Theory of operator-differential equations in abstract spaces that takes its origin in the papers of K. Yosida, E. Hille and R. Fillips, T. Kato and others, appeared as application of the methods of functional analysis in the theory of partial differential equations.
Interest to this problem is stipulated by the fact that it allows to investigate both systems of ordinary differential equations, integro-differential equations, quasidifferential equations and others from a unique point of view. It should be noted that the questions on solvability of operator-differential equations and boundary value problems for them are of independent scientific interest. Some significant results of the theory of operator- differential equation are cited in the books of S.G. Krein, A.A. Dezin, J.L. Lions and E. Majenes, V.I. Gorbachuk and M.L. Gorbachuk, S.A. Yakubov and others.
Interest to the investigations of solvability of the Cauchy problem and boundary value problems for operator-differential equations and also the increased amount of papers devoted to this theme proceed from the fact that these questions are closely mixed up with the problems of spectral theory of not selfadjoint operators and operator pencils that at the present time are one of developing sections of functional analysis. The beginning of devolepment of these theories is the known paper of the academician M.V. Keldysh. In this paper M.V. Keldysh introduced the notion of multiple completeness of eigen and adjoint vectors for a wide class of operator bundless, and also showed how the notion of n-fold completeness of eigen and adjoint vectors of operator pencil is associated with appropriate Cauchy problem. After this, there appears a great deal of papers in which significant theorems on multiple completeness, on busicity of a system of eigen and adjoint vectors and on multiple expansion in this system were obtained for different classes of operator bundless. Many problems of mechanics and mathematical physics are related to investigations of completeness of some part of eigen and adjoint vectors of operator pencils. A great deal of papers was devoted to these problems. There are some methods for solving these problems and one of them is the consideration of appropriate boundary value problems on semi-axis, that arises while investigating completeness of eigen and adjoint vectors responding to eigen values from the left half-plane. This method was suggested by academican M.G. Gasymov. He showed relation of completeness of a part of eigen and adjoint vectors and solvability a boundary value problem on a semi-axis with some analytic properties of a resolvent of an operator pencil, that was developed in the S.S. Mirzoevs paper.
The suggested book is devoted to similar questions of solvability of operator-differential equations of higher order and boundary value problems for them, to investigation of spectral properties of appropriate operator pencils. A theorem of Phragmen-Lindeloff type in some vector is proved. The principal part of the investigated operator-differential equations have multiple characteristics.