On the Solvability of a Class of Operator- Differential Equations of an Arbitrary Order

Abstract

The interest of studying initial value problems for operator-differential equations is increasing since the theory of operator-differential equations was established. This is due to the fact that these equations allow searching at ordinary differential operators in addition to partial differential operators. A large number of researches on various aspects of the theory of operator-differential equations (see [51-52], [60], [81-83]). Recently in correlation with the study of problem arising in applications greatly increased the interest of study of differential equations in Banach and Hilbert spaces, especially in the Sobolev spaces. Note also that these studies are a natural development of the theory of operator-differential equations and is closely related to the theory of semi-groups like in the works of T. Kato, E. Hille, R. Phillips, S. G. Krein, S. Agmon, and L. Nirenberg and reflected in the monograph of K-J. Engel and R. Nagel [53], [58]. The operator-differential equations often lead to the study of polynomial operator pencils, these pencils are the parameters of the investigated equation. It should be noted that the fundamental theory of operator pencils originates from the work of M. V. Keldysh [54] and developed subsequently in the works of Dzh. E. Allahverdiev, M. G. Gasymov, I. Ts. Gohberg, A. G. Kostyuchenko, M. G. Krein, S. G. Krein, G. K. Langer, V. B. Lidskogo, A.S. Markus, V. I. Matsaev, S. S. Mirzoev, G.V. Radzievskogo A. A. Shkalikov, S. Ya. Yakubov and others. During the study of operator-differential equations arising questions about the completeness, minimality, and the basis of their elementary solutions which can be closely related to the investigations of the completeness, minimality and the basis of the system of eigenvalues and associated eigenvectors of the resolvent operator pencils.