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báo cáo hóa học:" Research Article Homoclinic Solutions of Singular Nonautonomous 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 Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 959636 21 pages doi 10.1155 2009 959636 Research Article Homoclinic Solutions of Singular Nonautonomous Second-Order Differential Equations Irena Rachunkova and Jan TomeCek Department of Mathematical Analysis and Applications of Mathematics Faculty of Science Palacky University 17 listopadu 12 771 46 Olomouc Czech Republic Correspondence should be addressed to Irena Rachunkova rachunko@inf.upol.cz Received 27 April 2009 Revised 1 September 2009 Accepted 15 September 2009 Recommended by Donal O Regan This paper investigates the singular differential equation p t u f p f f u having a singularity at t 0. The existence of a strictly increasing solution a homoclinic solution satisfying u 0 0 u to L 0 is proved provided that f has two zeros and a linear behaviour near -TO. Copyright 2009 I. Rachunkova and J. Tomecek. 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 Having a positive parameter L we consider the problem p W p t f u M u 0 0 u to L 1.2 under the following basic assumptions for f and p f e Liploc -TO L f 0 f L 0 1.3 f x 0 for x e 0 L 1.4 there exists B 0 such that f x 0 for x e b 0 1.5 F b F L where F x - f z dz 1.6 0 p e C 0 to n C1 0 to p 0 0 1.7 p t _ . p ft 0 t e 0 to lim 0. 1.8 t TO p t 2 Boundary Value Problems Then problem 1.1 1.2 generalizes some models arising in hydrodynamics or in the nonlinear field theory see 1-5 . However 1.1 is singular at t 0 because p 0 0. Definition 1.1. If c 0 then a solution of 1.1 on 0 c is a function u e C1 0 c n C2 0 c satisfying 1.1 on 0 c . If u is a solution of 1.1 on 0 c for each c 0 then u is a solution of 1.1 on 0 to . Definition 1.2. Let u be a solution of 1.1 on 0 to . If u moreover fulfils conditions 1.2 it is called a solution of problem 1.1 1.2 . Clearly the