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báo cáo hóa học:" Research Article Positive Solutions of Singular Complementary Lidstone Boundary Value Problems"

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 Positive Solutions of Singular Complementary Lidstone Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 368169 15 pages doi 10.1155 2010 368169 Research Article Positive Solutions of Singular Complementary Lidstone Boundary Value Problems Ravi P. Agarwal 1 Donal O Regan 2 and Svatoslav Stanek3 1 Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901-6975 USA 2 Department of Mathematics National University of Ireland Galway Ireland 3 Department of Mathematical Analysis Faculty of Science Palacky University Tr. 17. listopadu 12 771 46 Olomouc Czech Republic Correspondence should be addressed to Ravi P. Agarwal agarwal@fit.edu Received 7 October 2010 Accepted 21 November 2010 Academic Editor Irena Rachunkova Copyright 2010 Ravi P. Agarwal 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. We investigate the existence of positive solutions of singular problem -1 mx 2m 1t f f x . . . xk2m f x 0 0 x 2i-1 0 x 2i-1l T 0 1 i m. Here m 1 and the Caratheodory function f t x0 . x2m may be singular in all its space variables x0 . x2m. The results are proved by regularization and sequential techniques. In limit processes the Vitali convergence theorem is used. 1. Introduction Let T be a positive constant J 0 T andR- -TO 0 R 0 TO JK0 R 0 . We consider the singular complementary Lidstone boundary value problem -1 mx 2m 1 f f t x t . x 2m t m 1 1.1 x 0 0 x 2i 0 x 2i 1J T 0 1 i m 1.2 where f satisfies the local Carathéodory function on J X D f e Car J X D with R2 X R0 X R_ X R0 X R X X R X R0 D 4k-1 R2 X R0 X R- X R0 X R X X R- X R0 if m 2k - 1 if m 2k. 1.3 4k 1 2 Boundary Value Problems The function f t X0 . X2m is positive and may be singular at the value zero of all its space variables X0 . . x2m. Let i e 0 1 . 2m . We say that f is singular at the value zero of its space variable Xi if for a.e. t e J and all