Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 720235 21 pages doi 2010 720235 Research Article Accurate Asymptotic Formulas for Eigenvalues and Eigenfunctions of a Boundary-Value Problem of Fourth Order Hamza Menken Mathematics Department Science and Arts Faculty Mersin University 33343 Mersin Turkey Correspondence should be addressed to Hamza Menken hmenken@ Received 7 July 2010 Accepted 9 November 2010 Academic Editor I. T. Kiguradze Copyright 2010 Hamza Menken. 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. In the present paper we consider a nonself-adjoint fourth-order differential operator with the periodic boundary conditions. We compute new accurate asymptotic expression of the fundamental solutions of the given equation. Then we obtain new accurate asymptotic formulas for eigenvalues and eigenfunctions. 1. Introduction In the present work we consider a nonself-adjoint fourth-order operator which is generated by the periodic boundary conditions y 4 q x y Ằy 0 x n y l 0 - yj x 0 j 0 1 2 3 where q x is a complex-valued function. Without lose of generality we can assume that zn q x dx 0. Spectral properties of Sturm-Liouville operator which is generated by the periodic and antiperiodic boundary conditions have been investigated by many authors the results on this direct and references are given details in the monographs 1-5 . In this paper we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the fourth-order boundary-value problem . For second-order differential equations similar asymptotic formulas were obtained in 6-9 . We note that in 6 10 11 2 Boundary Value Problems using the obtained asymptotic formulas for eigenvalues and eigenfunctions the basis properties of the root functions of the operators were .
