We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B(t)ut−L(t)u = f, where L(t) and B(t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L −λB is included into one of the half-planes Re λ < a or Re λ > a(a∈R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(a\in {\mathbb{R}})}$$\end{document}. Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions.