Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 710386, 20 pages doi:10.1155/2009/710386 Research Article On the Correct Solvability of the Boundary-Value Problem for One Class Operator-Differential Equations of the Fourth Order with Complex Characteristics Araz R. Aliev 1, 2 and Aydin A. Gasymov 1 1 Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9 F. Agaev street, Baku, AZ1141, Azerbaijan 2 Baku State University, 23 Z. Khalilov street, Baku, AZ1148, Azerbaijan Correspondence should be addressed to Araz R. Aliev, alievaraz@yahoo.com Received 19 February 2009; Revised 22 July 2009; Accepted 26 August 2009 Recommended by Ivan T. Kiguradze Sufficient coefficient conditions for the correct and unique solvability of the boundary-value problem for one class of operator-differential equations of the fourth order with complex characteristics, which cover the equations arising in solving the problems of stability of plastic plates, are obtained in this paper. Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspaces W 4 2 R  ; H in relation to the norms of the operator generated by the main part of this equation. It is noted that this problem has its own mathematical interest. Copyright q 2009 A. R. Aliev and A. A. Gasymov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction It is well known that a number of problems in mechanics lead to studying the completeness of all or part of the eigenvectors and joint vectors of certain polynomial operator groups and the completeness of elementary solutions of the operator-differential equations corresponding to these groups see, e.g., 1, 2, and their references. In this case, it is first necessary to investigate the correct solvability of Cauchy or boundary-value problems for these equations, and only after this it will be possible to proceed to the abovementioned problems. The present paper is dedicated to the problem of correct solvability of the boundary-value problem for one class of operator-differential equations of the fourth order, considered on a semiaxis. Let H be a separable Hilbert space and A be a self-adjoint positively defined operator in H.