tailieunhanh - báo cáo hóa học:" Research Article Limit Properties of Solutions of Singular Second-Order Differential Equations"

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 Limit Properties of Solutions of Singular Second-Order Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 905769 28 pages doi 2009 905769 Research Article Limit Properties of Solutions of Singular Second-Order Differential Equations Irena Rachunkova 1 Svatoslav Stanek 1 Ewa Weinmuller 2 and Michael Zenz2 1 Department of Mathematical Analysis Faculty of Science Palacky University Tomkova 40 779 00 Olomouc Czech Republic 2 Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8-10 1040 Wien Austria Correspondence should be addressed to Irena Rachunkova rachunko@ Received 23 April 2009 Accepted 28 May 2009 Recommended by Donal O Regan We discuss the properties of the differential equation u t a t u t f t u t u t . on 0 T where a e R 0 and f satisfies the Lp-Caratheodory conditions on 0 T X R2 for some p 1. A full description of the asymptotic behavior for t 0 of functions u satisfying the equation . on 0 T is given. We also describe the structure of boundary conditions which are necessary and sufficient for u to be at least in C1 0 T . As an application of the theory new existence and or uniqueness results for solutions of periodic boundary value problems are shown. Copyright 2009 Irena Rachunkova et al. 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. Motivation In this paper we study the analytical properties of the differential equation u t iu it f t u t u t . on 0 T where a e R 0 u 0 T R and the function f is defined for . t e 0 T and for all x y e D c R X R. The above equation is singular at t 0 because of the first term in the right-hand side which is in general unbounded for t 0. In this paper we will also alow the function f to be unbounded or bounded but discontinuous for certain values of the time variable t e 0 T . This form of f is motivated